id	sid	tid	token	lemma	pos
38536	1	1	Transcriber	Transcriber	NNP
38536	1	2	's	's	POS
38536	1	3	Note	note	NN
38536	1	4	:	:	:
38536	1	5	This	this	DT
38536	1	6	book	book	NN
38536	1	7	includes	include	VBZ
38536	1	8	extensive	extensive	JJ
38536	1	9	mathematical	mathematical	JJ
38536	1	10	expressions	expression	NNS
38536	1	11	and	and	CC
38536	1	12	equations	equation	NNS
38536	1	13	,	,	,
38536	1	14	which	which	WDT
38536	1	15	can	can	MD
38536	1	16	not	not	RB
38536	1	17	always	always	RB
38536	1	18	be	be	VB
38536	1	19	easily	easily	RB
38536	1	20	represented	represent	VBN
38536	1	21	in	in	IN
38536	1	22	plain	plain	JJ
38536	1	23	text	text	NN
38536	1	24	.	.	.
38536	2	1	The	the	DT
38536	2	2	reader	reader	NN
38536	2	3	is	be	VBZ
38536	2	4	encouraged	encourage	VBN
38536	2	5	to	to	TO
38536	2	6	download	download	VB
38536	2	7	the	the	DT
38536	2	8	HTML	HTML	NNP
38536	2	9	version	version	NN
38536	2	10	of	of	IN
38536	2	11	the	the	DT
38536	2	12	text	text	NN
38536	2	13	,	,	,
38536	2	14	which	which	WDT
38536	2	15	represents	represent	VBZ
38536	2	16	the	the	DT
38536	2	17	math	math	NN
38536	2	18	more	more	RBR
38536	2	19	clearly	clearly	RB
38536	2	20	.	.	.
38536	3	1	For	for	IN
38536	3	2	the	the	DT
38536	3	3	plain	plain	JJ
38536	3	4	text	text	NN
38536	3	5	version	version	NN
38536	3	6	,	,	,
38536	3	7	the	the	DT
38536	3	8	following	follow	VBG
38536	3	9	conventions	convention	NNS
38536	3	10	are	be	VBP
38536	3	11	used	use	VBN
38536	3	12	:	:	:
38536	3	13	Mixed	mix	VBN
38536	3	14	fractions	fraction	NNS
38536	3	15	are	be	VBP
38536	3	16	represented	represent	VBN
38536	3	17	by	by	IN
38536	3	18	a	a	DT
38536	3	19	dash	dash	NN
38536	3	20	with	with	IN
38536	3	21	no	no	DT
38536	3	22	spaces	space	NNS
38536	3	23	,	,	,
38536	3	24	while	while	IN
38536	3	25	subtraction	subtraction	NN
38536	3	26	is	be	VBZ
38536	3	27	represented	represent	VBN
38536	3	28	by	by	IN
38536	3	29	a	a	DT
38536	3	30	dash	dash	NN
38536	3	31	with	with	IN
38536	3	32	spaces	space	NNS
38536	3	33	on	on	IN
38536	3	34	either	either	DT
38536	3	35	side	side	NN
38536	3	36	.	.	.
38536	4	1	For	for	IN
38536	4	2	example	example	NN
38536	4	3	:	:	:
38536	4	4	1	1	CD
38536	4	5	-	-	SYM
38536	4	6	1/2	1/2	CD
38536	4	7	is	be	VBZ
38536	4	8	"	"	``
38536	4	9	one	one	CD
38536	4	10	and	and	CC
38536	4	11	one	one	CD
38536	4	12	half	half	NN
38536	4	13	.	.	.
38536	4	14	"	"	''
38536	5	1	1	1	LS
38536	5	2	-	-	SYM
38536	5	3	1/2	1/2	CD
38536	5	4	is	be	VBZ
38536	5	5	"	"	``
38536	5	6	one	one	CD
38536	5	7	minus	minus	CC
38536	5	8	one	one	CD
38536	5	9	half	half	NN
38536	5	10	.	.	.
38536	5	11	"	"	''
38536	6	1	The	the	DT
38536	6	2	"	"	``
38536	6	3	sideways-8	sideways-8	NNP
38536	6	4	"	"	''
38536	6	5	symbol	symbol	NN
38536	6	6	for	for	IN
38536	6	7	infinity	infinity	NN
38536	6	8	is	be	VBZ
38536	6	9	represented	represent	VBN
38536	6	10	as	as	IN
38536	6	11	[	[	-LRB-
38536	6	12	infinity	infinity	NN
38536	6	13	]	]	-RRB-
38536	6	14	.	.	.
38536	7	1	Square	square	JJ
38536	7	2	,	,	,
38536	7	3	cube	cube	NN
38536	7	4	,	,	,
38536	7	5	and	and	CC
38536	7	6	other	other	JJ
38536	7	7	roots	root	NNS
38536	7	8	are	be	VBP
38536	7	9	shown	show	VBN
38536	7	10	by	by	IN
38536	7	11	raising	raise	VBG
38536	7	12	a	a	DT
38536	7	13	quantity	quantity	NN
38536	7	14	to	to	IN
38536	7	15	the	the	DT
38536	7	16	appropriate	appropriate	JJ
38536	7	17	fractional	fractional	JJ
38536	7	18	power	power	NN
38536	7	19	.	.	.
38536	8	1	For	for	IN
38536	8	2	example	example	NN
38536	8	3	:	:	:
38536	8	4	[	[	-LRB-
38536	8	5	4]^(1/2	4]^(1/2	NN
38536	8	6	)	)	-RRB-
38536	8	7	is	be	VBZ
38536	8	8	"	"	``
38536	8	9	the	the	DT
38536	8	10	square	square	JJ
38536	8	11	root	root	NN
38536	8	12	of	of	IN
38536	8	13	4	4	CD
38536	8	14	.	.	.
38536	8	15	"	"	''
38536	9	1	[	[	-LRB-
38536	9	2	x]^(1	x]^(1	NNP
38536	9	3	/	/	SYM
38536	9	4	n	n	NNP
38536	9	5	)	)	-RRB-
38536	9	6	is	be	VBZ
38536	9	7	"	"	``
38536	9	8	the	the	DT
38536	9	9	nth	nth	NNP
38536	9	10	root	root	NN
38536	9	11	of	of	IN
38536	9	12	x.	x.	NNP
38536	9	13	"	"	''
38536	10	1	Extra	extra	JJ
38536	10	2	parentheses	parenthesis	NNS
38536	10	3	have	have	VBP
38536	10	4	been	be	VBN
38536	10	5	added	add	VBN
38536	10	6	as	as	IN
38536	10	7	needed	need	VBN
38536	10	8	to	to	TO
38536	10	9	clarify	clarify	VB
38536	10	10	the	the	DT
38536	10	11	correct	correct	JJ
38536	10	12	order	order	NN
38536	10	13	of	of	IN
38536	10	14	operations	operation	NNS
38536	10	15	.	.	.
38536	10	16	]	]	-RRB-
38536	11	1	A	a	DT
38536	11	2	REVIEW	REVIEW	NNP
38536	11	3	OF	of	IN
38536	11	4	ALGEBRA	ALGEBRA	NNP
38536	11	5	BY	by	IN
38536	11	6	ROMEYN	ROMEYN	NNS
38536	11	7	HENRY	HENRY	NNP
38536	11	8	RIVENBURG	RIVENBURG	NNP
38536	11	9	,	,	,
38536	11	10	A.M.	A.M.	NNP
38536	11	11	HEAD	HEAD	NNP
38536	11	12	OF	of	IN
38536	11	13	THE	the	DT
38536	11	14	DEPARTMENT	DEPARTMENT	NNP
38536	11	15	OF	of	IN
38536	11	16	MATHEMATICS	MATHEMATICS	NNP
38536	11	17	THE	the	DT
38536	11	18	PEDDIE	PEDDIE	NNP
38536	11	19	INSTITUTE	INSTITUTE	NNP
38536	11	20	,	,	,
38536	11	21	HIGHTSTOWN	HIGHTSTOWN	NNP
38536	11	22	,	,	,
38536	11	23	N.J.	New Jersey	NNP
38536	11	24	[	[	-LRB-
38536	11	25	Illustration	illustration	NN
38536	11	26	]	]	-RRB-
38536	11	27	AMERICAN	AMERICAN	NNP
38536	11	28	BOOK	BOOK	NNP
38536	11	29	COMPANY	COMPANY	NNP
38536	11	30	NEW	NEW	NNP
38536	11	31	YORK	YORK	NNP
38536	11	32	CINCINNATI	cincinnati	NN
38536	11	33	CHICAGO	CHICAGO	NNP
38536	11	34	COPYRIGHT	COPYRIGHT	NNP
38536	11	35	,	,	,
38536	11	36	1914	1914	CD
38536	11	37	,	,	,
38536	11	38	BY	by	IN
38536	11	39	ROMEYN	ROMEYN	NNP
38536	11	40	H.	H.	NNP
38536	11	41	RIVENBURG	RIVENBURG	NNP
38536	11	42	.	.	.
38536	12	1	COPYRIGHT	COPYRIGHT	NNP
38536	12	2	,	,	,
38536	12	3	1914	1914	CD
38536	12	4	,	,	,
38536	12	5	IN	in	IN
38536	12	6	GREAT	great	JJ
38536	12	7	BRITAIN	BRITAIN	NNP
38536	12	8	.	.	.
38536	13	1	A	a	DT
38536	13	2	REVIEW	REVIEW	NNP
38536	13	3	OF	of	IN
38536	13	4	ALGEBRA	ALGEBRA	NNP
38536	13	5	.	.	.
38536	14	1	E.	E.	NNP
38536	14	2	P.	P.	NNP
38536	14	3	6	6	CD
38536	14	4	PREFACE	PREFACE	NNP
38536	14	5	In	in	IN
38536	14	6	most	most	RBS
38536	14	7	high	high	JJ
38536	14	8	schools	school	NNS
38536	14	9	the	the	DT
38536	14	10	course	course	NN
38536	14	11	in	in	IN
38536	14	12	Elementary	Elementary	NNP
38536	14	13	Algebra	Algebra	NNP
38536	14	14	is	be	VBZ
38536	14	15	finished	finish	VBN
38536	14	16	by	by	IN
38536	14	17	the	the	DT
38536	14	18	end	end	NN
38536	14	19	of	of	IN
38536	14	20	the	the	DT
38536	14	21	second	second	JJ
38536	14	22	year	year	NN
38536	14	23	.	.	.
38536	15	1	By	by	IN
38536	15	2	the	the	DT
38536	15	3	senior	senior	JJ
38536	15	4	year	year	NN
38536	15	5	,	,	,
38536	15	6	most	most	JJS
38536	15	7	students	student	NNS
38536	15	8	have	have	VBP
38536	15	9	forgotten	forget	VBN
38536	15	10	many	many	JJ
38536	15	11	of	of	IN
38536	15	12	the	the	DT
38536	15	13	principles	principle	NNS
38536	15	14	,	,	,
38536	15	15	and	and	CC
38536	15	16	a	a	DT
38536	15	17	thorough	thorough	JJ
38536	15	18	review	review	NN
38536	15	19	is	be	VBZ
38536	15	20	necessary	necessary	JJ
38536	15	21	in	in	IN
38536	15	22	order	order	NN
38536	15	23	to	to	TO
38536	15	24	prepare	prepare	VB
38536	15	25	college	college	NN
38536	15	26	candidates	candidate	NNS
38536	15	27	for	for	IN
38536	15	28	the	the	DT
38536	15	29	entrance	entrance	NN
38536	15	30	examinations	examination	NNS
38536	15	31	and	and	CC
38536	15	32	for	for	IN
38536	15	33	effective	effective	JJ
38536	15	34	work	work	NN
38536	15	35	in	in	IN
38536	15	36	the	the	DT
38536	15	37	freshman	freshman	NN
38536	15	38	year	year	NN
38536	15	39	in	in	IN
38536	15	40	college	college	NN
38536	15	41	.	.	.
38536	16	1	Recognizing	recognize	VBG
38536	16	2	this	this	DT
38536	16	3	need	need	NN
38536	16	4	,	,	,
38536	16	5	many	many	JJ
38536	16	6	schools	school	NNS
38536	16	7	are	be	VBP
38536	16	8	devoting	devote	VBG
38536	16	9	at	at	RB
38536	16	10	least	least	RBS
38536	16	11	two	two	CD
38536	16	12	periods	period	NNS
38536	16	13	a	a	DT
38536	16	14	week	week	NN
38536	16	15	for	for	IN
38536	16	16	part	part	NN
38536	16	17	of	of	IN
38536	16	18	the	the	DT
38536	16	19	senior	senior	JJ
38536	16	20	year	year	NN
38536	16	21	to	to	IN
38536	16	22	a	a	DT
38536	16	23	review	review	NN
38536	16	24	of	of	IN
38536	16	25	algebra	algebra	NN
38536	16	26	.	.	.
38536	17	1	For	for	IN
38536	17	2	such	such	PDT
38536	17	3	a	a	DT
38536	17	4	review	review	NN
38536	17	5	the	the	DT
38536	17	6	regular	regular	JJ
38536	17	7	textbook	textbook	NN
38536	17	8	is	be	VBZ
38536	17	9	inadequate	inadequate	JJ
38536	17	10	.	.	.
38536	18	1	From	from	IN
38536	18	2	an	an	DT
38536	18	3	embarrassment	embarrassment	NN
38536	18	4	of	of	IN
38536	18	5	riches	rich	NNS
38536	18	6	the	the	DT
38536	18	7	teacher	teacher	NN
38536	18	8	finds	find	VBZ
38536	18	9	it	-PRON-	PRP
38536	18	10	laborious	laborious	JJ
38536	18	11	to	to	TO
38536	18	12	select	select	VB
38536	18	13	the	the	DT
38536	18	14	proper	proper	JJ
38536	18	15	examples	example	NNS
38536	18	16	,	,	,
38536	18	17	while	while	IN
38536	18	18	the	the	DT
38536	18	19	student	student	NN
38536	18	20	wastes	waste	VBZ
38536	18	21	time	time	NN
38536	18	22	in	in	IN
38536	18	23	searching	search	VBG
38536	18	24	for	for	IN
38536	18	25	scattered	scatter	VBN
38536	18	26	assignments	assignment	NNS
38536	18	27	.	.	.
38536	19	1	The	the	DT
38536	19	2	object	object	NN
38536	19	3	of	of	IN
38536	19	4	this	this	DT
38536	19	5	book	book	NN
38536	19	6	is	be	VBZ
38536	19	7	to	to	TO
38536	19	8	conserve	conserve	VB
38536	19	9	the	the	DT
38536	19	10	time	time	NN
38536	19	11	and	and	CC
38536	19	12	effort	effort	NN
38536	19	13	of	of	IN
38536	19	14	both	both	DT
38536	19	15	teacher	teacher	NN
38536	19	16	and	and	CC
38536	19	17	student	student	NN
38536	19	18	,	,	,
38536	19	19	by	by	IN
38536	19	20	providing	provide	VBG
38536	19	21	a	a	DT
38536	19	22	thorough	thorough	JJ
38536	19	23	and	and	CC
38536	19	24	effective	effective	JJ
38536	19	25	review	review	NN
38536	19	26	that	that	WDT
38536	19	27	can	can	MD
38536	19	28	readily	readily	RB
38536	19	29	be	be	VB
38536	19	30	completed	complete	VBN
38536	19	31	,	,	,
38536	19	32	if	if	IN
38536	19	33	need	need	NN
38536	19	34	be	be	VB
38536	19	35	,	,	,
38536	19	36	in	in	IN
38536	19	37	two	two	CD
38536	19	38	periods	period	NNS
38536	19	39	a	a	DT
38536	19	40	week	week	NN
38536	19	41	for	for	IN
38536	19	42	a	a	DT
38536	19	43	half	half	JJ
38536	19	44	year	year	NN
38536	19	45	.	.	.
38536	20	1	Each	each	DT
38536	20	2	student	student	NN
38536	20	3	is	be	VBZ
38536	20	4	expected	expect	VBN
38536	20	5	to	to	TO
38536	20	6	use	use	VB
38536	20	7	his	-PRON-	PRP$
38536	20	8	regular	regular	JJ
38536	20	9	textbook	textbook	NN
38536	20	10	in	in	IN
38536	20	11	algebra	algebra	NN
38536	20	12	for	for	IN
38536	20	13	reference	reference	NN
38536	20	14	,	,	,
38536	20	15	as	as	IN
38536	20	16	he	-PRON-	PRP
38536	20	17	would	would	MD
38536	20	18	use	use	VB
38536	20	19	a	a	DT
38536	20	20	dictionary,--to	dictionary,--to	NN
38536	20	21	recall	recall	NN
38536	20	22	a	a	DT
38536	20	23	definition	definition	NN
38536	20	24	,	,	,
38536	20	25	a	a	DT
38536	20	26	rule	rule	NN
38536	20	27	,	,	,
38536	20	28	or	or	CC
38536	20	29	a	a	DT
38536	20	30	process	process	NN
38536	20	31	that	that	WDT
38536	20	32	he	-PRON-	PRP
38536	20	33	has	have	VBZ
38536	20	34	forgotten	forget	VBN
38536	20	35	.	.	.
38536	21	1	He	-PRON-	PRP
38536	21	2	should	should	MD
38536	21	3	be	be	VB
38536	21	4	encouraged	encourage	VBN
38536	21	5	to	to	IN
38536	21	6	_	_	NNP
38536	21	7	think	think	VB
38536	21	8	_	_	NNP
38536	21	9	his	-PRON-	PRP$
38536	21	10	way	way	NN
38536	21	11	out	out	RB
38536	21	12	wherever	wherever	WRB
38536	21	13	possible	possible	JJ
38536	21	14	,	,	,
38536	21	15	however	however	RB
38536	21	16	,	,	,
38536	21	17	and	and	CC
38536	21	18	to	to	TO
38536	21	19	refer	refer	VB
38536	21	20	to	to	IN
38536	21	21	the	the	DT
38536	21	22	textbook	textbook	NN
38536	21	23	only	only	RB
38536	21	24	when	when	WRB
38536	21	25	_	_	NNP
38536	21	26	forced	force	VBD
38536	21	27	_	_	NNP
38536	21	28	to	to	TO
38536	21	29	do	do	VB
38536	21	30	so	so	RB
38536	21	31	as	as	IN
38536	21	32	a	a	DT
38536	21	33	last	last	JJ
38536	21	34	resort	resort	NN
38536	21	35	.	.	.
38536	22	1	The	the	DT
38536	22	2	definitions	definition	NNS
38536	22	3	given	give	VBN
38536	22	4	in	in	IN
38536	22	5	the	the	DT
38536	22	6	General	General	NNP
38536	22	7	Outline	Outline	NNP
38536	22	8	should	should	MD
38536	22	9	be	be	VB
38536	22	10	reviewed	review	VBN
38536	22	11	as	as	IN
38536	22	12	occasion	occasion	NN
38536	22	13	arises	arise	VBZ
38536	22	14	for	for	IN
38536	22	15	their	-PRON-	PRP$
38536	22	16	use	use	NN
38536	22	17	.	.	.
38536	23	1	The	the	DT
38536	23	2	whole	whole	JJ
38536	23	3	Outline	Outline	NNP
38536	23	4	can	can	MD
38536	23	5	be	be	VB
38536	23	6	profitably	profitably	RB
38536	23	7	employed	employ	VBN
38536	23	8	for	for	IN
38536	23	9	rapid	rapid	JJ
38536	23	10	class	class	NN
38536	23	11	reviews	review	NNS
38536	23	12	,	,	,
38536	23	13	by	by	IN
38536	23	14	covering	cover	VBG
38536	23	15	the	the	DT
38536	23	16	part	part	NN
38536	23	17	of	of	IN
38536	23	18	the	the	DT
38536	23	19	Outline	Outline	NNP
38536	23	20	that	that	WDT
38536	23	21	indicates	indicate	VBZ
38536	23	22	the	the	DT
38536	23	23	answer	answer	NN
38536	23	24	,	,	,
38536	23	25	the	the	DT
38536	23	26	method	method	NN
38536	23	27	,	,	,
38536	23	28	the	the	DT
38536	23	29	example	example	NN
38536	23	30	,	,	,
38536	23	31	or	or	CC
38536	23	32	the	the	DT
38536	23	33	formula	formula	NN
38536	23	34	,	,	,
38536	23	35	as	as	IN
38536	23	36	the	the	DT
38536	23	37	case	case	NN
38536	23	38	may	may	MD
38536	23	39	be	be	VB
38536	23	40	.	.	.
38536	24	1	The	the	DT
38536	24	2	whole	whole	JJ
38536	24	3	scheme	scheme	NN
38536	24	4	of	of	IN
38536	24	5	the	the	DT
38536	24	6	book	book	NN
38536	24	7	is	be	VBZ
38536	24	8	ordinarily	ordinarily	RB
38536	24	9	to	to	TO
38536	24	10	have	have	VB
38536	24	11	a	a	DT
38536	24	12	page	page	NN
38536	24	13	of	of	IN
38536	24	14	problems	problem	NNS
38536	24	15	represent	represent	VBP
38536	24	16	a	a	DT
38536	24	17	day	day	NN
38536	24	18	's	's	POS
38536	24	19	work	work	NN
38536	24	20	.	.	.
38536	25	1	This	this	DT
38536	25	2	,	,	,
38536	25	3	of	of	IN
38536	25	4	course	course	NN
38536	25	5	,	,	,
38536	25	6	does	do	VBZ
38536	25	7	not	not	RB
38536	25	8	apply	apply	VB
38536	25	9	to	to	IN
38536	25	10	the	the	DT
38536	25	11	Outlines	Outlines	NNP
38536	25	12	or	or	CC
38536	25	13	the	the	DT
38536	25	14	few	few	JJ
38536	25	15	pages	page	NNS
38536	25	16	of	of	IN
38536	25	17	theory	theory	NN
38536	25	18	,	,	,
38536	25	19	which	which	WDT
38536	25	20	can	can	MD
38536	25	21	be	be	VB
38536	25	22	covered	cover	VBN
38536	25	23	more	more	RBR
38536	25	24	rapidly	rapidly	RB
38536	25	25	.	.	.
38536	26	1	By	by	IN
38536	26	2	this	this	DT
38536	26	3	plan	plan	NN
38536	26	4	,	,	,
38536	26	5	making	make	VBG
38536	26	6	only	only	RB
38536	26	7	a	a	DT
38536	26	8	part	part	NN
38536	26	9	of	of	IN
38536	26	10	the	the	DT
38536	26	11	omissions	omission	NNS
38536	26	12	indicated	indicate	VBN
38536	26	13	in	in	IN
38536	26	14	the	the	DT
38536	26	15	next	next	JJ
38536	26	16	paragraph	paragraph	NN
38536	26	17	,	,	,
38536	26	18	the	the	DT
38536	26	19	essentials	essential	NNS
38536	26	20	of	of	IN
38536	26	21	the	the	DT
38536	26	22	algebra	algebra	NN
38536	26	23	can	can	MD
38536	26	24	be	be	VB
38536	26	25	readily	readily	RB
38536	26	26	covered	cover	VBN
38536	26	27	,	,	,
38536	26	28	if	if	IN
38536	26	29	need	need	NN
38536	26	30	be	be	VB
38536	26	31	,	,	,
38536	26	32	in	in	RB
38536	26	33	from	from	IN
38536	26	34	thirty	thirty	CD
38536	26	35	to	to	TO
38536	26	36	thirty	thirty	CD
38536	26	37	-	-	HYPH
38536	26	38	two	two	CD
38536	26	39	lessons	lesson	NNS
38536	26	40	,	,	,
38536	26	41	thus	thus	RB
38536	26	42	leaving	leave	VBG
38536	26	43	time	time	NN
38536	26	44	for	for	IN
38536	26	45	tests	test	NNS
38536	26	46	,	,	,
38536	26	47	even	even	RB
38536	26	48	if	if	IN
38536	26	49	only	only	RB
38536	26	50	eighteen	eighteen	CD
38536	26	51	weeks	week	NNS
38536	26	52	,	,	,
38536	26	53	of	of	IN
38536	26	54	two	two	CD
38536	26	55	periods	period	NNS
38536	26	56	each	each	DT
38536	26	57	,	,	,
38536	26	58	are	be	VBP
38536	26	59	allotted	allot	VBN
38536	26	60	to	to	IN
38536	26	61	the	the	DT
38536	26	62	course	course	NN
38536	26	63	.	.	.
38536	27	1	If	if	IN
38536	27	2	a	a	DT
38536	27	3	brief	brief	JJ
38536	27	4	course	course	NN
38536	27	5	is	be	VBZ
38536	27	6	desired	desire	VBN
38536	27	7	,	,	,
38536	27	8	the	the	DT
38536	27	9	Miscellaneous	Miscellaneous	NNP
38536	27	10	Examples	Examples	NNPS
38536	27	11	(	(	-LRB-
38536	27	12	pp	pp	NNP
38536	27	13	.	.	.
38536	28	1	31	31	CD
38536	28	2	to	to	TO
38536	28	3	35	35	CD
38536	28	4	,	,	,
38536	28	5	50	50	CD
38536	28	6	to	to	TO
38536	28	7	52	52	CD
38536	28	8	)	)	-RRB-
38536	28	9	,	,	,
38536	28	10	many	many	JJ
38536	28	11	of	of	IN
38536	28	12	the	the	DT
38536	28	13	problems	problem	NNS
38536	28	14	at	at	IN
38536	28	15	the	the	DT
38536	28	16	end	end	NN
38536	28	17	of	of	IN
38536	28	18	the	the	DT
38536	28	19	book	book	NN
38536	28	20	,	,	,
38536	28	21	and	and	CC
38536	28	22	the	the	DT
38536	28	23	College	College	NNP
38536	28	24	Entrance	Entrance	NNP
38536	28	25	Examinations	Examinations	NNPS
38536	28	26	may	may	MD
38536	28	27	be	be	VB
38536	28	28	omitted	omit	VBN
38536	28	29	without	without	IN
38536	28	30	marring	mar	VBG
38536	28	31	the	the	DT
38536	28	32	continuity	continuity	NN
38536	28	33	or	or	CC
38536	28	34	the	the	DT
38536	28	35	comprehensiveness	comprehensiveness	NN
38536	28	36	of	of	IN
38536	28	37	the	the	DT
38536	28	38	review	review	NN
38536	28	39	.	.	.
38536	29	1	ROMEYN	ROMEYN	NNP
38536	29	2	H.	H.	NNP
38536	29	3	RIVENBURG	RIVENBURG	NNP
38536	29	4	.	.	.
38536	30	1	CONTENTS	content	NNS
38536	30	2	PAGES	pages	VBP
38536	30	3	OUTLINE	OUTLINE	NNS
38536	30	4	OF	of	IN
38536	30	5	ELEMENTARY	ELEMENTARY	NNP
38536	30	6	AND	and	CC
38536	30	7	INTERMEDIATE	INTERMEDIATE	NNP
38536	30	8	ALGEBRA	ALGEBRA	NNP
38536	30	9	7	7	CD
38536	30	10	-	-	SYM
38536	30	11	13	13	CD
38536	30	12	ORDER	order	NN
38536	30	13	OF	of	IN
38536	30	14	OPERATIONS	OPERATIONS	NNP
38536	30	15	,	,	,
38536	30	16	EVALUATION	EVALUATION	NNP
38536	30	17	,	,	,
38536	30	18	PARENTHESES	parentheses	NN
38536	30	19	14	14	CD
38536	30	20	SPECIAL	special	NN
38536	30	21	RULES	rule	NNS
38536	30	22	OF	of	IN
38536	30	23	MULTIPLICATION	multiplication	NN
38536	30	24	AND	and	CC
38536	30	25	DIVISION	division	NN
38536	30	26	15	15	CD
38536	30	27	CASES	case	NNS
38536	30	28	IN	in	IN
38536	30	29	FACTORING	FACTORING	NNP
38536	30	30	16	16	CD
38536	30	31	,	,	,
38536	30	32	17	17	CD
38536	30	33	FACTORING	factoring	NN
38536	30	34	18	18	CD
38536	30	35	HIGHEST	HIGHEST	NNP
38536	30	36	COMMON	COMMON	NNP
38536	30	37	FACTOR	factor	NN
38536	30	38	AND	and	CC
38536	30	39	LOWEST	LOWEST	NNP
38536	30	40	COMMON	COMMON	NNP
38536	30	41	MULTIPLE	MULTIPLE	NNS
38536	30	42	19	19	CD
38536	30	43	FRACTIONS	fraction	NNS
38536	30	44	20	20	CD
38536	30	45	COMPLEX	complex	NN
38536	30	46	FRACTIONS	fraction	NNS
38536	30	47	AND	and	CC
38536	30	48	FRACTIONAL	fractional	JJ
38536	30	49	EQUATIONS	EQUATIONS	NNP
38536	30	50	21	21	CD
38536	30	51	,	,	,
38536	30	52	22	22	CD
38536	30	53	SIMULTANEOUS	simultaneous	NN
38536	30	54	EQUATIONS	EQUATIONS	NNP
38536	30	55	AND	and	CC
38536	30	56	INVOLUTION	involution	NN
38536	30	57	23	23	CD
38536	30	58	,	,	,
38536	30	59	24	24	CD
38536	30	60	SQUARE	square	NN
38536	30	61	ROOT	ROOT	VBZ
38536	30	62	25	25	CD
38536	30	63	THEORY	theory	NN
38536	30	64	OF	of	IN
38536	30	65	EXPONENTS	EXPONENTS	NNP
38536	30	66	26	26	CD
38536	30	67	-	-	SYM
38536	30	68	28	28	CD
38536	30	69	RADICALS	radical	NNS
38536	30	70	29	29	CD
38536	30	71	,	,	,
38536	30	72	30	30	CD
38536	30	73	MISCELLANEOUS	MISCELLANEOUS	NNP
38536	30	74	EXAMPLES	example	NNS
38536	30	75	,	,	,
38536	30	76	ALGEBRA	algebra	VBP
38536	30	77	TO	to	IN
38536	30	78	QUADRATICS	QUADRATICS	NNP
38536	30	79	31	31	CD
38536	30	80	-	-	SYM
38536	30	81	35	35	CD
38536	30	82	QUADRATIC	quadratic	NN
38536	30	83	EQUATIONS	EQUATIONS	NNP
38536	30	84	36	36	CD
38536	30	85	,	,	,
38536	30	86	37	37	CD
38536	30	87	THE	the	DT
38536	30	88	THEORY	theory	NN
38536	30	89	OF	of	IN
38536	30	90	QUADRATIC	QUADRATIC	NNP
38536	30	91	EQUATIONS	EQUATIONS	NNP
38536	30	92	38	38	CD
38536	30	93	-	-	SYM
38536	30	94	41	41	CD
38536	30	95	OUTLINE	OUTLINE	NNS
38536	30	96	OF	of	IN
38536	30	97	SIMULTANEOUS	SIMULTANEOUS	NNP
38536	30	98	QUADRATICS	QUADRATICS	NNP
38536	30	99	42	42	CD
38536	30	100	,	,	,
38536	30	101	43	43	CD
38536	30	102	SIMULTANEOUS	simultaneou	NNS
38536	30	103	QUADRATICS	quadratic	NNS
38536	30	104	44	44	CD
38536	30	105	RATIO	RATIO	NNS
38536	30	106	AND	and	CC
38536	30	107	PROPORTION	proportion	NN
38536	30	108	45	45	CD
38536	30	109	,	,	,
38536	30	110	46	46	CD
38536	30	111	ARITHMETICAL	ARITHMETICAL	NNP
38536	30	112	PROGRESSION	PROGRESSION	NNP
38536	30	113	47	47	CD
38536	30	114	GEOMETRICAL	GEOMETRICAL	NNP
38536	30	115	PROGRESSION	PROGRESSION	NNP
38536	30	116	48	48	CD
38536	30	117	THE	the	DT
38536	30	118	BINOMIAL	BINOMIAL	NNP
38536	30	119	THEOREM	theorem	NN
38536	30	120	49	49	CD
38536	30	121	MISCELLANEOUS	miscellaneous	NN
38536	30	122	EXAMPLES	EXAMPLES	NNP
38536	30	123	,	,	,
38536	30	124	QUADRATICS	QUADRATICS	NNPS
38536	30	125	AND	and	CC
38536	30	126	BEYOND	BEYOND	NNP
38536	30	127	50	50	CD
38536	30	128	-	-	SYM
38536	30	129	52	52	CD
38536	30	130	PROBLEMS	problem	NNS
38536	30	131	--	--	:
38536	30	132	LINEAR	LINEAR	NNP
38536	30	133	EQUATIONS	EQUATIONS	NNP
38536	30	134	,	,	,
38536	30	135	SIMULTANEOUS	SIMULTANEOUS	NNP
38536	30	136	EQUATIONS	EQUATIONS	NNP
38536	30	137	,	,	,
38536	30	138	QUADRATIC	QUADRATIC	NNP
38536	30	139	EQUATIONS	EQUATIONS	NNP
38536	30	140	,	,	,
38536	30	141	SIMULTANEOUS	SIMULTANEOUS	NNP
38536	30	142	QUADRATICS	QUADRATICS	NNP
38536	30	143	53	53	CD
38536	30	144	-	-	SYM
38536	30	145	57	57	CD
38536	30	146	COLLEGE	COLLEGE	NNP
38536	30	147	ENTRANCE	entrance	NN
38536	30	148	EXAMINATIONS	examination	VBZ
38536	30	149	58	58	CD
38536	30	150	-	-	SYM
38536	30	151	80	80	CD
38536	30	152	OUTLINE	OUTLINE	NNP
38536	30	153	OF	of	IN
38536	30	154	ELEMENTARY	ELEMENTARY	NNP
38536	30	155	AND	and	CC
38536	30	156	INTERMEDIATE	INTERMEDIATE	NNP
38536	30	157	ALGEBRA	ALGEBRA	NNP
38536	30	158	~Important	~Important	:
38536	30	159	Definitions~	Definitions~	NFP
38536	30	160	Factors	factor	NNS
38536	30	161	;	;	:
38536	30	162	coefficient	coefficient	NN
38536	30	163	;	;	,
38536	30	164	exponent	exponent	NN
38536	30	165	;	;	:
38536	30	166	power	power	NN
38536	30	167	;	;	,
38536	30	168	base	base	NN
38536	30	169	;	;	:
38536	30	170	term	term	NN
38536	30	171	;	;	,
38536	30	172	algebraic	algebraic	NNP
38536	30	173	sum	sum	NN
38536	30	174	;	;	,
38536	30	175	similar	similar	JJ
38536	30	176	terms	term	NNS
38536	30	177	;	;	:
38536	30	178	degree	degree	NN
38536	30	179	;	;	,
38536	30	180	homogeneous	homogeneous	JJ
38536	30	181	expression	expression	NN
38536	30	182	;	;	,
38536	30	183	linear	linear	JJ
38536	30	184	equation	equation	NN
38536	30	185	;	;	,
38536	30	186	root	root	NN
38536	30	187	of	of	IN
38536	30	188	an	an	DT
38536	30	189	equation	equation	NN
38536	30	190	;	;	:
38536	30	191	root	root	NN
38536	30	192	of	of	IN
38536	30	193	an	an	DT
38536	30	194	expression	expression	NN
38536	30	195	;	;	:
38536	30	196	identity	identity	NN
38536	30	197	;	;	,
38536	30	198	conditional	conditional	JJ
38536	30	199	equation	equation	NN
38536	30	200	;	;	:
38536	30	201	prime	prime	JJ
38536	30	202	quantity	quantity	NN
38536	30	203	;	;	:
38536	30	204	highest	high	JJS
38536	30	205	common	common	JJ
38536	30	206	factor	factor	NN
38536	30	207	(	(	-LRB-
38536	30	208	H.	H.	NNP
38536	30	209	C.	C.	NNP
38536	30	210	F.	F.	NNP
38536	30	211	)	)	-RRB-
38536	30	212	;	;	:
38536	30	213	lowest	low	JJS
38536	30	214	common	common	JJ
38536	30	215	multiple	multiple	NN
38536	30	216	(	(	-LRB-
38536	30	217	L.	L.	NNP
38536	30	218	C.	C.	NNP
38536	30	219	M.	M.	NNP
38536	30	220	)	)	-RRB-
38536	30	221	;	;	:
38536	30	222	involution	involution	NN
38536	30	223	;	;	:
38536	30	224	evolution	evolution	NN
38536	30	225	;	;	,
38536	30	226	imaginary	imaginary	JJ
38536	30	227	number	number	NN
38536	30	228	;	;	:
38536	30	229	real	real	JJ
38536	30	230	number	number	NN
38536	30	231	;	;	,
38536	30	232	rational	rational	JJ
38536	30	233	;	;	:
38536	30	234	similar	similar	JJ
38536	30	235	radicals	radical	NNS
38536	30	236	;	;	:
38536	30	237	binomial	binomial	JJ
38536	30	238	surd	surd	NN
38536	30	239	;	;	:
38536	30	240	pure	pure	JJ
38536	30	241	quadratic	quadratic	JJ
38536	30	242	equation	equation	NN
38536	30	243	;	;	,
38536	30	244	affected	affect	VBN
38536	30	245	quadratic	quadratic	JJ
38536	30	246	equation	equation	NN
38536	30	247	;	;	:
38536	30	248	equation	equation	NN
38536	30	249	in	in	IN
38536	30	250	the	the	DT
38536	30	251	quadratic	quadratic	JJ
38536	30	252	form	form	NN
38536	30	253	;	;	:
38536	30	254	simultaneous	simultaneous	JJ
38536	30	255	linear	linear	JJ
38536	30	256	equations	equation	NNS
38536	30	257	;	;	,
38536	30	258	simultaneous	simultaneous	JJ
38536	30	259	quadratic	quadratic	JJ
38536	30	260	equations	equation	NNS
38536	30	261	;	;	,
38536	30	262	discriminant	discriminant	JJ
38536	30	263	;	;	:
38536	30	264	symmetrical	symmetrical	JJ
38536	30	265	expression	expression	NN
38536	30	266	;	;	:
38536	30	267	ratio	ratio	NN
38536	30	268	;	;	:
38536	30	269	proportion	proportion	NN
38536	30	270	;	;	,
38536	30	271	fourth	fourth	JJ
38536	30	272	proportional	proportional	JJ
38536	30	273	;	;	:
38536	30	274	third	third	JJ
38536	30	275	proportional	proportional	JJ
38536	30	276	;	;	:
38536	30	277	mean	mean	VB
38536	30	278	proportional	proportional	JJ
38536	30	279	;	;	,
38536	30	280	arithmetic	arithmetic	JJ
38536	30	281	progression	progression	NN
38536	30	282	;	;	:
38536	30	283	geometric	geometric	JJ
38536	30	284	progression	progression	NN
38536	30	285	;	;	:
38536	30	286	S	S	NNP
38536	30	287	[	[	-LRB-
38536	30	288	infinity	infinity	NN
38536	30	289	]	]	-RRB-
38536	30	290	~Special	~Special	.
38536	30	291	Rules	rule	NNS
38536	30	292	for	for	IN
38536	30	293	Multiplication	Multiplication	NNP
38536	30	294	and	and	CC
38536	30	295	Division~	division~	NN
38536	30	296	1	1	CD
38536	30	297	.	.	.
38536	31	1	Square	square	NN
38536	31	2	of	of	IN
38536	31	3	the	the	DT
38536	31	4	sum	sum	NN
38536	31	5	of	of	IN
38536	31	6	two	two	CD
38536	31	7	quantities	quantity	NNS
38536	31	8	.	.	.
38536	32	1	(	(	-LRB-
38536	32	2	x	x	SYM
38536	32	3	+	+	CD
38536	32	4	y)^2	y)^2	NNS
38536	32	5	.	.	.
38536	33	1	2	2	LS
38536	33	2	.	.	.
38536	34	1	Square	square	NN
38536	34	2	of	of	IN
38536	34	3	the	the	DT
38536	34	4	difference	difference	NN
38536	34	5	of	of	IN
38536	34	6	two	two	CD
38536	34	7	quantities	quantity	NNS
38536	34	8	.	.	.
38536	35	1	(	(	-LRB-
38536	35	2	x	x	NNP
38536	35	3	-	-	:
38536	35	4	y)^2	y)^2	NNP
38536	35	5	.	.	.
38536	36	1	3	3	LS
38536	36	2	.	.	.
38536	37	1	Product	product	NN
38536	37	2	of	of	IN
38536	37	3	the	the	DT
38536	37	4	sum	sum	NN
38536	37	5	and	and	CC
38536	37	6	difference	difference	NN
38536	37	7	of	of	IN
38536	37	8	two	two	CD
38536	37	9	quantities	quantity	NNS
38536	37	10	.	.	.
38536	38	1	(	(	-LRB-
38536	38	2	s	s	NNP
38536	38	3	+	+	CC
38536	38	4	t)(s	t)(s	NNP
38536	38	5	-	-	HYPH
38536	38	6	t	t	NN
38536	38	7	)	)	-RRB-
38536	38	8	.	.	.
38536	39	1	4	4	LS
38536	39	2	.	.	.
38536	40	1	Product	product	NN
38536	40	2	of	of	IN
38536	40	3	two	two	CD
38536	40	4	binomials	binomial	NNS
38536	40	5	having	have	VBG
38536	40	6	a	a	DT
38536	40	7	common	common	JJ
38536	40	8	term	term	NN
38536	40	9	.	.	.
38536	41	1	(	(	-LRB-
38536	41	2	x	x	SYM
38536	41	3	+	+	SYM
38536	41	4	r)(x	r)(x	NN
38536	41	5	+	+	SYM
38536	41	6	m	m	NN
38536	41	7	)	)	-RRB-
38536	41	8	.	.	.
38536	42	1	5	5	CD
38536	42	2	.	.	.
38536	43	1	Product	product	NN
38536	43	2	of	of	IN
38536	43	3	two	two	CD
38536	43	4	binomials	binomial	NNS
38536	43	5	whose	whose	WP$
38536	43	6	corresponding	corresponding	JJ
38536	43	7	terms	term	NNS
38536	43	8	are	be	VBP
38536	43	9	similar	similar	JJ
38536	43	10	.	.	.
38536	44	1	(	(	-LRB-
38536	44	2	3x	3x	CD
38536	44	3	+	+	SYM
38536	44	4	2t)(2x	2t)(2x	CD
38536	44	5	-	-	SYM
38536	44	6	5	5	CD
38536	44	7	t	t	NN
38536	44	8	)	)	-RRB-
38536	44	9	.	.	.
38536	45	1	6	6	CD
38536	45	2	.	.	.
38536	46	1	Square	square	NN
38536	46	2	of	of	IN
38536	46	3	a	a	DT
38536	46	4	polynomial	polynomial	NN
38536	46	5	.	.	.
38536	47	1	(	(	-LRB-
38536	47	2	m	m	NN
38536	47	3	-	-	HYPH
38536	47	4	n/3	n/3	NNP
38536	47	5	+	+	CD
38536	47	6	k)^2	k)^2	RB
38536	47	7	.	.	.
38536	48	1	7	7	LS
38536	48	2	.	.	.
38536	49	1	Sum	Sum	NNP
38536	49	2	of	of	IN
38536	49	3	two	two	CD
38536	49	4	cubes	cube	NNS
38536	49	5	.	.	.
38536	50	1	(	(	-LRB-
38536	50	2	x^3	x^3	NNP
38536	50	3	+	+	SYM
38536	50	4	y^3)/(x	y^3)/(x	NNP
38536	50	5	+	+	SYM
38536	50	6	y	y	NN
38536	50	7	)	)	-RRB-
38536	50	8	=	=	SYM
38536	50	9	x^2	x^2	NNP
38536	50	10	-	-	HYPH
38536	50	11	xy	xy	NN
38536	50	12	+	+	CC
38536	50	13	y^2	y^2	PRP
38536	50	14	.	.	.
38536	51	1	8	8	LS
38536	51	2	.	.	.
38536	52	1	Difference	difference	NN
38536	52	2	of	of	IN
38536	52	3	two	two	CD
38536	52	4	cubes	cube	NNS
38536	52	5	.	.	.
38536	53	1	(	(	-LRB-
38536	53	2	x^3	x^3	NNP
38536	53	3	-	-	HYPH
38536	53	4	y^3)/(x	y^3)/(x	NNP
38536	53	5	-	-	HYPH
38536	53	6	y	y	NNP
38536	53	7	)	)	-RRB-
38536	53	8	=	=	NFP
38536	53	9	x^2	x^2	NNS
38536	53	10	+	+	SYM
38536	53	11	xy	xy	NN
38536	53	12	+	+	SYM
38536	53	13	y^2	y^2	PRP
38536	53	14	.	.	.
38536	54	1	9	9	CD
38536	54	2	.	.	.
38536	55	1	Sum	sum	NN
38536	55	2	or	or	CC
38536	55	3	difference	difference	NN
38536	55	4	of	of	IN
38536	55	5	two	two	CD
38536	55	6	like	like	JJ
38536	55	7	powers	power	NNS
38536	55	8	.	.	.
38536	56	1	(	(	-LRB-
38536	56	2	x^7	x^7	NNS
38536	56	3	+	+	SYM
38536	56	4	y^7)/(x	y^7)/(x	NN
38536	56	5	+	+	SYM
38536	56	6	y	y	NN
38536	56	7	)	)	-RRB-
38536	56	8	,	,	,
38536	56	9	(	(	-LRB-
38536	56	10	x^5	x^5	NNP
38536	56	11	-	-	HYPH
38536	56	12	y^5)/(x	y^5)/(x	NNP
38536	56	13	-	-	HYPH
38536	56	14	y	y	NNP
38536	56	15	)	)	-RRB-
38536	56	16	,	,	,
38536	56	17	(	(	-LRB-
38536	56	18	x^4	x^4	NNP
38536	56	19	-	-	HYPH
38536	56	20	y^4)/(x	y^4)/(x	NNP
38536	56	21	-	-	HYPH
38536	56	22	y	y	NNP
38536	56	23	)	)	-RRB-
38536	56	24	,	,	,
38536	56	25	(	(	-LRB-
38536	56	26	x^4	x^4	NNP
38536	56	27	-	-	HYPH
38536	56	28	y^4)/(x	y^4)/(x	NNP
38536	56	29	+	+	SYM
38536	56	30	y	y	NN
38536	56	31	)	)	-RRB-
38536	56	32	.	.	.
38536	57	1	~Cases	~Cases	NFP
38536	57	2	in	in	IN
38536	57	3	Factoring~	Factoring~	NNP
38536	57	4	1	1	CD
38536	57	5	.	.	.
38536	58	1	Common	common	JJ
38536	58	2	monomial	monomial	JJ
38536	58	3	factor	factor	NN
38536	58	4	.	.	.
38536	59	1	mx	mx	NNP
38536	59	2	+	+	CC
38536	59	3	my	-PRON-	PRP$
38536	59	4	-	-	HYPH
38536	59	5	mz	mz	NNP
38536	59	6	=	=	-RRB-
38536	59	7	m(x	m(x	NNP
38536	59	8	+	+	SYM
38536	59	9	y	y	NNP
38536	59	10	-	-	HYPH
38536	59	11	z	z	NNP
38536	59	12	)	)	-RRB-
38536	59	13	.	.	.
38536	60	1	2	2	LS
38536	60	2	.	.	.
38536	61	1	Trinomial	trinomial	JJ
38536	61	2	that	that	WDT
38536	61	3	is	be	VBZ
38536	61	4	a	a	DT
38536	61	5	perfect	perfect	JJ
38536	61	6	square	square	NN
38536	61	7	.	.	.
38536	62	1	x^2	x^2	NNP
38536	62	2	±	±	NNP
38536	62	3	2xy	2xy	NN
38536	62	4	+	+	SYM
38536	62	5	y^2	y^2	ADD
38536	62	6	=	=	NFP
38536	62	7	(	(	-LRB-
38536	62	8	x	x	SYM
38536	62	9	±	±	CD
38536	62	10	y)^2	y)^2	NNS
38536	62	11	.	.	.
38536	63	1	3	3	LS
38536	63	2	.	.	.
38536	64	1	The	the	DT
38536	64	2	difference	difference	NN
38536	64	3	of	of	IN
38536	64	4	two	two	CD
38536	64	5	squares	square	NNS
38536	64	6	.	.	.
38536	65	1	(	(	-LRB-
38536	65	2	a	a	LS
38536	65	3	)	)	-RRB-
38536	65	4	Two	two	CD
38536	65	5	terms	term	NNS
38536	65	6	.	.	.
38536	66	1	x^2	x^2	NNP
38536	66	2	-	-	HYPH
38536	66	3	y^2	y^2	NNP
38536	66	4	=	=	NFP
38536	66	5	(	(	-LRB-
38536	66	6	x	x	SYM
38536	66	7	+	+	SYM
38536	66	8	y)(x	y)(x	NNP
38536	66	9	-	-	HYPH
38536	66	10	y	y	NNP
38536	66	11	)	)	-RRB-
38536	66	12	.	.	.
38536	67	1	(	(	-LRB-
38536	67	2	b	b	LS
38536	67	3	)	)	-RRB-
38536	67	4	Four	four	CD
38536	67	5	terms	term	NNS
38536	67	6	.	.	.
38536	68	1	x^2	x^2	NNS
38536	68	2	+	+	SYM
38536	68	3	2xy	2xy	NN
38536	68	4	+	+	SYM
38536	68	5	y^2	y^2	PRP
38536	68	6	-	-	HYPH
38536	68	7	m^2	m^2	CD
38536	68	8	=	=	NFP
38536	68	9	(	(	-LRB-
38536	68	10	x	x	SYM
38536	68	11	+	+	SYM
38536	68	12	y	y	NN
38536	68	13	+	+	CC
38536	68	14	m)(x	m)(x	NNP
38536	68	15	+	+	NNP
38536	68	16	y	y	NNP
38536	68	17	-	-	HYPH
38536	68	18	m	m	NNP
38536	68	19	)	)	-RRB-
38536	68	20	.	.	.
38536	69	1	(	(	-LRB-
38536	69	2	c	c	NN
38536	69	3	)	)	-RRB-
38536	69	4	Six	six	CD
38536	69	5	terms	term	NNS
38536	69	6	.	.	.
38536	70	1	x^2	x^2	NNS
38536	70	2	+	+	SYM
38536	70	3	2xy	2xy	NN
38536	70	4	+	+	SYM
38536	70	5	y^2	y^2	NN
38536	70	6	-	-	HYPH
38536	70	7	p^2	p^2	JJ
38536	70	8	-	-	HYPH
38536	70	9	2pq	2pq	JJ
38536	70	10	-	-	HYPH
38536	70	11	q^2	q^2	NN
38536	70	12	=	=	NFP
38536	70	13	[	[	-LRB-
38536	70	14	(	(	-LRB-
38536	70	15	x	x	SYM
38536	70	16	+	+	SYM
38536	70	17	y	y	NN
38536	70	18	)	)	-RRB-
38536	70	19	+	+	CC
38536	70	20	(	(	-LRB-
38536	70	21	p	p	NN
38536	70	22	+	+	CC
38536	70	23	q)][(x	q)][(x	NN
38536	70	24	+	+	SYM
38536	70	25	y	y	NN
38536	70	26	)	)	-RRB-
38536	70	27	-	-	:
38536	70	28	(	(	-LRB-
38536	70	29	p	p	NN
38536	70	30	+	+	SYM
38536	70	31	q	q	NN
38536	70	32	)	)	-RRB-
38536	70	33	]	]	-RRB-
38536	70	34	.	.	.
38536	71	1	(	(	-LRB-
38536	71	2	d	d	LS
38536	71	3	)	)	-RRB-
38536	71	4	Incomplete	Incomplete	NNP
38536	71	5	square	square	NN
38536	71	6	.	.	.
38536	72	1	x^4	x^4	NNP
38536	72	2	+	+	SYM
38536	72	3	x^2	x^2	NNP
38536	72	4	y^2	y^2	NNS
38536	72	5	+	+	SYM
38536	72	6	y^4	y^4	XX
38536	72	7	=	=	SYM
38536	72	8	x^4	x^4	NNP
38536	72	9	+	+	CC
38536	72	10	2x^2	2x^2	CD
38536	72	11	y^2	y^2	CD
38536	72	12	+	+	SYM
38536	72	13	y^4	y^4	CD
38536	72	14	-	-	HYPH
38536	72	15	x^2	x^2	JJ
38536	72	16	y^2	y^2	NNS
38536	72	17	=	=	NFP
38536	72	18	(	(	-LRB-
38536	72	19	x^2	x^2	NNS
38536	72	20	+	+	SYM
38536	72	21	y^2	y^2	NNS
38536	72	22	+	+	SYM
38536	72	23	xy)(x^2	xy)(x^2	NN
38536	72	24	+	+	SYM
38536	72	25	y^2	y^2	NNP
38536	72	26	-	-	HYPH
38536	72	27	xy	xy	NN
38536	72	28	)	)	-RRB-
38536	72	29	.	.	.
38536	73	1	4	4	LS
38536	73	2	.	.	.
38536	74	1	Trinomial	trinomial	NN
38536	74	2	of	of	IN
38536	74	3	the	the	DT
38536	74	4	form	form	NN
38536	74	5	x^2	x^2	NNS
38536	74	6	+	+	SYM
38536	74	7	bx	bx	NN
38536	74	8	+	+	SYM
38536	74	9	c.	c.	NNP
38536	74	10	x^2	x^2	NNP
38536	74	11	-	-	HYPH
38536	74	12	5x	5x	NNP
38536	74	13	+	+	SYM
38536	74	14	6	6	CD
38536	74	15	=	=	SYM
38536	74	16	(	(	-LRB-
38536	74	17	x	x	SYM
38536	74	18	-	-	CD
38536	74	19	2)(x	2)(x	CD
38536	74	20	-	-	SYM
38536	74	21	3	3	CD
38536	74	22	)	)	-RRB-
38536	74	23	.	.	.
38536	75	1	5	5	CD
38536	75	2	.	.	.
38536	76	1	Trinomial	trinomial	NN
38536	76	2	of	of	IN
38536	76	3	the	the	DT
38536	76	4	form	form	NN
38536	76	5	ax^2	ax^2	NN
38536	76	6	+	+	SYM
38536	76	7	bx	bx	NN
38536	76	8	+	+	SYM
38536	76	9	c.	c.	NN
38536	76	10	20x^2	20x^2	CD
38536	76	11	+	+	SYM
38536	76	12	7x	7x	NNP
38536	76	13	-	-	SYM
38536	76	14	6	6	CD
38536	76	15	=	=	SYM
38536	76	16	(	(	-LRB-
38536	76	17	4x	4x	NNS
38536	76	18	+	+	SYM
38536	76	19	3)(5x	3)(5x	CD
38536	76	20	-	-	HYPH
38536	76	21	2	2	CD
38536	76	22	)	)	-RRB-
38536	76	23	.	.	.
38536	77	1	6	6	CD
38536	77	2	.	.	.
38536	78	1	Sum	sum	NN
38536	78	2	or	or	CC
38536	78	3	difference	difference	NN
38536	78	4	of	of	IN
38536	78	5	two	two	CD
38536	78	6	cubes	cube	NNS
38536	78	7	.	.	.
38536	79	1	See	see	VB
38536	79	2	"	"	``
38536	79	3	Special	special	JJ
38536	79	4	Rules	rule	NNS
38536	79	5	,	,	,
38536	79	6	"	"	''
38536	79	7	7	7	CD
38536	79	8	and	and	CC
38536	79	9	8	8	CD
38536	79	10	.	.	.
38536	79	11	two	two	CD
38536	79	12	like	like	IN
38536	79	13	powers	power	NNS
38536	79	14	.	.	.
38536	80	1	See	see	VB
38536	80	2	"	"	``
38536	80	3	Special	special	JJ
38536	80	4	Rules	rule	NNS
38536	80	5	,	,	,
38536	80	6	"	"	''
38536	80	7	9	9	CD
38536	80	8	.	.	.
38536	81	1	7	7	LS
38536	81	2	.	.	.
38536	82	1	Common	common	JJ
38536	82	2	polynomial	polynomial	JJ
38536	82	3	factor	factor	NN
38536	82	4	.	.	.
38536	83	1	Grouping	group	VBG
38536	83	2	.	.	.
38536	84	1	t^2	t^2	CD
38536	84	2	p	p	NN
38536	84	3	+	+	CC
38536	84	4	t^2	t^2	CD
38536	84	5	q	q	NN
38536	84	6	-	-	HYPH
38536	84	7	2mp	2mp	JJ
38536	84	8	-	-	HYPH
38536	84	9	2mq	2mq	JJ
38536	84	10	=	=	SYM
38536	84	11	t^2(p	t^2(p	NN
38536	84	12	+	+	CC
38536	84	13	q	q	NN
38536	84	14	)	)	-RRB-
38536	84	15	-	-	:
38536	84	16	2m(p	2m(p	CD
38536	84	17	+	+	SYM
38536	84	18	q	q	NN
38536	84	19	)	)	-RRB-
38536	84	20	=	=	NFP
38536	84	21	(	(	-LRB-
38536	84	22	p	p	NN
38536	84	23	+	+	CC
38536	84	24	q)(t^2	q)(t^2	NN
38536	84	25	-	-	HYPH
38536	84	26	2	2	CD
38536	84	27	m	m	NN
38536	84	28	)	)	-RRB-
38536	84	29	.	.	.
38536	85	1	8	8	LS
38536	85	2	.	.	.
38536	86	1	Factor	Factor	NNP
38536	86	2	Theorem	Theorem	NNP
38536	86	3	.	.	.
38536	87	1	x^3	x^3	NNS
38536	87	2	+	+	SYM
38536	87	3	17x	17x	CD
38536	87	4	-	-	SYM
38536	87	5	18	18	CD
38536	87	6	=	=	SYM
38536	87	7	(	(	-LRB-
38536	87	8	x	x	SYM
38536	87	9	-	-	SYM
38536	87	10	1)(x^2	1)(x^2	CD
38536	87	11	+	+	SYM
38536	87	12	x	x	SYM
38536	87	13	+	+	SYM
38536	87	14	18	18	CD
38536	87	15	)	)	-RRB-
38536	87	16	.	.	.
38536	88	1	~H.	~H.	NNP
38536	89	1	C.	C.	NNP
38536	89	2	F.	F.	NNP
38536	89	3	and	and	CC
38536	89	4	L.	L.	NNP
38536	89	5	C.	C.	NNP
38536	89	6	M.~	M.~	NNP
38536	89	7	a^2	a^2	CD
38536	89	8	+	+	SYM
38536	89	9	2a	2a	CD
38536	89	10	-	-	SYM
38536	89	11	3	3	CD
38536	89	12	=	=	SYM
38536	89	13	(	(	-LRB-
38536	89	14	a	a	DT
38536	89	15	+	+	SYM
38536	89	16	3)(a	3)(a	CD
38536	89	17	-	-	SYM
38536	89	18	1	1	CD
38536	89	19	)	)	-RRB-
38536	89	20	.	.	.
38536	90	1	a^2	a^2	RB
38536	90	2	+	+	CC
38536	90	3	7a	7a	NN
38536	90	4	+	+	SYM
38536	90	5	12	12	CD
38536	90	6	=	=	SYM
38536	90	7	(	(	-LRB-
38536	90	8	a	a	DT
38536	90	9	+	+	$
38536	90	10	3)(a	3)(a	CD
38536	90	11	+	+	SYM
38536	90	12	4	4	CD
38536	90	13	)	)	-RRB-
38536	90	14	.	.	.
38536	91	1	a^4	a^4	CD
38536	91	2	+	+	SYM
38536	91	3	27a	27a	NNS
38536	91	4	=	=	SYM
38536	91	5	a(a	a(a	CD
38536	91	6	+	+	CC
38536	91	7	3)(a^2	3)(a^2	CD
38536	91	8	-	-	HYPH
38536	91	9	3a	3a	CD
38536	91	10	+	+	SYM
38536	91	11	9	9	CD
38536	91	12	)	)	-RRB-
38536	91	13	.	.	.
38536	92	1	H.	H.	NNP
38536	92	2	C.	C.	NNP
38536	92	3	F.	F.	NNP
38536	92	4	=	=	SYM
38536	92	5	a	a	NN
38536	92	6	+	+	SYM
38536	92	7	3	3	CD
38536	92	8	.	.	.
38536	93	1	L.	L.	NNP
38536	93	2	C.	C.	NNP
38536	93	3	M.	M.	NNP
38536	93	4	=	=	NNP
38536	93	5	(	(	-LRB-
38536	93	6	a	a	DT
38536	93	7	+	+	$
38536	93	8	3)(a	3)(a	CD
38536	93	9	-	-	HYPH
38536	93	10	1)(a	1)(a	CD
38536	93	11	+	+	SYM
38536	93	12	4)a(a^2	4)a(a^2	CD
38536	93	13	-	-	HYPH
38536	93	14	3a	3a	CD
38536	93	15	+	+	SYM
38536	93	16	9	9	CD
38536	93	17	)	)	-RRB-
38536	93	18	.	.	.
38536	94	1	~Fractions~	~Fractions~	NFP
38536	94	2	Reduction	reduction	NN
38536	94	3	to	to	IN
38536	94	4	lowest	low	JJS
38536	94	5	terms	term	NNS
38536	94	6	.	.	.
38536	95	1	Reduction	reduction	NN
38536	95	2	of	of	IN
38536	95	3	a	a	DT
38536	95	4	mixed	mixed	JJ
38536	95	5	number	number	NN
38536	95	6	to	to	IN
38536	95	7	an	an	DT
38536	95	8	improper	improper	JJ
38536	95	9	fraction	fraction	NN
38536	95	10	.	.	.
38536	96	1	Reduction	reduction	NN
38536	96	2	of	of	IN
38536	96	3	an	an	DT
38536	96	4	improper	improper	JJ
38536	96	5	fraction	fraction	NN
38536	96	6	to	to	IN
38536	96	7	a	a	DT
38536	96	8	mixed	mixed	JJ
38536	96	9	number	number	NN
38536	96	10	.	.	.
38536	97	1	Addition	addition	NN
38536	97	2	and	and	CC
38536	97	3	subtraction	subtraction	NN
38536	97	4	of	of	IN
38536	97	5	fractions	fraction	NNS
38536	97	6	.	.	.
38536	98	1	Multiplication	Multiplication	NNP
38536	98	2	and	and	CC
38536	98	3	division	division	NN
38536	98	4	of	of	IN
38536	98	5	fractions	fraction	NNS
38536	98	6	.	.	.
38536	99	1	Law	law	NN
38536	99	2	of	of	IN
38536	99	3	signs	sign	NNS
38536	99	4	in	in	IN
38536	99	5	division	division	NN
38536	99	6	,	,	,
38536	99	7	changing	change	VBG
38536	99	8	signs	sign	NNS
38536	99	9	of	of	IN
38536	99	10	factors	factor	NNS
38536	99	11	,	,	,
38536	99	12	etc	etc	FW
38536	99	13	.	.	.
38536	100	1	Complex	complex	JJ
38536	100	2	fractions	fraction	NNS
38536	100	3	.	.	.
38536	101	1	~Simultaneous	~Simultaneous	NFP
38536	101	2	Equations~	equations~	NN
38536	101	3	Solved	solve	VBN
38536	101	4	by	by	IN
38536	101	5	addition	addition	NN
38536	101	6	or	or	CC
38536	101	7	subtraction	subtraction	NN
38536	101	8	.	.	.
38536	102	1	substitution	substitution	NN
38536	102	2	.	.	.
38536	103	1	comparison	comparison	NN
38536	103	2	.	.	.
38536	104	1	Graphical	graphical	JJ
38536	104	2	representation	representation	NN
38536	104	3	.	.	.
38536	105	1	~Involution~	~Involution~	NFP
38536	105	2	Law	law	NN
38536	105	3	of	of	IN
38536	105	4	signs	sign	NNS
38536	105	5	.	.	.
38536	106	1	Binomial	binomial	JJ
38536	106	2	theorem	theorem	NN
38536	106	3	laws	law	NNS
38536	106	4	.	.	.
38536	107	1	Expansion	expansion	NN
38536	107	2	of	of	IN
38536	107	3	monomials	monomial	NNS
38536	107	4	and	and	CC
38536	107	5	fractions	fraction	NNS
38536	107	6	.	.	.
38536	108	1	binomials	binomial	NNS
38536	108	2	.	.	.
38536	109	1	trinomials	trinomial	NNS
38536	109	2	.	.	.
38536	110	1	~Evolution~	~Evolution~	NFP
38536	110	2	Law	law	NN
38536	110	3	of	of	IN
38536	110	4	signs	sign	NNS
38536	110	5	.	.	.
38536	111	1	Evolution	evolution	NN
38536	111	2	of	of	IN
38536	111	3	monomials	monomial	NNS
38536	111	4	and	and	CC
38536	111	5	fractions	fraction	NNS
38536	111	6	.	.	.
38536	112	1	Square	square	JJ
38536	112	2	root	root	NN
38536	112	3	of	of	IN
38536	112	4	algebraic	algebraic	NN
38536	112	5	expressions	expression	NNS
38536	112	6	.	.	.
38536	113	1	Square	square	JJ
38536	113	2	root	root	NN
38536	113	3	of	of	IN
38536	113	4	arithmetical	arithmetical	JJ
38536	113	5	numbers	number	NNS
38536	113	6	.	.	.
38536	114	1	Optional	optional	JJ
38536	114	2	Cube	Cube	NNP
38536	114	3	root	root	NN
38536	114	4	of	of	IN
38536	114	5	algebraic	algebraic	NN
38536	114	6	expressions	expression	NNS
38536	114	7	.	.	.
38536	115	1	Cube	Cube	NNP
38536	115	2	root	root	NN
38536	115	3	of	of	IN
38536	115	4	arithmetical	arithmetical	JJ
38536	115	5	numbers	number	NNS
38536	115	6	.	.	.
38536	116	1	~Theory	~Theory	NFP
38536	116	2	of	of	IN
38536	116	3	Exponents~	Exponents~	NNP
38536	116	4	Proofs	Proofs	NNP
38536	116	5	:	:	:
38536	116	6	a^m	a^m	NNP
38536	116	7	×	×	NN
38536	116	8	a^n	a^n	UH
38536	116	9	=	=	NN
38536	116	10	a^(m	a^(m	NN
38536	116	11	+	+	SYM
38536	116	12	n	n	NN
38536	116	13	)	)	-RRB-
38536	116	14	;	;	:
38536	116	15	(	(	-LRB-
38536	116	16	a^m)/(a^n	a^m)/(a^n	NN
38536	116	17	)	)	-RRB-
38536	116	18	=	=	SYM
38536	116	19	a^(m	a^(m	CD
38536	116	20	-	-	HYPH
38536	116	21	n	n	NN
38536	116	22	)	)	-RRB-
38536	116	23	;	;	:
38536	116	24	(	(	-LRB-
38536	116	25	a^m)^n	a^m)^n	NNP
38536	116	26	=	=	NFP
38536	116	27	a^(mn	a^(mn	NNP
38536	116	28	)	)	-RRB-
38536	116	29	;	;	:
38536	116	30	[	[	-LRB-
38536	116	31	a^(mn)]^(1	a^(mn)]^(1	NN
38536	116	32	/	/	SYM
38536	116	33	n	n	NN
38536	116	34	)	)	-RRB-
38536	116	35	=	=	NFP
38536	116	36	a^m	a^m	NNP
38536	116	37	;	;	:
38536	116	38	(	(	-LRB-
38536	116	39	a	a	DT
38536	116	40	/	/	SYM
38536	116	41	b)^n	b)^n	JJ
38536	116	42	=	=	NFP
38536	116	43	(	(	-LRB-
38536	116	44	a^n)/(b^n	a^n)/(b^n	NN
38536	116	45	)	)	-RRB-
38536	116	46	;	;	:
38536	116	47	(	(	-LRB-
38536	116	48	abc)^n	abc)^n	ADD
38536	116	49	=	=	SYM
38536	116	50	a^n	a^n	NNP
38536	116	51	b^n	b^n	NN
38536	116	52	c^n	c^n	NNS
38536	116	53	.	.	.
38536	117	1	Meaning	mean	VBG
38536	117	2	of	of	IN
38536	117	3	fractional	fractional	JJ
38536	117	4	exponent	exponent	NN
38536	117	5	.	.	.
38536	118	1	zero	zero	CD
38536	118	2	exponent	exponent	NN
38536	118	3	.	.	.
38536	119	1	negative	negative	JJ
38536	119	2	exponent	exponent	NN
38536	119	3	.	.	.
38536	120	1	Four	four	CD
38536	120	2	rules	rule	NNS
38536	120	3	To	to	TO
38536	120	4	multiply	multiply	VB
38536	120	5	quantities	quantity	NNS
38536	120	6	having	have	VBG
38536	120	7	the	the	DT
38536	120	8	same	same	JJ
38536	120	9	base	base	NN
38536	120	10	,	,	,
38536	120	11	add	add	VB
38536	120	12	exponents	exponent	NNS
38536	120	13	.	.	.
38536	121	1	To	to	TO
38536	121	2	divide	divide	VB
38536	121	3	quantities	quantity	NNS
38536	121	4	having	have	VBG
38536	121	5	the	the	DT
38536	121	6	same	same	JJ
38536	121	7	base	base	NN
38536	121	8	,	,	,
38536	121	9	subtract	subtract	NN
38536	121	10	exponents	exponent	NNS
38536	121	11	.	.	.
38536	122	1	To	to	TO
38536	122	2	raise	raise	VB
38536	122	3	to	to	IN
38536	122	4	a	a	DT
38536	122	5	power	power	NN
38536	122	6	,	,	,
38536	122	7	multiply	multiply	JJ
38536	122	8	exponents	exponent	NNS
38536	122	9	.	.	.
38536	123	1	To	to	TO
38536	123	2	extract	extract	VB
38536	123	3	a	a	DT
38536	123	4	root	root	NN
38536	123	5	,	,	,
38536	123	6	divide	divide	VB
38536	123	7	the	the	DT
38536	123	8	exponent	exponent	NN
38536	123	9	of	of	IN
38536	123	10	the	the	DT
38536	123	11	power	power	NN
38536	123	12	by	by	IN
38536	123	13	the	the	DT
38536	123	14	index	index	NN
38536	123	15	of	of	IN
38536	123	16	the	the	DT
38536	123	17	root	root	NN
38536	123	18	.	.	.
38536	124	1	~Radicals~	~Radicals~	NFP
38536	124	2	Radical	Radical	NNP
38536	124	3	in	in	IN
38536	124	4	its	-PRON-	PRP$
38536	124	5	simplest	simple	JJS
38536	124	6	form	form	NN
38536	124	7	.	.	.
38536	125	1	Transformation	transformation	NN
38536	125	2	of	of	IN
38536	125	3	radicals	radical	NNS
38536	125	4	Fraction	Fraction	NNP
38536	125	5	under	under	IN
38536	125	6	the	the	DT
38536	125	7	radical	radical	JJ
38536	125	8	sign	sign	NN
38536	125	9	.	.	.
38536	126	1	Reduction	reduction	NN
38536	126	2	to	to	IN
38536	126	3	an	an	DT
38536	126	4	entire	entire	JJ
38536	126	5	surd	surd	NN
38536	126	6	.	.	.
38536	127	1	Changing	change	VBG
38536	127	2	to	to	IN
38536	127	3	surds	surd	NNS
38536	127	4	of	of	IN
38536	127	5	different	different	JJ
38536	127	6	order	order	NN
38536	127	7	.	.	.
38536	128	1	Reduction	reduction	NN
38536	128	2	to	to	IN
38536	128	3	simplest	simple	JJS
38536	128	4	form	form	NN
38536	128	5	.	.	.
38536	129	1	Addition	addition	NN
38536	129	2	and	and	CC
38536	129	3	subtraction	subtraction	NN
38536	129	4	of	of	IN
38536	129	5	radicals	radical	NNS
38536	129	6	.	.	.
38536	130	1	Multiplication	Multiplication	NNP
38536	130	2	and	and	CC
38536	130	3	division	division	NN
38536	130	4	of	of	IN
38536	130	5	radicals	radical	NNS
38536	130	6	a^(1	a^(1	ADD
38536	130	7	/	/	SYM
38536	130	8	n	n	NN
38536	130	9	)	)	-RRB-
38536	130	10	·	·	NFP
38536	130	11	b^(1	b^(1	NNS
38536	130	12	/	/	SYM
38536	130	13	n	n	NN
38536	130	14	)	)	-RRB-
38536	130	15	=	=	NFP
38536	130	16	[	[	-LRB-
38536	130	17	ab]^(1	ab]^(1	NNP
38536	130	18	/	/	SYM
38536	130	19	n	n	NNP
38536	130	20	)	)	-RRB-
38536	130	21	.	.	.
38536	131	1	(	(	-LRB-
38536	131	2	[	[	-LRB-
38536	131	3	ab]^(1	ab]^(1	NNP
38536	131	4	/	/	SYM
38536	131	5	n))/(a^(1	n))/(a^(1	NNP
38536	131	6	/	/	SYM
38536	131	7	n	n	NNP
38536	131	8	)	)	-RRB-
38536	131	9	)	)	-RRB-
38536	131	10	=	=	NFP
38536	131	11	b^(1	b^(1	ADD
38536	131	12	/	/	SYM
38536	131	13	n	n	NN
38536	131	14	)	)	-RRB-
38536	131	15	.	.	.
38536	132	1	Rationalization	rationalization	NN
38536	132	2	Monomial	Monomial	NNP
38536	132	3	denominator	denominator	NN
38536	132	4	.	.	.
38536	133	1	Binomial	binomial	JJ
38536	133	2	denominator	denominator	NN
38536	133	3	.	.	.
38536	134	1	Trinomial	trinomial	JJ
38536	134	2	denominator	denominator	NN
38536	134	3	.	.	.
38536	135	1	Square	square	JJ
38536	135	2	root	root	NN
38536	135	3	of	of	IN
38536	135	4	a	a	DT
38536	135	5	binomial	binomial	JJ
38536	135	6	surd	surd	NN
38536	135	7	.	.	.
38536	136	1	Radical	radical	JJ
38536	136	2	equations	equation	NNS
38536	136	3	.	.	.
38536	137	1	_	_	NNP
38536	137	2	Always	always	RB
38536	137	3	_	_	NNP
38536	137	4	check	check	VBP
38536	137	5	results	result	NNS
38536	137	6	to	to	TO
38536	137	7	avoid	avoid	VB
38536	137	8	extraneous	extraneous	JJ
38536	137	9	roots	root	NNS
38536	137	10	.	.	.
38536	138	1	~Quadratic	~Quadratic	NFP
38536	138	2	Equations~	Equations~	.
38536	138	3	Pure	pure	JJ
38536	138	4	.	.	.
38536	139	1	x^2	x^2	NNP
38536	139	2	=	=	SYM
38536	139	3	a.	a.	NN
38536	140	1	Affected	affect	VBN
38536	140	2	.	.	.
38536	141	1	ax^2	ax^2	NN
38536	141	2	+	+	SYM
38536	141	3	bx	bx	NN
38536	141	4	+	+	NNS
38536	141	5	c	c	NN
38536	141	6	=	=	SYM
38536	141	7	0	0	NFP
38536	141	8	.	.	.
38536	142	1	Methods	method	NNS
38536	142	2	of	of	IN
38536	142	3	solving	solve	VBG
38536	142	4	Completing	complete	VBG
38536	142	5	the	the	DT
38536	142	6	square	square	NN
38536	142	7	.	.	.
38536	143	1	Formula	formula	NN
38536	143	2	.	.	.
38536	144	1	Developed	develop	VBN
38536	144	2	from	from	IN
38536	144	3	ax^2	ax^2	CD
38536	144	4	+	+	SYM
38536	144	5	bx	bx	NN
38536	144	6	+	+	NNS
38536	144	7	c	c	NN
38536	144	8	=	=	SYM
38536	144	9	0	0	NFP
38536	144	10	.	.	.
38536	145	1	Factoring	factor	VBG
38536	145	2	.	.	.
38536	146	1	Equations	equation	NNS
38536	146	2	in	in	IN
38536	146	3	the	the	DT
38536	146	4	quadratic	quadratic	JJ
38536	146	5	form	form	NN
38536	146	6	.	.	.
38536	147	1	Properties	property	NNS
38536	147	2	of	of	IN
38536	147	3	quadratics	quadratic	NNS
38536	147	4	r_1	r_1	NNP
38536	147	5	=	=	SYM
38536	147	6	-b/2a	-b/2a	:
38536	147	7	+	+	NFP
38536	147	8	(	(	-LRB-
38536	147	9	[	[	-LRB-
38536	147	10	b^2	b^2	NNP
38536	147	11	-	-	SYM
38536	147	12	4ac]^(1/2))/(2a	4ac]^(1/2))/(2a	CD
38536	147	13	)	)	-RRB-
38536	147	14	.	.	.
38536	148	1	r_2	r_2	NN
38536	148	2	=	=	SYM
38536	148	3	-b/2a	-b/2a	:
38536	148	4	-	-	:
38536	148	5	(	(	-LRB-
38536	148	6	[	[	-LRB-
38536	148	7	b^2	b^2	NNP
38536	148	8	-	-	SYM
38536	148	9	4ac]^(1/2))/(2a	4ac]^(1/2))/(2a	CD
38536	148	10	)	)	-RRB-
38536	148	11	.	.	.
38536	149	1	Then	then	RB
38536	149	2	r_1	r_1	VB
38536	149	3	+	+	CC
38536	149	4	r_2	r_2	CD
38536	149	5	=	=	SYM
38536	149	6	-b	-b	:
38536	149	7	/	/	SYM
38536	149	8	a	a	NN
38536	149	9	.	.	.
38536	150	1	r_1	r_1	VB
38536	150	2	·	·	NFP
38536	150	3	r_2	r_2	NN
38536	150	4	=	=	SYM
38536	150	5	c	c	NN
38536	150	6	/	/	SYM
38536	150	7	a	a	NN
38536	150	8	.	.	.
38536	151	1	Discriminant	discriminant	JJ
38536	151	2	,	,	,
38536	151	3	b^2	b^2	NNS
38536	151	4	-	-	HYPH
38536	151	5	4ac	4ac	NN
38536	151	6	,	,	,
38536	151	7	and	and	CC
38536	151	8	its	-PRON-	PRP$
38536	151	9	discussion	discussion	NN
38536	151	10	.	.	.
38536	152	1	Nature	nature	NN
38536	152	2	or	or	CC
38536	152	3	character	character	NN
38536	152	4	of	of	IN
38536	152	5	the	the	DT
38536	152	6	roots	root	NNS
38536	152	7	.	.	.
38536	153	1	~Simultaneous	~Simultaneous	NFP
38536	153	2	Quadratics~	Quadratics~	JJR
38536	153	3	CASE	CASE	NNP
38536	153	4	I.	I.	NNP
38536	154	1	One	one	CD
38536	154	2	equation	equation	NN
38536	154	3	linear	linear	NN
38536	154	4	.	.	.
38536	155	1	The	the	DT
38536	155	2	other	other	JJ
38536	155	3	quadratic	quadratic	NN
38536	155	4	.	.	.
38536	156	1	3x	3x	NNP
38536	156	2	-	-	HYPH
38536	156	3	y	y	NN
38536	156	4	=	=	SYM
38536	156	5	12	12	CD
38536	156	6	,	,	,
38536	156	7	x^2	x^2	NNPS
38536	156	8	-	-	HYPH
38536	156	9	y^2	y^2	NNP
38536	156	10	=	=	SYM
38536	156	11	16	16	CD
38536	156	12	.	.	.
38536	157	1	CASE	CASE	NNP
38536	157	2	II	II	NNP
38536	157	3	.	.	.
38536	158	1	Both	both	DT
38536	158	2	equations	equation	NNS
38536	158	3	homogeneous	homogeneous	JJ
38536	158	4	and	and	CC
38536	158	5	of	of	IN
38536	158	6	the	the	DT
38536	158	7	second	second	JJ
38536	158	8	degree	degree	NN
38536	158	9	.	.	.
38536	159	1	x^2	x^2	NNP
38536	159	2	-	-	HYPH
38536	159	3	xy	xy	NNP
38536	159	4	+	+	CC
38536	159	5	y^2	y^2	PRP
38536	159	6	=	=	SYM
38536	159	7	21	21	CD
38536	159	8	,	,	,
38536	159	9	y^2	y^2	NNP
38536	159	10	-	-	HYPH
38536	159	11	2xy	2xy	NNP
38536	159	12	=	=	SYM
38536	159	13	-15	-15	XX
38536	159	14	.	.	.
38536	160	1	CASE	CASE	NNP
38536	160	2	III	III	NNP
38536	160	3	.	.	.
38536	161	1	Any	any	DT
38536	161	2	two	two	CD
38536	161	3	of	of	IN
38536	161	4	the	the	DT
38536	161	5	quantities	quantity	NNS
38536	161	6	x	x	SYM
38536	161	7	+	+	SYM
38536	161	8	y	y	NN
38536	161	9	,	,	,
38536	161	10	x^2	x^2	NNP
38536	161	11	+	+	SYM
38536	161	12	y^2	y^2	UH
38536	161	13	,	,	,
38536	161	14	xy	xy	NNP
38536	161	15	,	,	,
38536	161	16	x^3	x^3	NNP
38536	161	17	+	+	SYM
38536	161	18	y^3	y^3	NNS
38536	161	19	,	,	,
38536	161	20	x^3	x^3	NNP
38536	161	21	-	-	HYPH
38536	161	22	y^3	y^3	NNP
38536	161	23	,	,	,
38536	161	24	x	x	NNP
38536	161	25	-	-	NNP
38536	161	26	y	y	NNP
38536	161	27	,	,	,
38536	161	28	x^2	x^2	NNP
38536	161	29	±	±	NNP
38536	161	30	xy	xy	NNP
38536	161	31	+	+	SYM
38536	161	32	y^2	y^2	PRP
38536	161	33	,	,	,
38536	161	34	etc	etc	FW
38536	161	35	.	.	NN
38536	161	36	,	,	,
38536	161	37	given	give	VBN
38536	161	38	.	.	.
38536	162	1	x^2	x^2	NNP
38536	162	2	+	+	SYM
38536	162	3	y^2	y^2	NNS
38536	162	4	=	=	SYM
38536	162	5	41	41	CD
38536	162	6	,	,	,
38536	162	7	x	x	SYM
38536	162	8	+	+	SYM
38536	162	9	y	y	NN
38536	162	10	=	=	SYM
38536	162	11	9	9	CD
38536	162	12	.	.	.
38536	163	1	CASE	CASE	NNP
38536	163	2	IV	IV	NNP
38536	163	3	.	.	.
38536	164	1	Both	both	DT
38536	164	2	equations	equation	NNS
38536	164	3	symmetrical	symmetrical	JJ
38536	164	4	or	or	CC
38536	164	5	symmetrical	symmetrical	JJ
38536	164	6	except	except	IN
38536	164	7	for	for	IN
38536	164	8	sign	sign	NN
38536	164	9	.	.	.
38536	165	1	Usually	usually	RB
38536	165	2	one	one	CD
38536	165	3	equation	equation	NN
38536	165	4	of	of	IN
38536	165	5	high	high	JJ
38536	165	6	degree	degree	NN
38536	165	7	,	,	,
38536	165	8	the	the	DT
38536	165	9	other	other	JJ
38536	165	10	of	of	IN
38536	165	11	the	the	DT
38536	165	12	first	first	JJ
38536	165	13	degree	degree	NN
38536	165	14	.	.	.
38536	166	1	x^5	x^5	NNP
38536	166	2	+	+	SYM
38536	166	3	y^5	y^5	NNP
38536	166	4	=	=	SYM
38536	166	5	242	242	CD
38536	166	6	,	,	,
38536	166	7	x	x	NNS
38536	166	8	+	+	SYM
38536	166	9	y	y	NN
38536	166	10	=	=	SYM
38536	166	11	2	2	CD
38536	166	12	.	.	.
38536	167	1	CASE	CASE	NNP
38536	167	2	V.	V.	NNP
38536	167	3	Special	Special	NNP
38536	167	4	Devices	Devices	NNPS
38536	167	5	I.	I.	NNP
38536	168	1	Solve	solve	VB
38536	168	2	for	for	IN
38536	168	3	a	a	DT
38536	168	4	compound	compound	NN
38536	168	5	unknown	unknown	NN
38536	168	6	,	,	,
38536	168	7	like	like	IN
38536	168	8	xy	xy	NNP
38536	168	9	,	,	,
38536	168	10	x	x	NNP
38536	168	11	+	+	SYM
38536	168	12	y	y	NN
38536	168	13	,	,	,
38536	168	14	(	(	-LRB-
38536	168	15	1)/(xy	1)/(xy	NN
38536	168	16	)	)	-RRB-
38536	168	17	,	,	,
38536	168	18	etc	etc	FW
38536	168	19	.	.	FW
38536	168	20	,	,	,
38536	168	21	first	first	RB
38536	168	22	.	.	.
38536	169	1	x^2y^2	x^2y^2	ADD
38536	169	2	+	+	NFP
38536	169	3	xy	xy	NNP
38536	169	4	=	=	SYM
38536	169	5	6	6	CD
38536	169	6	,	,	,
38536	169	7	x	x	NNS
38536	169	8	+	+	SYM
38536	169	9	2y	2y	CD
38536	169	10	=	=	SYM
38536	169	11	-5	-5	ADD
38536	169	12	.	.	.
38536	170	1	II	ii	CD
38536	170	2	.	.	.
38536	171	1	Divide	divide	VB
38536	171	2	the	the	DT
38536	171	3	equations	equation	NNS
38536	171	4	,	,	,
38536	171	5	member	member	NN
38536	171	6	by	by	IN
38536	171	7	member	member	NN
38536	171	8	.	.	.
38536	172	1	x^4	x^4	NNP
38536	172	2	-	-	HYPH
38536	172	3	y^4	y^4	NNP
38536	172	4	=	=	SYM
38536	172	5	20	20	CD
38536	172	6	,	,	,
38536	172	7	x^2	x^2	NNPS
38536	172	8	-	-	HYPH
38536	172	9	y^2	y^2	NNP
38536	172	10	=	=	SYM
38536	172	11	5	5	CD
38536	172	12	.	.	.
38536	173	1	III	iii	CD
38536	173	2	.	.	.
38536	174	1	Eliminate	eliminate	VB
38536	174	2	the	the	DT
38536	174	3	quadratic	quadratic	JJ
38536	174	4	terms	term	NNS
38536	174	5	.	.	.
38536	175	1	4x	4x	NNS
38536	175	2	+	+	SYM
38536	175	3	3y	3y	CD
38536	175	4	=	=	SYM
38536	175	5	2xy	2xy	JJ
38536	175	6	,	,	,
38536	175	7	7x	7x	NNP
38536	175	8	-	-	HYPH
38536	175	9	5y	5y	NNP
38536	175	10	=	=	SYM
38536	175	11	5xy	5xy	NN
38536	175	12	.	.	.
38536	176	1	~Ratio	~Ratio	:
38536	176	2	and	and	CC
38536	176	3	Proportion~	proportion~	NN
38536	176	4	Proportionals	proportional	NNS
38536	176	5	mean	mean	VBP
38536	176	6	,	,	,
38536	176	7	third	third	JJ
38536	176	8	,	,	,
38536	176	9	fourth	fourth	JJ
38536	176	10	.	.	.
38536	177	1	Theorems	theorem	NNS
38536	177	2	1	1	CD
38536	177	3	.	.	.
38536	178	1	Product	product	NN
38536	178	2	of	of	IN
38536	178	3	means	mean	NNS
38536	178	4	equals	equal	VBZ
38536	178	5	product	product	NN
38536	178	6	of	of	IN
38536	178	7	extremes	extreme	NNS
38536	178	8	.	.	.
38536	179	1	2	2	LS
38536	179	2	.	.	.
38536	180	1	If	if	IN
38536	180	2	the	the	DT
38536	180	3	product	product	NN
38536	180	4	of	of	IN
38536	180	5	two	two	CD
38536	180	6	numbers	number	NNS
38536	180	7	equals	equal	VBZ
38536	180	8	the	the	DT
38536	180	9	product	product	NN
38536	180	10	of	of	IN
38536	180	11	two	two	CD
38536	180	12	other	other	JJ
38536	180	13	numbers	number	NNS
38536	180	14	,	,	,
38536	180	15	either	either	DT
38536	180	16	pair	pair	NN
38536	180	17	,	,	,
38536	180	18	etc	etc	FW
38536	180	19	.	.	.
38536	181	1	3	3	LS
38536	181	2	.	.	.
38536	182	1	Alternation	alternation	NN
38536	182	2	.	.	.
38536	183	1	4	4	LS
38536	183	2	.	.	.
38536	184	1	Inversion	inversion	NN
38536	184	2	.	.	.
38536	185	1	5	5	CD
38536	185	2	.	.	.
38536	186	1	Composition	composition	NN
38536	186	2	.	.	.
38536	187	1	6	6	CD
38536	187	2	.	.	.
38536	188	1	Division	division	NN
38536	188	2	.	.	.
38536	189	1	7	7	LS
38536	189	2	.	.	.
38536	190	1	Composition	composition	NN
38536	190	2	and	and	CC
38536	190	3	division	division	NN
38536	190	4	.	.	.
38536	191	1	8	8	LS
38536	191	2	.	.	.
38536	192	1	In	in	IN
38536	192	2	a	a	DT
38536	192	3	series	series	NN
38536	192	4	of	of	IN
38536	192	5	equal	equal	JJ
38536	192	6	ratios	ratio	NNS
38536	192	7	,	,	,
38536	192	8	the	the	DT
38536	192	9	sum	sum	NN
38536	192	10	of	of	IN
38536	192	11	the	the	DT
38536	192	12	antecedents	antecedent	NNS
38536	192	13	is	be	VBZ
38536	192	14	to	to	IN
38536	192	15	the	the	DT
38536	192	16	sum	sum	NN
38536	192	17	of	of	IN
38536	192	18	the	the	DT
38536	192	19	consequents	consequent	NNS
38536	192	20	as	as	IN
38536	192	21	any	any	DT
38536	192	22	antecedent	antecedent	NN
38536	192	23	,	,	,
38536	192	24	etc	etc	FW
38536	192	25	.	.	.
38536	193	1	Special	special	JJ
38536	193	2	method	method	NN
38536	193	3	of	of	IN
38536	193	4	proving	prove	VBG
38536	193	5	four	four	CD
38536	193	6	quantities	quantity	NNS
38536	193	7	in	in	IN
38536	193	8	proportion	proportion	NN
38536	193	9	.	.	.
38536	194	1	Let	let	VB
38536	194	2	a	a	NN
38536	194	3	/	/	SYM
38536	194	4	b	b	NN
38536	194	5	=	=	SYM
38536	194	6	x	x	NNS
38536	194	7	,	,	,
38536	194	8	a	a	DT
38536	194	9	=	=	XX
38536	194	10	bx	bx	NN
38536	194	11	,	,	,
38536	194	12	etc	etc	FW
38536	194	13	.	.	.
38536	195	1	~Progressions~	~Progressions~	NFP
38536	195	2	Development	Development	NNP
38536	195	3	of	of	IN
38536	195	4	formulas	formula	NNS
38536	195	5	.	.	.
38536	196	1	{	{	-LRB-
38536	196	2	l	l	NN
38536	196	3	=	=	SYM
38536	196	4	ar^(n	ar^(n	NNP
38536	196	5	-	-	HYPH
38536	196	6	1	1	CD
38536	196	7	)	)	-RRB-
38536	196	8	.	.	.
38536	197	1	{	{	-LRB-
38536	197	2	l	l	NN
38536	197	3	=	=	SYM
38536	197	4	a	a	NN
38536	197	5	+	+	SYM
38536	197	6	(	(	-LRB-
38536	197	7	n	n	NN
38536	197	8	-	-	HYPH
38536	197	9	1)d	1)d	CD
38536	197	10	.	.	.
38536	197	11	{	{	-LRB-
38536	197	12	S	s	NN
38536	197	13	=	=	NFP
38536	197	14	(	(	-LRB-
38536	197	15	ar^n	ar^n	NFP
38536	197	16	-	-	HYPH
38536	197	17	a)/(r	a)/(r	NNP
38536	197	18	-	-	HYPH
38536	197	19	1	1	CD
38536	197	20	)	)	-RRB-
38536	197	21	.	.	.
38536	198	1	{	{	-LRB-
38536	198	2	S	S	NNP
38536	198	3	=	=	NFP
38536	198	4	(	(	-LRB-
38536	198	5	n/2)(a	n/2)(a	NNP
38536	198	6	+	+	NNP
38536	198	7	l	l	NN
38536	198	8	)	)	-RRB-
38536	198	9	.	.	.
38536	199	1	{	{	-LRB-
38536	199	2	S	S	NNP
38536	199	3	=	=	NFP
38536	199	4	(	(	-LRB-
38536	199	5	rl	rl	JJ
38536	199	6	-	-	JJ
38536	199	7	a)/(r	a)/(r	NNP
38536	199	8	-	-	HYPH
38536	199	9	1	1	CD
38536	199	10	)	)	-RRB-
38536	199	11	.	.	.
38536	200	1	{	{	-LRB-
38536	200	2	S	S	NNP
38536	200	3	=	=	NFP
38536	200	4	(	(	-LRB-
38536	200	5	n/2)[2a	n/2)[2a	NNP
38536	200	6	+	+	CC
38536	200	7	(	(	-LRB-
38536	200	8	n	n	CD
38536	200	9	-	-	HYPH
38536	200	10	1)d	1)d	CD
38536	200	11	]	]	-RRB-
38536	200	12	.	.	.
38536	201	1	{	{	-LRB-
38536	201	2	S[infinity	s[infinity	NN
38536	201	3	]	]	-RRB-
38536	201	4	=	=	NN
38536	201	5	(	(	-LRB-
38536	201	6	a)/(1	a)/(1	NNP
38536	201	7	-	-	HYPH
38536	201	8	r	r	NNP
38536	201	9	)	)	-RRB-
38536	201	10	.	.	.
38536	202	1	Insertion	insertion	NN
38536	202	2	of	of	IN
38536	202	3	means	mean	VBZ
38536	202	4	Arithmetical	Arithmetical	NNP
38536	202	5	.	.	.
38536	203	1	Geometrical	Geometrical	NNP
38536	203	2	.	.	.
38536	204	1	~Binomial	~Binomial	NFP
38536	204	2	Theorem~	theorem~	NN
38536	204	3	Review	Review	NNP
38536	204	4	of	of	IN
38536	204	5	binomial	binomial	JJ
38536	204	6	theorem	theorem	NN
38536	204	7	laws	law	NNS
38536	204	8	.	.	.
38536	205	1	See	see	VB
38536	205	2	Involution	involution	NN
38536	205	3	.	.	.
38536	206	1	Expansion	expansion	NN
38536	206	2	of	of	IN
38536	206	3	(	(	-LRB-
38536	206	4	a	a	DT
38536	206	5	+	+	SYM
38536	206	6	b)^n	b)^n	NNP
38536	206	7	.	.	.
38536	207	1	Finding	find	VBG
38536	207	2	any	any	DT
38536	207	3	term	term	NN
38536	207	4	by	by	IN
38536	207	5	key	key	JJ
38536	207	6	number	number	NN
38536	207	7	method	method	NN
38536	207	8	.	.	.
38536	208	1	r^(th	r^(th	NNP
38536	208	2	)	)	-RRB-
38536	208	3	or	or	CC
38536	208	4	(	(	-LRB-
38536	208	5	r	r	NN
38536	208	6	+	+	CD
38536	208	7	1)^(th	1)^(th	CD
38536	208	8	)	)	-RRB-
38536	208	9	term	term	NN
38536	208	10	method	method	NN
38536	208	11	.	.	.
38536	209	1	A	a	DT
38536	209	2	REVIEW	REVIEW	NNP
38536	209	3	OF	of	IN
38536	209	4	ALGEBRA	ALGEBRA	NNP
38536	209	5	ORDER	order	NN
38536	209	6	OF	of	IN
38536	209	7	OPERATIONS	OPERATIONS	NNP
38536	209	8	,	,	,
38536	209	9	EVALUATION	EVALUATION	NNP
38536	209	10	,	,	,
38536	209	11	PARENTHESES	PARENTHESES	NNP
38536	209	12	Order	order	NN
38536	209	13	of	of	IN
38536	209	14	operations	operation	NNS
38536	209	15	:	:	:
38536	209	16	First	first	RB
38536	209	17	of	of	IN
38536	209	18	all	all	DT
38536	209	19	,	,	,
38536	209	20	raising	raise	VBG
38536	209	21	to	to	IN
38536	209	22	a	a	DT
38536	209	23	power	power	NN
38536	209	24	and	and	CC
38536	209	25	extracting	extract	VBG
38536	209	26	a	a	DT
38536	209	27	root	root	NN
38536	209	28	.	.	.
38536	210	1	Next	next	RB
38536	210	2	,	,	,
38536	210	3	multiplication	multiplication	NN
38536	210	4	and	and	CC
38536	210	5	division	division	NN
38536	210	6	.	.	.
38536	211	1	Last	last	JJ
38536	211	2	of	of	IN
38536	211	3	all	all	DT
38536	211	4	,	,	,
38536	211	5	addition	addition	NN
38536	211	6	and	and	CC
38536	211	7	subtraction	subtraction	NN
38536	211	8	.	.	.
38536	212	1	Find	find	VB
38536	212	2	the	the	DT
38536	212	3	value	value	NN
38536	212	4	of	of	IN
38536	212	5	:	:	:
38536	212	6	1	1	CD
38536	212	7	.	.	.
38536	213	1	5	5	CD
38536	213	2	·	·	SYM
38536	213	3	2	2	CD
38536	213	4	^	^	SYM
38536	213	5	2	2	CD
38536	213	6	-	-	SYM
38536	213	7	25^(1/2	25^(1/2	CD
38536	213	8	)	)	-RRB-
38536	213	9	÷	÷	NN
38536	213	10	5	5	CD
38536	213	11	+	+	SYM
38536	213	12	2	2	CD
38536	213	13	^	^	NN
38536	213	14	2	2	CD
38536	213	15	·	·	SYM
38536	213	16	8	8	CD
38536	213	17	÷	÷	NN
38536	213	18	4	4	CD
38536	213	19	-	-	SYM
38536	213	20	2	2	CD
38536	213	21	.	.	.
38536	214	1	2	2	LS
38536	214	2	.	.	.
38536	215	1	(	(	-LRB-
38536	215	2	3	3	CD
38536	215	3	×	×	CD
38536	215	4	6	6	CD
38536	215	5	÷	÷	NN
38536	215	6	9)/2	9)/2	CD
38536	215	7	-	-	HYPH
38536	215	8	2[100^(1/2	2[100^(1/2	CD
38536	215	9	)	)	-RRB-
38536	215	10	]	]	-RRB-
38536	215	11	÷	÷	NNP
38536	215	12	5	5	CD
38536	215	13	+	+	SYM
38536	215	14	4	4	CD
38536	215	15	·	·	SYM
38536	215	16	2	2	CD
38536	215	17	^	^	SYM
38536	215	18	3	3	CD
38536	215	19	-	-	HYPH
38536	215	20	(	(	-LRB-
38536	215	21	14	14	CD
38536	215	22	·	·	SYM
38536	215	23	2)/28	2)/28	CD
38536	215	24	.	.	.
38536	216	1	3	3	LS
38536	216	2	.	.	.
38536	217	1	9	9	CD
38536	217	2	·	·	SYM
38536	217	3	2	2	CD
38536	217	4	÷	÷	NN
38536	217	5	6	6	CD
38536	217	6	+	+	SYM
38536	217	7	3	3	CD
38536	217	8	-	-	SYM
38536	217	9	2	2	CD
38536	217	10	·	·	SYM
38536	217	11	4	4	CD
38536	217	12	^	^	SYM
38536	217	13	2	2	CD
38536	217	14	÷	÷	NN
38536	217	15	8^(1/3	8^(1/3	CD
38536	217	16	)	)	-RRB-
38536	217	17	-	-	:
38536	217	18	4	4	CD
38536	217	19	+	+	SYM
38536	217	20	(	(	-LRB-
38536	217	21	3	3	CD
38536	217	22	·	·	SYM
38536	217	23	2	2	CD
38536	217	24	^	^	SYM
38536	217	25	2)/6	2)/6	CD
38536	217	26	.	.	.
38536	218	1	Evaluate	evaluate	NN
38536	218	2	:	:	:
38536	218	3	4	4	CD
38536	218	4	.	.	.
38536	219	1	(	(	-LRB-
38536	219	2	a^4	a^4	NNP
38536	219	3	-	-	HYPH
38536	219	4	a^3	a^3	NNP
38536	219	5	+	+	SYM
38536	219	6	b^3)/([a^2	b^3)/([a^2	NNP
38536	219	7	b^2]^(1/2	b^2]^(1/2	CD
38536	219	8	)	)	-RRB-
38536	219	9	)	)	-RRB-
38536	219	10	+	+	NFP
38536	219	11	(	(	-LRB-
38536	219	12	c[a^(1/2	c[a^(1/2	NNP
38536	219	13	)	)	-RRB-
38536	219	14	]	]	-RRB-
38536	219	15	+	+	$
38536	219	16	a^3bc)/(abc	a^3bc)/(abc	UH
38536	219	17	)	)	-RRB-
38536	219	18	,	,	,
38536	219	19	if	if	IN
38536	219	20	a	a	DT
38536	219	21	=	=	NN
38536	219	22	1	1	CD
38536	219	23	,	,	,
38536	219	24	b	b	NN
38536	219	25	=	=	SYM
38536	219	26	2	2	CD
38536	219	27	,	,	,
38536	219	28	c	c	NN
38536	219	29	=	=	SYM
38536	219	30	3	3	CD
38536	219	31	.	.	.
38536	220	1	5	5	LS
38536	220	2	.	.	.
38536	220	3	t^(1/3	t^(1/3	NN
38536	220	4	)	)	-RRB-
38536	220	5	+	+	CC
38536	220	6	[	[	-LRB-
38536	220	7	tm]^(1/3	tm]^(1/3	NNP
38536	220	8	)	)	-RRB-
38536	220	9	+	+	CC
38536	220	10	m^(1/3	m^(1/3	NNP
38536	220	11	)	)	-RRB-
38536	220	12	,	,	,
38536	220	13	if	if	IN
38536	220	14	t	t	NNP
38536	220	15	=	=	SYM
38536	220	16	8	8	CD
38536	220	17	,	,	,
38536	220	18	m	m	NN
38536	220	19	=	=	SYM
38536	220	20	27	27	CD
38536	220	21	.	.	.
38536	221	1	6	6	CD
38536	221	2	.	.	.
38536	222	1	(	(	-LRB-
38536	222	2	2[3	2[3	CD
38536	222	3	+	+	SYM
38536	222	4	2d	2d	CD
38536	222	5	+	+	SYM
38536	222	6	a]^(1/2))/(3[a	a]^(1/2))/(3[a	NN
38536	222	7	+	+	SYM
38536	222	8	b	b	NN
38536	222	9	-	-	HYPH
38536	222	10	cx	cx	NNP
38536	222	11	-	-	HYPH
38536	222	12	c]^(1/2	c]^(1/2	NN
38536	222	13	)	)	-RRB-
38536	222	14	)	)	-RRB-
38536	222	15	+	+	NFP
38536	222	16	(	(	-LRB-
38536	222	17	(	(	-LRB-
38536	222	18	3c	3c	IN
38536	222	19	-	-	HYPH
38536	222	20	d)x)/(7ad	d)x)/(7ad	NN
38536	222	21	-	-	HYPH
38536	222	22	[	[	-LRB-
38536	222	23	abc]^(1/2	abc]^(1/2	NN
38536	222	24	)	)	-RRB-
38536	222	25	)	)	-RRB-
38536	222	26	,	,	,
38536	222	27	if	if	IN
38536	222	28	a	a	DT
38536	222	29	=	=	NN
38536	222	30	5	5	CD
38536	222	31	,	,	,
38536	222	32	b	b	NN
38536	222	33	=	=	SYM
38536	222	34	3	3	CD
38536	222	35	,	,	,
38536	222	36	c	c	NNP
38536	222	37	=	=	SYM
38536	222	38	-1	-1	NNP
38536	222	39	,	,	,
38536	222	40	d	d	NNP
38536	222	41	=	=	SYM
38536	222	42	-2	-2	NNP
38536	222	43	,	,	,
38536	222	44	x	x	NNS
38536	222	45	=	=	SYM
38536	222	46	0	0	CD
38536	222	47	.	.	.
38536	223	1	7	7	LS
38536	223	2	.	.	.
38536	223	3	a	a	DT
38536	223	4	-	-	HYPH
38536	223	5	{	{	-LRB-
38536	223	6	5b	5b	NN
38536	223	7	-	-	:
38536	223	8	[	[	-LRB-
38536	223	9	a	a	FW
38536	223	10	-	-	HYPH
38536	223	11	(	(	-LRB-
38536	223	12	3c	3c	NN
38536	223	13	-	-	SYM
38536	223	14	3b	3b	JJ
38536	223	15	)	)	-RRB-
38536	223	16	+	+	CC
38536	223	17	2c	2c	CD
38536	223	18	-	-	HYPH
38536	223	19	3(a	3(a	CD
38536	223	20	-	-	HYPH
38536	223	21	2b	2b	CD
38536	223	22	-	-	HYPH
38536	223	23	c	c	NNP
38536	223	24	)	)	-RRB-
38536	223	25	]	]	-RRB-
38536	223	26	}	}	-RRB-
38536	223	27	,	,	,
38536	223	28	if	if	IN
38536	223	29	a	a	DT
38536	223	30	=	=	SYM
38536	223	31	-3	-3	NN
38536	223	32	,	,	,
38536	223	33	b	b	NN
38536	223	34	=	=	SYM
38536	223	35	4	4	CD
38536	223	36	,	,	,
38536	223	37	c	c	NN
38536	223	38	=	=	SYM
38536	223	39	-5	-5	ADD
38536	223	40	.	.	.
38536	224	1	(	(	-LRB-
38536	224	2	_	_	NNP
38536	224	3	Yale	Yale	NNP
38536	224	4	.	.	.
38536	224	5	_	_	NNP
38536	224	6	)	)	-RRB-
38536	224	7	Simplify	Simplify	NNP
38536	224	8	:	:	:
38536	224	9	8	8	CD
38536	224	10	.	.	.
38536	224	11	m	m	LS
38536	224	12	-	-	HYPH
38536	224	13	[	[	-LRB-
38536	224	14	2	2	CD
38536	224	15	m	m	CD
38536	224	16	-	-	HYPH
38536	224	17	{	{	-LRB-
38536	224	18	3r	3r	NNPS
38536	224	19	-	-	HYPH
38536	224	20	(	(	-LRB-
38536	224	21	4r	4r	NN
38536	224	22	-	-	HYPH
38536	224	23	2	2	CD
38536	224	24	m	m	NN
38536	224	25	)	)	-RRB-
38536	224	26	}	}	-RRB-
38536	224	27	]	]	-RRB-
38536	224	28	.	.	.
38536	225	1	9	9	CD
38536	225	2	.	.	.
38536	226	1	2a	2a	LS
38536	226	2	-	-	:
38536	226	3	[	[	-LRB-
38536	226	4	5d	5d	CD
38536	226	5	+	+	SYM
38536	226	6	{	{	-LRB-
38536	226	7	3c	3c	CD
38536	226	8	-	-	:
38536	226	9	(	(	-LRB-
38536	226	10	a	a	NN
38536	226	11	+	+	SYM
38536	226	12	[	[	-LRB-
38536	226	13	2d	2d	CD
38536	226	14	-	-	HYPH
38536	226	15	3a	3a	NNP
38536	226	16	+	+	NNP
38536	226	17	4c	4c	NNP
38536	226	18	]	]	-RRB-
38536	226	19	)	)	-RRB-
38536	226	20	}	}	-RRB-
38536	226	21	]	]	-RRB-
38536	226	22	.	.	.
38536	227	1	10	10	CD
38536	227	2	.	.	.
38536	228	1	3c^2	3c^2	CD
38536	228	2	+	+	CD
38536	228	3	c(2a	c(2a	NN
38536	228	4	-	-	HYPH
38536	228	5	[	[	-LRB-
38536	228	6	6c	6c	NN
38536	228	7	-	-	HYPH
38536	228	8	{	{	-LRB-
38536	228	9	3a	3a	NN
38536	228	10	+	+	NNP
38536	228	11	c	c	NNP
38536	228	12	-	-	HYPH
38536	228	13	4a	4a	NNP
38536	228	14	}	}	-RRB-
38536	228	15	]	]	-RRB-
38536	228	16	)	)	-RRB-
38536	228	17	.	.	.
38536	229	1	SPECIAL	special	NN
38536	229	2	RULES	rule	NNS
38536	229	3	OF	of	IN
38536	229	4	MULTIPLICATION	multiplication	NN
38536	229	5	AND	and	CC
38536	229	6	DIVISION	division	NN
38536	229	7	Give	give	VBP
38536	229	8	results	result	NNS
38536	229	9	by	by	IN
38536	229	10	inspection	inspection	NN
38536	229	11	:	:	:
38536	229	12	1	1	LS
38536	229	13	.	.	.
38536	230	1	(	(	-LRB-
38536	230	2	g	g	NN
38536	230	3	+	+	CC
38536	230	4	1/2	1/2	CD
38536	230	5	k)^2	k)^2	NN
38536	230	6	.	.	.
38536	231	1	2	2	LS
38536	231	2	.	.	.
38536	232	1	(	(	-LRB-
38536	232	2	s	s	NNP
38536	232	3	-	-	HYPH
38536	232	4	(	(	-LRB-
38536	232	5	2m)/3)^2	2m)/3)^2	CD
38536	232	6	.	.	.
38536	233	1	3	3	LS
38536	233	2	.	.	.
38536	234	1	(	(	-LRB-
38536	234	2	2v	2v	CD
38536	234	3	+	+	SYM
38536	234	4	3w)(2v	3w)(2v	NNP
38536	234	5	-	-	HYPH
38536	234	6	3w	3w	NNP
38536	234	7	)	)	-RRB-
38536	234	8	.	.	.
38536	235	1	4	4	LS
38536	235	2	.	.	.
38536	236	1	(	(	-LRB-
38536	236	2	x	x	NN
38536	236	3	+	+	SYM
38536	236	4	3ts)(x	3ts)(x	NNP
38536	236	5	-	-	HYPH
38536	236	6	7ts	7ts	NNP
38536	236	7	)	)	-RRB-
38536	236	8	.	.	.
38536	237	1	5	5	CD
38536	237	2	.	.	.
38536	238	1	(	(	-LRB-
38536	238	2	2l	2l	CD
38536	238	3	+	+	SYM
38536	238	4	3g)(4l	3g)(4l	NNP
38536	238	5	-	-	HYPH
38536	238	6	11	11	CD
38536	238	7	g	g	NN
38536	238	8	)	)	-RRB-
38536	238	9	.	.	.
38536	239	1	6	6	CD
38536	239	2	.	.	.
38536	240	1	(	(	-LRB-
38536	240	2	a	a	LS
38536	240	3	-	-	:
38536	240	4	(	(	-LRB-
38536	240	5	2b)/3	2b)/3	CD
38536	240	6	+	+	CC
38536	240	7	c	c	NN
38536	240	8	-	-	HYPH
38536	240	9	d)^2	d)^2	NN
38536	240	10	.	.	.
38536	241	1	7	7	LS
38536	241	2	.	.	.
38536	242	1	(	(	-LRB-
38536	242	2	x^3	x^3	NNP
38536	242	3	+	+	CC
38536	242	4	8m^3)/(x	8m^3)/(x	CD
38536	242	5	+	+	SYM
38536	242	6	2	2	CD
38536	242	7	m	m	NN
38536	242	8	)	)	-RRB-
38536	242	9	.	.	.
38536	243	1	8	8	LS
38536	243	2	.	.	.
38536	244	1	(	(	-LRB-
38536	244	2	y^3	y^3	NNP
38536	244	3	-	-	HYPH
38536	244	4	27k^(3m))/(y	27k^(3m))/(y	CD
38536	244	5	-	-	HYPH
38536	244	6	3k^m	3k^m	NNP
38536	244	7	)	)	-RRB-
38536	244	8	.	.	.
38536	245	1	9	9	CD
38536	245	2	.	.	.
38536	246	1	(	(	-LRB-
38536	246	2	c^5	c^5	NNP
38536	246	3	-	-	HYPH
38536	246	4	d^5)/(c	d^5)/(c	NNP
38536	246	5	-	-	HYPH
38536	246	6	d	d	NN
38536	246	7	)	)	-RRB-
38536	246	8	.	.	.
38536	247	1	10	10	CD
38536	247	2	.	.	.
38536	248	1	(	(	-LRB-
38536	248	2	e^5	e^5	NNS
38536	248	3	+	+	SYM
38536	248	4	d^5)/(e	d^5)/(e	NN
38536	248	5	+	+	SYM
38536	248	6	d	d	NN
38536	248	7	)	)	-RRB-
38536	248	8	.	.	.
38536	249	1	11	11	CD
38536	249	2	.	.	.
38536	250	1	(	(	-LRB-
38536	250	2	x^4	x^4	NNP
38536	250	3	-	-	HYPH
38536	250	4	y^4)/(x	y^4)/(x	NNP
38536	250	5	-	-	HYPH
38536	250	6	y	y	NNP
38536	250	7	)	)	-RRB-
38536	250	8	.	.	.
38536	251	1	12	12	CD
38536	251	2	.	.	.
38536	252	1	(	(	-LRB-
38536	252	2	x^4	x^4	NNP
38536	252	3	-	-	HYPH
38536	252	4	y^4)/(x	y^4)/(x	NNP
38536	252	5	+	+	SYM
38536	252	6	y	y	NN
38536	252	7	)	)	-RRB-
38536	252	8	.	.	.
38536	253	1	13	13	CD
38536	253	2	.	.	.
38536	254	1	(	(	-LRB-
38536	254	2	a	a	DT
38536	254	3	-	-	HYPH
38536	254	4	.03)(a	.03)(a	:
38536	254	5	-	-	HYPH
38536	254	6	.0007	.0007	NN
38536	254	7	)	)	-RRB-
38536	254	8	.	.	.
38536	255	1	14	14	CD
38536	255	2	.	.	.
38536	256	1	(	(	-LRB-
38536	256	2	g^n	g^n	NNP
38536	256	3	-	-	HYPH
38536	256	4	1/2)(g^n	1/2)(g^n	CD
38536	256	5	+	+	CD
38536	256	6	3/4	3/4	CD
38536	256	7	)	)	-RRB-
38536	256	8	.	.	.
38536	257	1	15	15	CD
38536	257	2	.	.	.
38536	258	1	(	(	-LRB-
38536	258	2	t^7	t^7	CD
38536	258	3	-	-	HYPH
38536	258	4	v^(7/2))/(t	v^(7/2))/(t	NN
38536	258	5	-	-	HYPH
38536	258	6	v^(1/2	v^(1/2	NN
38536	258	7	)	)	-RRB-
38536	258	8	)	)	-RRB-
38536	258	9	.	.	.
38536	259	1	16	16	CD
38536	259	2	.	.	.
38536	260	1	(	(	-LRB-
38536	260	2	k^32	k^32	NNS
38536	260	3	+	+	SYM
38536	260	4	1)(k^16	1)(k^16	CD
38536	260	5	+1)(k^8	+1)(k^8	CD
38536	260	6	+	+	SYM
38536	260	7	1)(k^4	1)(k^4	CD
38536	260	8	+1)(k^2	+1)(k^2	CD
38536	260	9	+1)(k	+1)(k	NNS
38536	260	10	+	+	CC
38536	260	11	1)(k	1)(k	CD
38536	260	12	-	-	SYM
38536	260	13	1	1	CD
38536	260	14	)	)	-RRB-
38536	260	15	.	.	.
38536	261	1	17	17	CD
38536	261	2	.	.	.
38536	262	1	[	[	-LRB-
38536	262	2	(	(	-LRB-
38536	262	3	a	a	DT
38536	262	4	+	+	SYM
38536	262	5	b	b	NN
38536	262	6	)	)	-RRB-
38536	262	7	+	+	NFP
38536	262	8	(	(	-LRB-
38536	262	9	c	c	NN
38536	262	10	+	+	SYM
38536	262	11	d)][(a	d)][(a	NN
38536	262	12	+	+	SYM
38536	262	13	b	b	NN
38536	262	14	)	)	-RRB-
38536	262	15	-	-	:
38536	262	16	(	(	-LRB-
38536	262	17	c	c	NN
38536	262	18	+	+	SYM
38536	262	19	d	d	NN
38536	262	20	)	)	-RRB-
38536	262	21	]	]	-RRB-
38536	262	22	.	.	.
38536	263	1	18	18	CD
38536	263	2	.	.	.
38536	264	1	(	(	-LRB-
38536	264	2	p	p	NNP
38536	264	3	-	-	HYPH
38536	264	4	q	q	NNP
38536	264	5	+	+	CC
38536	264	6	r	r	NNP
38536	264	7	-	-	HYPH
38536	264	8	s)(p	s)(p	NNP
38536	264	9	-	-	HYPH
38536	264	10	q	q	NNP
38536	264	11	-	-	HYPH
38536	264	12	r	r	NNP
38536	264	13	+	+	CC
38536	264	14	s	s	NN
38536	264	15	)	)	-RRB-
38536	264	16	.	.	.
38536	265	1	19	19	CD
38536	265	2	.	.	.
38536	266	1	(	(	-LRB-
38536	266	2	3	3	CD
38536	266	3	m	m	NNP
38536	266	4	-	-	HYPH
38536	266	5	n	n	NNP
38536	266	6	-	-	HYPH
38536	266	7	l	l	NN
38536	266	8	+	+	CC
38536	266	9	2r)(3	2r)(3	CD
38536	266	10	m	m	NN
38536	266	11	+	+	CC
38536	266	12	n	n	NNP
38536	266	13	-	-	HYPH
38536	266	14	l	l	NN
38536	266	15	-	-	HYPH
38536	266	16	2r	2r	NNP
38536	266	17	)	)	-RRB-
38536	266	18	.	.	.
38536	267	1	20	20	CD
38536	267	2	.	.	.
38536	268	1	(	(	-LRB-
38536	268	2	x	x	SYM
38536	268	3	+	+	SYM
38536	268	4	5)(x	5)(x	CD
38536	268	5	-	-	SYM
38536	268	6	2)(x	2)(x	CD
38536	268	7	-	-	HYPH
38536	268	8	5)(x	5)(x	CD
38536	268	9	+	+	SYM
38536	268	10	2	2	CD
38536	268	11	)	)	-RRB-
38536	268	12	.	.	.
38536	269	1	21	21	CD
38536	269	2	.	.	.
38536	270	1	(	(	-LRB-
38536	270	2	a^2	a^2	CD
38536	270	3	+	+	DT
38536	270	4	b^2	b^2	NNS
38536	270	5	-	-	HYPH
38536	270	6	c	c	NN
38536	270	7	-	-	HYPH
38536	270	8	2d	2d	CD
38536	270	9	+	+	CC
38536	270	10	3e)^2	3e)^2	CD
38536	270	11	.	.	.
38536	271	1	22	22	CD
38536	271	2	.	.	.
38536	272	1	(	(	-LRB-
38536	272	2	s	s	NNP
38536	272	3	+	+	CC
38536	272	4	t	t	NNP
38536	272	5	-	-	HYPH
38536	272	6	2v/5	2v/5	CD
38536	272	7	+	+	SYM
38536	272	8	3w/6	3w/6	CD
38536	272	9	+	+	SYM
38536	272	10	z^2)^2	z^2)^2	NN
38536	272	11	.	.	.
38536	273	1	23	23	CD
38536	273	2	.	.	.
38536	274	1	(	(	-LRB-
38536	274	2	x^5	x^5	NNS
38536	274	3	+	+	SYM
38536	274	4	32)/(x	32)/(x	CD
38536	274	5	+	+	SYM
38536	274	6	2	2	CD
38536	274	7	)	)	-RRB-
38536	274	8	.	.	.
38536	275	1	~References:~	~References:~	NNP
38536	275	2	The	the	DT
38536	275	3	chapter	chapter	NN
38536	275	4	on	on	IN
38536	275	5	Special	Special	NNP
38536	275	6	Rules	Rules	NNPS
38536	275	7	of	of	IN
38536	275	8	Multiplication	Multiplication	NNP
38536	275	9	and	and	CC
38536	275	10	Division	Division	NNP
38536	275	11	in	in	IN
38536	275	12	any	any	DT
38536	275	13	algebra	algebra	NN
38536	275	14	.	.	.
38536	276	1	Special	special	JJ
38536	276	2	Rules	Rules	NNPS
38536	276	3	of	of	IN
38536	276	4	Multiplication	Multiplication	NNP
38536	276	5	and	and	CC
38536	276	6	Division	Division	NNP
38536	276	7	in	in	IN
38536	276	8	the	the	DT
38536	276	9	Outline	Outline	NNP
38536	276	10	in	in	IN
38536	276	11	the	the	DT
38536	276	12	front	front	NN
38536	276	13	of	of	IN
38536	276	14	the	the	DT
38536	276	15	book	book	NN
38536	276	16	.	.	.
38536	277	1	CASES	case	NNS
38536	277	2	IN	in	IN
38536	277	3	FACTORING	factor	VBG
38536	277	4	The	the	DT
38536	277	5	number	number	NN
38536	277	6	of	of	IN
38536	277	7	terms	term	NNS
38536	277	8	in	in	IN
38536	277	9	an	an	DT
38536	277	10	expression	expression	NN
38536	277	11	usually	usually	RB
38536	277	12	gives	give	VBZ
38536	277	13	the	the	DT
38536	277	14	clue	clue	NN
38536	277	15	to	to	IN
38536	277	16	the	the	DT
38536	277	17	possible	possible	JJ
38536	277	18	cases	case	NNS
38536	277	19	under	under	IN
38536	277	20	which	which	WDT
38536	277	21	it	-PRON-	PRP
38536	277	22	may	may	MD
38536	277	23	come	come	VB
38536	277	24	.	.	.
38536	278	1	By	by	IN
38536	278	2	applying	apply	VBG
38536	278	3	the	the	DT
38536	278	4	_	_	NNP
38536	278	5	test	test	NN
38536	278	6	_	_	NNP
38536	278	7	for	for	IN
38536	278	8	each	each	DT
38536	278	9	and	and	CC
38536	278	10	eliminating	eliminate	VBG
38536	278	11	the	the	DT
38536	278	12	_	_	NNP
38536	278	13	possible	possible	JJ
38536	278	14	_	_	NNP
38536	278	15	cases	case	NNS
38536	278	16	one	one	CD
38536	278	17	by	by	IN
38536	278	18	one	one	CD
38536	278	19	,	,	,
38536	278	20	the	the	DT
38536	278	21	right	right	JJ
38536	278	22	case	case	NN
38536	278	23	is	be	VBZ
38536	278	24	readily	readily	RB
38536	278	25	found	find	VBN
38536	278	26	.	.	.
38536	279	1	Hence	hence	RB
38536	279	2	,	,	,
38536	279	3	the	the	DT
38536	279	4	number	number	NN
38536	279	5	of	of	IN
38536	279	6	terms	term	NNS
38536	279	7	in	in	IN
38536	279	8	the	the	DT
38536	279	9	expression	expression	NN
38536	279	10	and	and	CC
38536	279	11	a	a	DT
38536	279	12	ready	ready	JJ
38536	279	13	and	and	CC
38536	279	14	accurate	accurate	JJ
38536	279	15	knowledge	knowledge	NN
38536	279	16	of	of	IN
38536	279	17	the	the	DT
38536	279	18	Cases	case	NNS
38536	279	19	in	in	IN
38536	279	20	Factoring	factor	VBG
38536	279	21	are	be	VBP
38536	279	22	the	the	DT
38536	279	23	real	real	JJ
38536	279	24	keys	key	NNS
38536	279	25	to	to	IN
38536	279	26	success	success	NN
38536	279	27	in	in	IN
38536	279	28	this	this	DT
38536	279	29	vitally	vitally	RB
38536	279	30	important	important	JJ
38536	279	31	part	part	NN
38536	279	32	of	of	IN
38536	279	33	algebra	algebra	NN
38536	279	34	.	.	.
38536	280	1	CASE	CASE	NNP
38536	280	2	I.	I.	NNP
38536	281	1	A	a	DT
38536	281	2	common	common	JJ
38536	281	3	monomial	monomial	JJ
38536	281	4	factor	factor	NN
38536	281	5	.	.	.
38536	282	1	Applies	apply	VBZ
38536	282	2	to	to	IN
38536	282	3	any	any	DT
38536	282	4	number	number	NN
38536	282	5	of	of	IN
38536	282	6	terms	term	NNS
38536	282	7	.	.	.
38536	283	1	5cx	5cx	JJ
38536	283	2	-	-	HYPH
38536	283	3	5ct	5ct	JJ
38536	283	4	+	+	CC
38536	283	5	5cv	5cv	JJ
38536	283	6	-	-	HYPH
38536	283	7	15	15	CD
38536	283	8	c^2	c^2	NNS
38536	283	9	m	m	NN
38536	283	10	+	+	SYM
38536	283	11	25	25	CD
38536	283	12	c^3	c^3	CD
38536	283	13	m^2	m^2	CD
38536	283	14	=	=	SYM
38536	283	15	5c(x	5c(x	CD
38536	283	16	-	-	HYPH
38536	283	17	t	t	NN
38536	283	18	+	+	CC
38536	283	19	v	v	NN
38536	283	20	-	-	SYM
38536	283	21	3	3	CD
38536	283	22	cm	cm	NNS
38536	283	23	+	+	SYM
38536	283	24	5c^2	5c^2	CD
38536	283	25	m^2	m^2	CD
38536	283	26	)	)	-RRB-
38536	283	27	.	.	.
38536	284	1	CASE	CASE	NNP
38536	284	2	II	II	NNP
38536	284	3	.	.	.
38536	285	1	A	a	DT
38536	285	2	trinomial	trinomial	NN
38536	285	3	that	that	WDT
38536	285	4	is	be	VBZ
38536	285	5	a	a	DT
38536	285	6	perfect	perfect	JJ
38536	285	7	square	square	NN
38536	285	8	.	.	.
38536	286	1	Three	three	CD
38536	286	2	terms	term	NNS
38536	286	3	.	.	.
38536	287	1	x^2	x^2	NNP
38536	287	2	±	±	NNP
38536	287	3	2xm	2xm	CD
38536	287	4	+	+	SYM
38536	287	5	m^2	m^2	CD
38536	287	6	=	=	SYM
38536	287	7	(	(	-LRB-
38536	287	8	x	x	SYM
38536	287	9	±	±	CD
38536	287	10	m)^2	m)^2	NN
38536	287	11	.	.	.
38536	288	1	CASE	CASE	NNP
38536	288	2	III	III	NNP
38536	288	3	.	.	.
38536	289	1	The	the	DT
38536	289	2	difference	difference	NN
38536	289	3	of	of	IN
38536	289	4	two	two	CD
38536	289	5	squares	square	NNS
38536	289	6	.	.	.
38536	290	1	_	_	NNP
38536	290	2	A.	A.	NNP
38536	290	3	_	_	NNP
38536	290	4	Two	two	CD
38536	290	5	terms	term	NNS
38536	290	6	.	.	.
38536	291	1	x^2	x^2	NNP
38536	291	2	-	-	HYPH
38536	291	3	y^2	y^2	NNP
38536	291	4	=	=	NFP
38536	291	5	(	(	-LRB-
38536	291	6	x	x	SYM
38536	291	7	+	+	SYM
38536	291	8	y)(x	y)(x	NNP
38536	291	9	-	-	HYPH
38536	291	10	y	y	NNP
38536	291	11	)	)	-RRB-
38536	291	12	.	.	.
38536	292	1	_	_	NNP
38536	292	2	B.	B.	NNP
38536	292	3	_	_	NNP
38536	292	4	Four	Four	NNP
38536	292	5	terms	term	NNS
38536	292	6	.	.	.
38536	293	1	x^2	x^2	NNS
38536	293	2	+	+	SYM
38536	293	3	2xy	2xy	NN
38536	293	4	+	+	SYM
38536	293	5	y^2	y^2	PRP
38536	293	6	-	-	HYPH
38536	293	7	m^2	m^2	CD
38536	293	8	=	=	NFP
38536	293	9	(	(	-LRB-
38536	293	10	x^2	x^2	NNS
38536	293	11	+	+	SYM
38536	293	12	2xy	2xy	NN
38536	293	13	+	+	SYM
38536	293	14	y^2	y^2	ADD
38536	293	15	)	)	-RRB-
38536	293	16	-	-	:
38536	293	17	m^2	m^2	CD
38536	293	18	=	=	NFP
38536	293	19	(	(	-LRB-
38536	293	20	x	x	SYM
38536	293	21	+	+	SYM
38536	293	22	y	y	NN
38536	293	23	+	+	CC
38536	293	24	m)(x	m)(x	NNP
38536	293	25	+	+	NNP
38536	293	26	y	y	NNP
38536	293	27	-	-	HYPH
38536	293	28	m	m	NNP
38536	293	29	)	)	-RRB-
38536	293	30	_	_	NNP
38536	293	31	C.	C.	NNP
38536	293	32	_	_	NNP
38536	293	33	Six	six	CD
38536	293	34	terms	term	NNS
38536	293	35	.	.	.
38536	294	1	x^2	x^2	JJ
38536	294	2	-	-	HYPH
38536	294	3	2xy	2xy	JJ
38536	294	4	+	+	SYM
38536	294	5	y^2	y^2	NN
38536	294	6	-	-	HYPH
38536	294	7	m^2	m^2	CD
38536	294	8	-	-	HYPH
38536	294	9	2mn	2mn	CD
38536	294	10	-	-	HYPH
38536	294	11	n^2	n^2	NNS
38536	294	12	=	=	NFP
38536	294	13	(	(	-LRB-
38536	294	14	x^2	x^2	JJ
38536	294	15	-	-	HYPH
38536	294	16	2xy	2xy	NNP
38536	294	17	+	+	SYM
38536	294	18	y^2	y^2	UH
38536	294	19	)	)	-RRB-
38536	294	20	-	-	:
38536	294	21	(	(	-LRB-
38536	294	22	m^2	m^2	CD
38536	294	23	+	+	SYM
38536	294	24	2mn	2mn	NN
38536	294	25	+	+	SYM
38536	294	26	n^2	n^2	NNS
38536	294	27	)	)	-RRB-
38536	294	28	=	=	NFP
38536	294	29	(	(	-LRB-
38536	294	30	x	x	SYM
38536	294	31	-	-	:
38536	294	32	y)^2	y)^2	NNP
38536	294	33	-	-	HYPH
38536	294	34	(	(	-LRB-
38536	294	35	m	m	NN
38536	294	36	+	+	SYM
38536	294	37	n)^2	n)^2	.
38536	294	38	=	=	NFP
38536	294	39	[	[	-LRB-
38536	294	40	(	(	-LRB-
38536	294	41	x	x	SYM
38536	294	42	-	-	NN
38536	294	43	y	y	NN
38536	294	44	)	)	-RRB-
38536	294	45	+	+	CC
38536	294	46	(	(	-LRB-
38536	294	47	m	m	NN
38536	294	48	+	+	CC
38536	294	49	n)][(x	n)][(x	NNP
38536	294	50	-	-	HYPH
38536	294	51	y	y	NN
38536	294	52	)	)	-RRB-
38536	294	53	-	-	:
38536	294	54	(	(	-LRB-
38536	294	55	m	m	NN
38536	294	56	+	+	CC
38536	294	57	n	n	NN
38536	294	58	)	)	-RRB-
38536	294	59	]	]	-RRB-
38536	294	60	.	.	.
38536	295	1	_	_	NNP
38536	295	2	D.	D.	NNP
38536	295	3	_	_	NNP
38536	295	4	An	an	DT
38536	295	5	incomplete	incomplete	JJ
38536	295	6	square	square	NN
38536	295	7	.	.	.
38536	296	1	Three	three	CD
38536	296	2	terms	term	NNS
38536	296	3	,	,	,
38536	296	4	and	and	CC
38536	296	5	4th	4th	JJ
38536	296	6	powers	power	NNS
38536	296	7	or	or	CC
38536	296	8	multiples	multiple	NNS
38536	296	9	of	of	IN
38536	296	10	4	4	CD
38536	296	11	.	.	.
38536	296	12	c^4	c^4	NNS
38536	296	13	+	+	SYM
38536	296	14	c^2	c^2	NNS
38536	296	15	d^2	d^2	NNS
38536	296	16	+	+	CC
38536	296	17	d^4	d^4	NNP
38536	296	18	=	=	SYM
38536	296	19	c^4	c^4	CD
38536	296	20	+	+	SYM
38536	296	21	2c^2	2c^2	CD
38536	296	22	d^2	d^2	NNS
38536	296	23	+	+	CC
38536	296	24	d^4	d^4	CD
38536	296	25	-	-	HYPH
38536	296	26	c^2	c^2	NNP
38536	296	27	d^2	d^2	ADD
38536	296	28	=	=	NFP
38536	296	29	(	(	-LRB-
38536	296	30	c^2	c^2	NNP
38536	296	31	+	+	CC
38536	296	32	d^2)^2	d^2)^2	NNP
38536	296	33	-	-	HYPH
38536	296	34	c^2	c^2	NNP
38536	296	35	d^2	d^2	ADD
38536	296	36	=	=	NFP
38536	296	37	(	(	-LRB-
38536	296	38	c^2	c^2	NNP
38536	296	39	+	+	SYM
38536	296	40	d^2	d^2	NNS
38536	296	41	+	+	SYM
38536	296	42	cd)(c^2	cd)(c^2	NN
38536	296	43	+	+	CC
38536	296	44	d^2	d^2	NNP
38536	296	45	-	-	HYPH
38536	296	46	cd	cd	NN
38536	296	47	)	)	-RRB-
38536	296	48	.	.	.
38536	297	1	CASE	CASE	NNP
38536	297	2	IV	IV	NNP
38536	297	3	.	.	.
38536	298	1	A	a	DT
38536	298	2	trinomial	trinomial	NN
38536	298	3	of	of	IN
38536	298	4	the	the	DT
38536	298	5	form	form	NN
38536	298	6	x^2	x^2	NNP
38536	298	7	+	+	SYM
38536	298	8	bx	bx	NN
38536	298	9	+	+	SYM
38536	298	10	c.	c.	NN
38536	298	11	Three	three	CD
38536	298	12	terms	term	NNS
38536	298	13	.	.	.
38536	299	1	x^2	x^2	NNP
38536	299	2	+	+	SYM
38536	299	3	x	x	SYM
38536	299	4	-	-	SYM
38536	299	5	30	30	CD
38536	299	6	=	=	SYM
38536	299	7	(	(	-LRB-
38536	299	8	x	x	SYM
38536	299	9	+	+	SYM
38536	299	10	6)(x	6)(x	CD
38536	299	11	-	-	HYPH
38536	299	12	5	5	CD
38536	299	13	)	)	-RRB-
38536	299	14	.	.	.
38536	300	1	CASE	CASE	NNP
38536	300	2	V.	V.	NNP
38536	300	3	A	a	DT
38536	300	4	trinomial	trinomial	NN
38536	300	5	of	of	IN
38536	300	6	the	the	DT
38536	300	7	form	form	NN
38536	300	8	ax^2	ax^2	NN
38536	300	9	+	+	SYM
38536	300	10	bx	bx	NN
38536	300	11	+	+	SYM
38536	300	12	c.	c.	NN
38536	300	13	Three	three	CD
38536	300	14	terms	term	NNS
38536	300	15	.	.	.
38536	301	1	20x^2	20x^2	CD
38536	301	2	+	+	SYM
38536	301	3	7x	7x	CD
38536	301	4	-	-	SYM
38536	301	5	6	6	CD
38536	301	6	=	=	SYM
38536	301	7	(	(	-LRB-
38536	301	8	4x	4x	NNS
38536	301	9	+	+	SYM
38536	301	10	3)(5x	3)(5x	CD
38536	301	11	-	-	HYPH
38536	301	12	2	2	CD
38536	301	13	)	)	-RRB-
38536	301	14	.	.	.
38536	302	1	CASE	CASE	NNP
38536	302	2	VI	VI	NNP
38536	302	3	.	.	.
38536	303	1	_	_	NNP
38536	303	2	A.	A.	NNP
38536	303	3	_	_	NNP
38536	303	4	The	the	DT
38536	303	5	sum	sum	NN
38536	303	6	or	or	CC
38536	303	7	difference	difference	NN
38536	303	8	of	of	IN
38536	303	9	two	two	CD
38536	303	10	cubes	cube	NNS
38536	303	11	.	.	.
38536	304	1	Two	two	CD
38536	304	2	terms	term	NNS
38536	304	3	.	.	.
38536	305	1	x^3	x^3	NNS
38536	305	2	+	+	SYM
38536	305	3	y^3	y^3	NNS
38536	305	4	=	=	NFP
38536	305	5	(	(	-LRB-
38536	305	6	x	x	SYM
38536	305	7	+	+	CD
38536	305	8	y)(x^2	y)(x^2	NN
38536	305	9	-	-	HYPH
38536	305	10	xy	xy	NN
38536	305	11	+	+	CC
38536	305	12	y^2	y^2	PRP
38536	305	13	)	)	-RRB-
38536	305	14	;	;	:
38536	305	15	x^3	x^3	NNP
38536	305	16	-	-	HYPH
38536	305	17	y^3	y^3	NNP
38536	305	18	=	=	NFP
38536	305	19	(	(	-LRB-
38536	305	20	x	x	SYM
38536	305	21	-	-	SYM
38536	305	22	y)(x^2	y)(x^2	NN
38536	305	23	+	+	CD
38536	305	24	xy	xy	NN
38536	305	25	+	+	SYM
38536	305	26	y^2	y^2	PRP
38536	305	27	)	)	-RRB-
38536	305	28	.	.	.
38536	306	1	_	_	NNP
38536	306	2	B.	B.	NNP
38536	306	3	_	_	NNP
38536	306	4	The	the	DT
38536	306	5	sum	sum	NN
38536	306	6	or	or	CC
38536	306	7	difference	difference	NN
38536	306	8	of	of	IN
38536	306	9	two	two	CD
38536	306	10	like	like	JJ
38536	306	11	powers	power	NNS
38536	306	12	.	.	.
38536	307	1	Two	two	CD
38536	307	2	terms	term	NNS
38536	307	3	.	.	.
38536	308	1	x^4	x^4	NNP
38536	308	2	-	-	HYPH
38536	308	3	y^4	y^4	NNP
38536	308	4	=	=	NFP
38536	308	5	(	(	-LRB-
38536	308	6	x	x	NN
38536	308	7	-	-	NN
38536	308	8	y)(x^3	y)(x^3	NN
38536	308	9	+	+	SYM
38536	308	10	x^2y	x^2y	NN
38536	308	11	+	+	XX
38536	308	12	xy^2	xy^2	XX
38536	308	13	+	+	NFP
38536	308	14	y^3	y^3	NN
38536	308	15	)	)	-RRB-
38536	308	16	;	;	:
38536	308	17	x^5	x^5	NNP
38536	308	18	+	+	SYM
38536	308	19	y^5	y^5	NNP
38536	308	20	=	=	NFP
38536	308	21	(	(	-LRB-
38536	308	22	x	x	SYM
38536	308	23	+	+	SYM
38536	308	24	y)(x^4	y)(x^4	NN
38536	308	25	-	-	HYPH
38536	308	26	x^3y	x^3y	NN
38536	308	27	+	+	CC
38536	308	28	x^2	x^2	JJ
38536	308	29	y^2	y^2	NNS
38536	308	30	-	-	HYPH
38536	308	31	xy^3	xy^3	NN
38536	308	32	+	+	SYM
38536	308	33	y^4	y^4	UH
38536	308	34	)	)	-RRB-
38536	308	35	.	.	.
38536	309	1	CASE	CASE	NNP
38536	309	2	VII	VII	NNP
38536	309	3	.	.	.
38536	310	1	A	a	DT
38536	310	2	common	common	JJ
38536	310	3	polynomial	polynomial	JJ
38536	310	4	factor	factor	NN
38536	310	5	.	.	.
38536	311	1	Any	any	DT
38536	311	2	_	_	NNP
38536	311	3	composite	composite	JJ
38536	311	4	_	_	NNP
38536	311	5	number	number	NN
38536	311	6	of	of	IN
38536	311	7	terms	term	NNS
38536	311	8	.	.	.
38536	312	1	t^2	t^2	CD
38536	312	2	p	p	NN
38536	312	3	+	+	CC
38536	312	4	t^2	t^2	CD
38536	312	5	q	q	NN
38536	312	6	-	-	HYPH
38536	312	7	t^2	t^2	CD
38536	312	8	r	r	NNP
38536	312	9	-	-	:
38536	312	10	g^2	g^2	NNP
38536	312	11	p	p	NNP
38536	312	12	-	-	HYPH
38536	312	13	g^2	g^2	NNP
38536	312	14	q	q	NN
38536	312	15	+	+	NNS
38536	312	16	g^2	g^2	NNP
38536	312	17	r	r	NN
38536	312	18	=	=	SYM
38536	312	19	t^2	t^2	NN
38536	312	20	(	(	-LRB-
38536	312	21	p	p	NN
38536	312	22	+	+	CC
38536	312	23	q	q	NNP
38536	312	24	-	-	HYPH
38536	312	25	r	r	NNP
38536	312	26	)	)	-RRB-
38536	312	27	-	-	:
38536	312	28	g^2	g^2	FW
38536	312	29	(	(	-LRB-
38536	312	30	p	p	NN
38536	312	31	+	+	CC
38536	312	32	q	q	NN
38536	312	33	-r	-r	NN
38536	312	34	)	)	-RRB-
38536	312	35	=	=	NFP
38536	312	36	(	(	-LRB-
38536	312	37	p	p	NN
38536	312	38	+	+	CC
38536	312	39	q	q	NN
38536	312	40	-	-	HYPH
38536	312	41	r)(t^2	r)(t^2	CD
38536	312	42	-	-	HYPH
38536	312	43	g^2	g^2	NN
38536	312	44	)	)	-RRB-
38536	312	45	=	=	NFP
38536	312	46	(	(	-LRB-
38536	312	47	p	p	NN
38536	312	48	+	+	CC
38536	312	49	q	q	NN
38536	312	50	-	-	HYPH
38536	312	51	r)(t	r)(t	NNP
38536	312	52	+	+	CC
38536	312	53	g)(t	g)(t	NNP
38536	312	54	-	-	HYPH
38536	312	55	g	g	NNP
38536	312	56	)	)	-RRB-
38536	312	57	.	.	.
38536	313	1	CASE	case	NN
38536	313	2	VIII	VIII	NNP
38536	313	3	.	.	.
38536	314	1	The	the	DT
38536	314	2	Factor	Factor	NNP
38536	314	3	Theorem	Theorem	NNP
38536	314	4	.	.	.
38536	315	1	Any	any	DT
38536	315	2	number	number	NN
38536	315	3	of	of	IN
38536	315	4	terms	term	NNS
38536	315	5	.	.	.
38536	316	1	x^3	x^3	NNS
38536	316	2	+	+	SYM
38536	316	3	17x	17x	CD
38536	316	4	-	-	SYM
38536	316	5	18	18	CD
38536	316	6	=	=	SYM
38536	316	7	(	(	-LRB-
38536	316	8	x	x	SYM
38536	316	9	-	-	SYM
38536	316	10	1)(x^2	1)(x^2	CD
38536	316	11	+	+	SYM
38536	316	12	x	x	SYM
38536	316	13	+	+	SYM
38536	316	14	18	18	CD
38536	316	15	)	)	-RRB-
38536	316	16	.	.	.
38536	317	1	FACTORING	FACTORING	NNP
38536	317	2	Review	Review	NNP
38536	317	3	the	the	DT
38536	317	4	_	_	NNP
38536	317	5	Cases	Cases	NNPS
38536	317	6	in	in	IN
38536	317	7	Factoring	Factoring	NNP
38536	317	8	_	_	NNP
38536	317	9	(	(	-LRB-
38536	317	10	see	see	VB
38536	317	11	Outline	Outline	NNP
38536	317	12	on	on	IN
38536	317	13	preceding	precede	VBG
38536	317	14	pages	page	NNS
38536	317	15	)	)	-RRB-
38536	317	16	and	and	CC
38536	317	17	write	write	VB
38536	317	18	out	out	RP
38536	317	19	the	the	DT
38536	317	20	prime	prime	JJ
38536	317	21	factors	factor	NNS
38536	317	22	of	of	IN
38536	317	23	the	the	DT
38536	317	24	following	follow	VBG
38536	317	25	:	:	:
38536	317	26	1	1	CD
38536	317	27	.	.	.
38536	318	1	8a^(13	8a^(13	CD
38536	318	2	)	)	-RRB-
38536	318	3	+	+	SYM
38536	318	4	am^(12	am^(12	NN
38536	318	5	)	)	-RRB-
38536	318	6	.	.	.
38536	319	1	2	2	LS
38536	319	2	.	.	.
38536	319	3	x^7	x^7	NNS
38536	319	4	+	+	SYM
38536	319	5	y^7	y^7	NNS
38536	319	6	.	.	.
38536	320	1	3	3	LS
38536	320	2	.	.	.
38536	321	1	4x^2	4x^2	CD
38536	321	2	+	+	SYM
38536	321	3	11x	11x	CD
38536	321	4	-	-	SYM
38536	321	5	3	3	CD
38536	321	6	.	.	.
38536	322	1	4	4	LS
38536	322	2	.	.	.
38536	322	3	m^2	m^2	CD
38536	322	4	+	+	SYM
38536	322	5	n^2	n^2	NNS
38536	322	6	-	-	,
38536	322	7	(	(	-LRB-
38536	322	8	1	1	CD
38536	322	9	+	+	CD
38536	322	10	2mn	2mn	NN
38536	322	11	)	)	-RRB-
38536	322	12	.	.	.
38536	323	1	5	5	CD
38536	323	2	.	.	.
38536	324	1	-x^2	-x^2	:
38536	324	2	+	+	CC
38536	324	3	2x	2x	CD
38536	324	4	-	-	SYM
38536	324	5	1	1	CD
38536	324	6	+	+	SYM
38536	324	7	x^4	x^4	NNS
38536	324	8	.	.	.
38536	325	1	6	6	LS
38536	325	2	.	.	.
38536	325	3	x^(16	x^(16	NNP
38536	325	4	)	)	-RRB-
38536	325	5	-	-	:
38536	325	6	y^(16	y^(16	NNP
38536	325	7	)	)	-RRB-
38536	325	8	.	.	.
38536	326	1	(	(	-LRB-
38536	326	2	Five	five	CD
38536	326	3	factors	factor	NNS
38536	326	4	.	.	.
38536	326	5	)	)	-RRB-
38536	327	1	7	7	LS
38536	327	2	.	.	.
38536	328	1	(	(	-LRB-
38536	328	2	x	x	SYM
38536	328	3	+	+	NNP
38536	328	4	1)^2	1)^2	NNP
38536	328	5	-	-	HYPH
38536	328	6	5x	5x	CD
38536	328	7	-	-	HYPH
38536	328	8	29	29	CD
38536	328	9	.	.	.
38536	329	1	8	8	LS
38536	329	2	.	.	.
38536	329	3	x^4	x^4	NNP
38536	329	4	+	+	SYM
38536	329	5	x^2	x^2	NNP
38536	329	6	y^2	y^2	NNS
38536	329	7	+	+	SYM
38536	329	8	y^4	y^4	XX
38536	329	9	.	.	.
38536	330	1	9	9	LS
38536	330	2	.	.	.
38536	330	3	x^4	x^4	NNP
38536	330	4	-	-	:
38536	330	5	11x^2	11x^2	CD
38536	330	6	+	+	SYM
38536	330	7	1	1	CD
38536	330	8	.	.	.
38536	331	1	10	10	CD
38536	331	2	.	.	.
38536	331	3	x^(2	x^(2	ADD
38536	331	4	m	m	LS
38536	331	5	)	)	-RRB-
38536	331	6	+	+	CC
38536	331	7	2	2	CD
38536	331	8	+	+	SYM
38536	331	9	1/(x^(2	1/(x^(2	CD
38536	331	10	m	m	NN
38536	331	11	)	)	-RRB-
38536	331	12	)	)	-RRB-
38536	331	13	.	.	.
38536	332	1	11	11	CD
38536	332	2	.	.	.
38536	332	3	x^(6	x^(6	NFP
38536	332	4	m	m	LS
38536	332	5	)	)	-RRB-
38536	332	6	+	+	CC
38536	332	7	13x^(3	13x^(3	CD
38536	332	8	m	m	NN
38536	332	9	)	)	-RRB-
38536	332	10	+	+	CC
38536	332	11	12	12	CD
38536	332	12	.	.	.
38536	333	1	12	12	CD
38536	333	2	.	.	.
38536	334	1	4a^2	4a^2	CD
38536	334	2	b^2	b^2	NNS
38536	334	3	-	-	,
38536	334	4	(	(	-LRB-
38536	334	5	a^2	a^2	NNS
38536	334	6	+	+	DT
38536	334	7	b^2	b^2	NNS
38536	334	8	-	-	HYPH
38536	334	9	c^2)^2	c^2)^2	NN
38536	334	10	.	.	.
38536	335	1	13	13	CD
38536	335	2	.	.	.
38536	336	1	(	(	-LRB-
38536	336	2	x^2	x^2	JJ
38536	336	3	-	-	HYPH
38536	336	4	x	x	NNP
38536	336	5	-	-	HYPH
38536	336	6	6)(x^2	6)(x^2	CD
38536	336	7	-	-	HYPH
38536	336	8	x	x	NN
38536	336	9	-	-	CD
38536	336	10	20	20	CD
38536	336	11	)	)	-RRB-
38536	336	12	.	.	.
38536	337	1	14	14	CD
38536	337	2	.	.	.
38536	337	3	a^4	a^4	CD
38536	337	4	-	-	HYPH
38536	337	5	8a	8a	NNP
38536	337	6	-	-	HYPH
38536	337	7	a^3	a^3	NNP
38536	337	8	+	+	SYM
38536	337	9	8	8	CD
38536	337	10	.	.	.
38536	338	1	15	15	CD
38536	338	2	.	.	.
38536	338	3	p^3	p^3	CD
38536	338	4	+	+	SYM
38536	338	5	7p^2	7p^2	CD
38536	338	6	+	+	SYM
38536	338	7	14p	14p	NNS
38536	338	8	+	+	SYM
38536	338	9	8	8	CD
38536	338	10	.	.	.
38536	339	1	16	16	CD
38536	339	2	.	.	.
38536	340	1	18a^2	18a^2	CD
38536	340	2	b	b	NN
38536	340	3	+	+	CC
38536	340	4	60ab^2	60ab^2	CD
38536	340	5	+	+	CD
38536	340	6	50b^3	50b^3	CD
38536	340	7	.	.	.
38536	341	1	17	17	CD
38536	341	2	.	.	.
38536	341	3	x^3	x^3	CD
38536	341	4	-	-	:
38536	341	5	7x	7x	NNP
38536	341	6	+	+	CC
38536	341	7	6	6	CD
38536	341	8	.	.	.
38536	342	1	18	18	CD
38536	342	2	.	.	.
38536	343	1	24c^2	24c^2	CD
38536	343	2	d^2	d^2	NN
38536	343	3	-	-	HYPH
38536	343	4	47cd	47cd	NN
38536	343	5	-	-	HYPH
38536	343	6	75	75	CD
38536	343	7	.	.	.
38536	344	1	19	19	CD
38536	344	2	.	.	.
38536	345	1	(	(	-LRB-
38536	345	2	a^2	a^2	RB
38536	345	3	-	-	HYPH
38536	345	4	b^2)^2	b^2)^2	NN
38536	345	5	-	-	,
38536	345	6	(	(	-LRB-
38536	345	7	a^2	a^2	NNP
38536	345	8	-	-	HYPH
38536	345	9	ab)^2	ab)^2	JJ
38536	345	10	.	.	.
38536	346	1	20	20	CD
38536	346	2	.	.	.
38536	346	3	a^2	a^2	PRP$
38536	346	4	x^3	x^3	NNS
38536	346	5	-	-	,
38536	346	6	(	(	-LRB-
38536	346	7	8a^2)/(y^3	8a^2)/(y^3	NNP
38536	346	8	)	)	-RRB-
38536	346	9	-	-	:
38536	346	10	x^3	x^3	NNS
38536	346	11	+	+	SYM
38536	346	12	8/(y^3	8/(y^3	CD
38536	346	13	)	)	-RRB-
38536	346	14	.	.	.
38536	347	1	21	21	LS
38536	347	2	.	.	.
38536	347	3	gt	gt	NNP
38536	347	4	-	-	HYPH
38536	347	5	gk	gk	NNS
38536	347	6	+	+	SYM
38536	347	7	gl^2	gl^2	NN
38536	347	8	+	+	CC
38536	347	9	xt	xt	NNP
38536	347	10	-	-	HYPH
38536	347	11	xk	xk	NNP
38536	347	12	+	+	NFP
38536	347	13	xl^2	xl^2	XX
38536	347	14	.	.	.
38536	348	1	22	22	CD
38536	348	2	.	.	.
38536	349	1	(	(	-LRB-
38536	349	2	m	m	NNP
38536	349	3	-	-	HYPH
38536	349	4	n)(2a^2	n)(2a^2	NNP
38536	349	5	-	-	HYPH
38536	349	6	2ab	2ab	NN
38536	349	7	)	)	-RRB-
38536	349	8	+	+	CC
38536	349	9	(	(	-LRB-
38536	349	10	n	n	JJ
38536	349	11	-	-	HYPH
38536	349	12	m)(2ab	m)(2ab	NN
38536	349	13	-	-	HYPH
38536	349	14	2b^2	2b^2	CD
38536	349	15	)	)	-RRB-
38536	349	16	.	.	.
38536	350	1	23	23	CD
38536	350	2	.	.	.
38536	350	3	a^2	a^2	CD
38536	350	4	-	-	HYPH
38536	350	5	x^2	x^2	JJ
38536	350	6	-	-	HYPH
38536	350	7	y^2	y^2	NN
38536	350	8	+	+	SYM
38536	350	9	b^2	b^2	NNS
38536	350	10	+	+	SYM
38536	350	11	2ab	2ab	JJ
38536	350	12	+	+	SYM
38536	350	13	2xy	2xy	NN
38536	350	14	.	.	.
38536	351	1	24	24	CD
38536	351	2	.	.	.
38536	352	1	(	(	-LRB-
38536	352	2	2c^2	2c^2	CD
38536	352	3	+	+	CC
38536	352	4	3d^2)a	3d^2)a	CD
38536	352	5	+	+	SYM
38536	352	6	(	(	-LRB-
38536	352	7	2a^2	2a^2	CD
38536	352	8	+	+	SYM
38536	352	9	3c^2)d	3c^2)d	CD
38536	352	10	.	.	NNP
38536	352	11	25	25	CD
38536	352	12	.	.	.
38536	353	1	(	(	-LRB-
38536	353	2	n(n	n(n	NN
38536	353	3	-	-	HYPH
38536	353	4	1))/(1	1))/(1	CD
38536	353	5	·	·	NFP
38536	353	6	2	2	CD
38536	353	7	)	)	-RRB-
38536	353	8	a^(n	a^(n	CD
38536	353	9	-	-	HYPH
38536	353	10	2	2	CD
38536	353	11	)	)	-RRB-
38536	353	12	b^2	b^2	NNS
38536	353	13	+	+	NFP
38536	353	14	(	(	-LRB-
38536	353	15	n(n	n(n	NNP
38536	353	16	-	-	HYPH
38536	353	17	1)(n	1)(n	CD
38536	353	18	-	-	HYPH
38536	353	19	2))/(1	2))/(1	CD
38536	353	20	·	·	NFP
38536	353	21	2	2	CD
38536	353	22	·	·	SYM
38536	353	23	3	3	CD
38536	353	24	)	)	-RRB-
38536	353	25	a^(n	a^(n	CD
38536	353	26	-	-	SYM
38536	353	27	3	3	CD
38536	353	28	)	)	-RRB-
38536	353	29	b^3	b^3	NNS
38536	353	30	.	.	.
38536	354	1	26	26	CD
38536	354	2	.	.	.
38536	355	1	(	(	-LRB-
38536	355	2	x	x	NN
38536	355	3	-	-	NN
38536	355	4	x^2)^3	x^2)^3	NN
38536	355	5	+	+	NFP
38536	355	6	(	(	-LRB-
38536	355	7	x^2	x^2	NNP
38536	355	8	-	-	HYPH
38536	355	9	1)^3	1)^3	JJ
38536	355	10	+	+	SYM
38536	355	11	(	(	-LRB-
38536	355	12	1	1	CD
38536	355	13	-	-	HYPH
38536	355	14	x)^3	x)^3	NNP
38536	355	15	.	.	.
38536	356	1	(	(	-LRB-
38536	356	2	_	_	NNP
38536	356	3	M.	M.	NNP
38536	357	1	I.	I.	NNP
38536	357	2	T.	T.	NNP
38536	357	3	_	_	NNP
38536	357	4	)	)	-RRB-
38536	357	5	27	27	CD
38536	357	6	.	.	.
38536	358	1	(	(	-LRB-
38536	358	2	27y^3)^2	27y^3)^2	CD
38536	358	3	-	-	:
38536	358	4	2(27y^3)(8b^3	2(27y^3)(8b^3	CD
38536	358	5	)	)	-RRB-
38536	358	6	+	+	NFP
38536	358	7	(	(	-LRB-
38536	358	8	8b^3)^2	8b^3)^2	CD
38536	358	9	.	.	.
38536	359	1	(	(	-LRB-
38536	359	2	_	_	NNP
38536	359	3	Princeton	Princeton	NNP
38536	359	4	.	.	.
38536	359	5	_	_	NNP
38536	359	6	)	)	-RRB-
38536	359	7	28	28	CD
38536	359	8	.	.	.
38536	360	1	(	(	-LRB-
38536	360	2	a^3	a^3	NNP
38536	360	3	+	+	SYM
38536	360	4	8b^3)(a	8b^3)(a	CD
38536	360	5	+	+	SYM
38536	360	6	b	b	NN
38536	360	7	)	)	-RRB-
38536	360	8	-	-	:
38536	360	9	6ab(a^2	6ab(a^2	NNP
38536	360	10	-	-	HYPH
38536	360	11	2ab	2ab	NNP
38536	360	12	+	+	SYM
38536	360	13	4b^2	4b^2	CD
38536	360	14	)	)	-RRB-
38536	360	15	.	.	.
38536	361	1	(	(	-LRB-
38536	361	2	_	_	NNP
38536	361	3	M.	M.	NNP
38536	362	1	I.	I.	NNP
38536	362	2	T.	T.	NNP
38536	362	3	_	_	NNP
38536	362	4	)	)	-RRB-
38536	362	5	Solve	solve	NN
38536	362	6	by	by	IN
38536	362	7	factoring	factor	VBG
38536	362	8	:	:	:
38536	362	9	29	29	CD
38536	362	10	.	.	.
38536	362	11	x^3	x^3	NNP
38536	362	12	=	=	SYM
38536	362	13	x.	x.	NNP
38536	363	1	30	30	CD
38536	363	2	.	.	.
38536	363	3	z^2	z^2	CD
38536	363	4	-	-	HYPH
38536	363	5	4z	4z	CD
38536	363	6	-	-	HYPH
38536	363	7	45	45	CD
38536	363	8	=	=	SYM
38536	363	9	0	0	CD
38536	363	10	.	.	.
38536	364	1	31	31	CD
38536	364	2	.	.	.
38536	364	3	x^3	x^3	NNP
38536	364	4	-	-	:
38536	364	5	x^2	x^2	NNP
38536	364	6	=	=	SYM
38536	364	7	4x	4x	NNP
38536	364	8	-	-	SYM
38536	364	9	4	4	CD
38536	364	10	.	.	.
38536	365	1	~Reference:~	~reference:~	VB
38536	365	2	The	the	DT
38536	365	3	chapter	chapter	NN
38536	365	4	on	on	IN
38536	365	5	Factoring	factor	VBG
38536	365	6	in	in	IN
38536	365	7	any	any	DT
38536	365	8	algebra	algebra	NN
38536	365	9	.	.	.
38536	366	1	HIGHEST	HIGHEST	NNP
38536	366	2	COMMON	COMMON	NNP
38536	366	3	FACTOR	FACTOR	NNP
38536	366	4	AND	and	CC
38536	366	5	LOWEST	LOWEST	NNP
38536	366	6	COMMON	COMMON	NNP
38536	366	7	MULTIPLE	MULTIPLE	NNS
38536	366	8	Define	Define	NNP
38536	366	9	H.	H.	NNP
38536	366	10	C.	C.	NNP
38536	366	11	F.	F.	NNP
38536	366	12	and	and	CC
38536	366	13	L.	L.	NNP
38536	366	14	C.	C.	NNP
38536	366	15	M.	M.	NNP
38536	366	16	Find	Find	NNP
38536	366	17	by	by	IN
38536	366	18	factoring	factor	VBG
38536	366	19	the	the	DT
38536	366	20	H.	H.	NNP
38536	366	21	C.	C.	NNP
38536	366	22	F.	F.	NNP
38536	366	23	and	and	CC
38536	366	24	L.	L.	NNP
38536	366	25	C.	C.	NNP
38536	366	26	M.	M.	NNP
38536	366	27	:	:	:
38536	366	28	1	1	CD
38536	366	29	.	.	.
38536	367	1	3x^2	3x^2	CD
38536	367	2	-	-	SYM
38536	367	3	3x	3x	CD
38536	367	4	,	,	,
38536	367	5	12x^2	12x^2	CD
38536	367	6	(	(	-LRB-
38536	367	7	x^2	x^2	NNP
38536	367	8	-	-	SYM
38536	367	9	1	1	CD
38536	367	10	)	)	-RRB-
38536	367	11	,	,	,
38536	367	12	18x^3	18x^3	NNP
38536	367	13	(	(	-LRB-
38536	367	14	x^3	x^3	NNP
38536	367	15	-	-	SYM
38536	367	16	1	1	CD
38536	367	17	)	)	-RRB-
38536	367	18	.	.	.
38536	368	1	2	2	LS
38536	368	2	.	.	.
38536	369	1	(	(	-LRB-
38536	369	2	x^2	x^2	NNP
38536	369	3	-	-	SYM
38536	369	4	1)(x^2	1)(x^2	CD
38536	369	5	+	+	SYM
38536	369	6	5x	5x	CD
38536	369	7	+	+	SYM
38536	369	8	6	6	CD
38536	369	9	)	)	-RRB-
38536	369	10	,	,	,
38536	369	11	(	(	-LRB-
38536	369	12	x^2	x^2	NNP
38536	369	13	+	+	SYM
38536	369	14	3x)(x^2	3x)(x^2	JJ
38536	369	15	-	-	HYPH
38536	369	16	x	x	NN
38536	369	17	-	-	SYM
38536	369	18	6	6	CD
38536	369	19	)	)	-RRB-
38536	369	20	.	.	.
38536	370	1	(	(	-LRB-
38536	370	2	_	_	NNP
38536	370	3	Harvard	Harvard	NNP
38536	370	4	.	.	.
38536	370	5	_	_	NNP
38536	370	6	)	)	-RRB-
38536	370	7	3	3	CD
38536	370	8	.	.	.
38536	370	9	x^2	x^2	NNP
38536	370	10	-	-	HYPH
38536	370	11	y^2	y^2	NNP
38536	370	12	,	,	,
38536	370	13	x^2	x^2	NNS
38536	370	14	+	+	SYM
38536	370	15	y^2	y^2	PRP
38536	370	16	,	,	,
38536	370	17	x^3	x^3	NNS
38536	370	18	+	+	SYM
38536	370	19	y^3	y^3	NNS
38536	370	20	,	,	,
38536	370	21	x^6	x^6	NNS
38536	370	22	+	+	SYM
38536	370	23	y^6	y^6	NNS
38536	370	24	,	,	,
38536	370	25	x^6	x^6	NNP
38536	370	26	-	-	HYPH
38536	370	27	y^6	y^6	NNP
38536	370	28	.	.	.
38536	371	1	(	(	-LRB-
38536	371	2	_	_	NNP
38536	371	3	College	College	NNP
38536	371	4	Entrance	Entrance	NNP
38536	371	5	Board	Board	NNP
38536	371	6	.	.	.
38536	371	7	_	_	NNP
38536	371	8	)	)	-RRB-
38536	371	9	4	4	CD
38536	371	10	.	.	.
38536	371	11	x^3	x^3	CD
38536	371	12	+	+	SYM
38536	371	13	x^2	x^2	NNP
38536	371	14	-	-	SYM
38536	371	15	2	2	CD
38536	371	16	,	,	,
38536	371	17	x^3	x^3	CD
38536	371	18	+	+	SYM
38536	371	19	2x^2	2x^2	CD
38536	371	20	-	-	SYM
38536	371	21	3	3	CD
38536	371	22	.	.	.
38536	372	1	(	(	-LRB-
38536	372	2	_	_	NNP
38536	372	3	Cornell	Cornell	NNP
38536	372	4	.	.	.
38536	372	5	_	_	NNP
38536	372	6	)	)	-RRB-
38536	372	7	5	5	CD
38536	372	8	.	.	.
38536	372	9	x^5	x^5	CD
38536	372	10	-	-	HYPH
38536	372	11	2x^4	2x^4	CD
38536	372	12	+	+	SYM
38536	372	13	x^2	x^2	NNP
38536	372	14	,	,	,
38536	372	15	2x^4	2x^4	CD
38536	372	16	-	-	HYPH
38536	372	17	4x^3	4x^3	CD
38536	372	18	-	-	HYPH
38536	372	19	4x	4x	NNS
38536	372	20	+	+	SYM
38536	372	21	6	6	CD
38536	372	22	.	.	.
38536	373	1	(	(	-LRB-
38536	373	2	_	_	NNP
38536	373	3	Yale	Yale	NNP
38536	373	4	.	.	.
38536	373	5	_	_	NNP
38536	373	6	)	)	-RRB-
38536	373	7	6	6	CD
38536	373	8	.	.	.
38536	373	9	x^2	x^2	NNS
38536	373	10	+	+	SYM
38536	373	11	a^2	a^2	CD
38536	373	12	-	-	HYPH
38536	373	13	b^2	b^2	NNS
38536	373	14	+	+	CC
38536	373	15	2ax	2ax	NN
38536	373	16	,	,	,
38536	373	17	x^2	x^2	NNP
38536	373	18	-	-	HYPH
38536	373	19	a^2	a^2	NNS
38536	373	20	+	+	SYM
38536	373	21	b^2	b^2	NNS
38536	373	22	+	+	SYM
38536	373	23	2bx	2bx	NN
38536	373	24	,	,	,
38536	373	25	x^2	x^2	JJ
38536	373	26	-	-	HYPH
38536	373	27	a^2	a^2	IN
38536	373	28	-	-	HYPH
38536	373	29	b^2	b^2	NNP
38536	373	30	-	-	HYPH
38536	373	31	2ab	2ab	NN
38536	373	32	.	.	.
38536	374	1	(	(	-LRB-
38536	374	2	_	_	NNP
38536	374	3	Harvard	Harvard	NNP
38536	374	4	.	.	.
38536	374	5	_	_	NNP
38536	374	6	)	)	-RRB-
38536	374	7	7	7	CD
38536	374	8	.	.	.
38536	375	1	2x^2	2x^2	CD
38536	375	2	-	-	HYPH
38536	375	3	x	x	NN
38536	375	4	-	-	HYPH
38536	375	5	15	15	CD
38536	375	6	,	,	,
38536	375	7	3x^2	3x^2	CD
38536	375	8	-	-	SYM
38536	375	9	11x	11x	CD
38536	375	10	+	+	SYM
38536	375	11	6	6	CD
38536	375	12	,	,	,
38536	375	13	2x^3	2x^3	CD
38536	375	14	-	-	HYPH
38536	375	15	x^2	x^2	NNP
38536	375	16	-	-	HYPH
38536	375	17	13x	13x	CD
38536	375	18	-	-	SYM
38536	375	19	6	6	CD
38536	375	20	.	.	.
38536	376	1	(	(	-LRB-
38536	376	2	_	_	NNP
38536	376	3	College	College	NNP
38536	376	4	Entrance	Entrance	NNP
38536	376	5	Board	Board	NNP
38536	376	6	.	.	.
38536	376	7	_	_	NNP
38536	376	8	)	)	-RRB-
38536	376	9	8	8	CD
38536	376	10	.	.	.
38536	377	1	(	(	-LRB-
38536	377	2	tv	tv	NN
38536	377	3	-	-	HYPH
38536	377	4	v^2)^3	v^2)^3	RB
38536	377	5	,	,	,
38536	377	6	v^3	v^3	NNP
38536	377	7	-	-	HYPH
38536	377	8	t^2v	t^2v	NNP
38536	377	9	,	,	,
38536	377	10	t^3	t^3	NNP
38536	377	11	-	-	HYPH
38536	377	12	v^3	v^3	NNP
38536	377	13	,	,	,
38536	377	14	v^2	v^2	NN
38536	377	15	-	-	HYPH
38536	377	16	2vt	2vt	NN
38536	377	17	+	+	SYM
38536	377	18	t^2	t^2	CD
38536	377	19	.	.	.
38536	378	1	Pick	pick	VB
38536	378	2	out	out	RP
38536	378	3	the	the	DT
38536	378	4	H.	H.	NNP
38536	378	5	C.	C.	NNP
38536	378	6	F.	F.	NNP
38536	378	7	and	and	CC
38536	378	8	the	the	DT
38536	378	9	L.	L.	NNP
38536	378	10	C.	C.	NNP
38536	378	11	M.	M.	NNP
38536	378	12	of	of	IN
38536	378	13	the	the	DT
38536	378	14	following	follow	VBG
38536	378	15	:	:	:
38536	378	16	9	9	CD
38536	378	17	.	.	.
38536	379	1	8(x^2	8(x^2	JJ
38536	379	2	+	+	SYM
38536	379	3	y)^(32	y)^(32	NNS
38536	379	4	)	)	-RRB-
38536	379	5	(	(	-LRB-
38536	379	6	t^2	t^2	CD
38536	379	7	+	+	SYM
38536	379	8	z)^(19	z)^(19	NNS
38536	379	9	)	)	-RRB-
38536	379	10	(	(	-LRB-
38536	379	11	m	m	NN
38536	379	12	-	-	HYPH
38536	379	13	n^3)^(14	n^3)^(14	NNP
38536	379	14	)	)	-RRB-
38536	379	15	,	,	,
38536	379	16	12(x^2	12(x^2	CD
38536	379	17	+	+	SYM
38536	379	18	y)^(23	y)^(23	NN
38536	379	19	)	)	-RRB-
38536	379	20	(	(	-LRB-
38536	379	21	t^2	t^2	CD
38536	379	22	+	+	SYM
38536	379	23	z)^(41	z)^(41	NNP
38536	379	24	)	)	-RRB-
38536	379	25	(	(	-LRB-
38536	379	26	m	m	NN
38536	379	27	-	-	HYPH
38536	379	28	n^3)^(17	n^3)^(17	NN
38536	379	29	)	)	-RRB-
38536	379	30	,	,	,
38536	379	31	18(m	18(m	CD
38536	379	32	-	-	HYPH
38536	379	33	n^3)^(11	n^3)^(11	JJ
38536	379	34	)	)	-RRB-
38536	379	35	(	(	-LRB-
38536	379	36	x^2	x^2	NNP
38536	379	37	+	+	SYM
38536	379	38	y)^(39	y)^(39	NFP
38536	379	39	)	)	-RRB-
38536	379	40	(	(	-LRB-
38536	379	41	t^2	t^2	CD
38536	379	42	+	+	SYM
38536	379	43	z)^(37	z)^(37	NFP
38536	379	44	)	)	-RRB-
38536	379	45	.	.	.
38536	380	1	10	10	CD
38536	380	2	.	.	.
38536	381	1	17ax^3	17ax^3	CD
38536	381	2	(	(	-LRB-
38536	381	3	y	y	NNP
38536	381	4	+	+	CC
38536	381	5	z)^(10	z)^(10	NNP
38536	381	6	)	)	-RRB-
38536	381	7	(	(	-LRB-
38536	381	8	y	y	NNP
38536	381	9	-	-	HYPH
38536	381	10	x)^(19	x)^(19	NNP
38536	381	11	)	)	-RRB-
38536	381	12	(	(	-LRB-
38536	381	13	x	x	SYM
38536	381	14	+	+	SYM
38536	381	15	z)^(27	z)^(27	FW
38536	381	16	)	)	-RRB-
38536	381	17	,	,	,
38536	381	18	34a^2	34a^2	NNP
38536	381	19	x^4	x^4	NNP
38536	381	20	(	(	-LRB-
38536	381	21	y	y	NNP
38536	381	22	+	+	SYM
38536	381	23	z)^(11	z)^(11	NNS
38536	381	24	)	)	-RRB-
38536	381	25	(	(	-LRB-
38536	381	26	y	y	NNP
38536	381	27	-	-	HYPH
38536	381	28	x)^(21	x)^(21	NNP
38536	381	29	)	)	-RRB-
38536	381	30	(	(	-LRB-
38536	381	31	x	x	SYM
38536	381	32	+	+	SYM
38536	381	33	z)^(13	z)^(13	NN
38536	381	34	)	)	-RRB-
38536	381	35	,	,	,
38536	381	36	51a^3	51a^3	CD
38536	381	37	x^5	x^5	XX
38536	381	38	(	(	-LRB-
38536	381	39	y	y	NNP
38536	381	40	+	+	NNP
38536	381	41	z)^4	z)^4	NNP
38536	381	42	(	(	-LRB-
38536	381	43	x	x	NN
38536	381	44	+	+	SYM
38536	381	45	z)^(32	z)^(32	NNS
38536	381	46	)	)	-RRB-
38536	381	47	(	(	-LRB-
38536	381	48	y	y	NNP
38536	381	49	-	-	HYPH
38536	381	50	x)^(29	x)^(29	NNP
38536	381	51	)	)	-RRB-
38536	381	52	.	.	.
38536	382	1	~Reference:~	~reference:~	VB
38536	382	2	The	the	DT
38536	382	3	chapter	chapter	NN
38536	382	4	on	on	IN
38536	382	5	H.	H.	NNP
38536	382	6	C.	C.	NNP
38536	382	7	F.	F.	NNP
38536	382	8	and	and	CC
38536	382	9	L.	L.	NNP
38536	382	10	C.	C.	NNP
38536	382	11	M.	M.	NNP
38536	382	12	in	in	IN
38536	382	13	any	any	DT
38536	382	14	algebra	algebra	NN
38536	382	15	.	.	.
38536	383	1	FRACTIONS	fraction	NNS
38536	383	2	Define	Define	NNP
38536	383	3	:	:	:
38536	383	4	fraction	fraction	NN
38536	383	5	,	,	,
38536	383	6	terms	term	NNS
38536	383	7	of	of	IN
38536	383	8	a	a	DT
38536	383	9	fraction	fraction	NN
38536	383	10	,	,	,
38536	383	11	reciprocal	reciprocal	JJ
38536	383	12	of	of	IN
38536	383	13	a	a	DT
38536	383	14	number	number	NN
38536	383	15	.	.	.
38536	384	1	Look	look	VB
38536	384	2	up	up	RP
38536	384	3	_	_	IN
38536	384	4	the	the	DT
38536	384	5	law	law	NN
38536	384	6	of	of	IN
38536	384	7	signs	sign	NNS
38536	384	8	_	_	NNP
38536	384	9	as	as	IN
38536	384	10	it	-PRON-	PRP
38536	384	11	applies	apply	VBZ
38536	384	12	to	to	IN
38536	384	13	fractions	fraction	NNS
38536	384	14	.	.	.
38536	385	1	Except	except	IN
38536	385	2	for	for	IN
38536	385	3	this	this	DT
38536	385	4	,	,	,
38536	385	5	fractions	fraction	NNS
38536	385	6	in	in	IN
38536	385	7	algebra	algebra	NN
38536	385	8	are	be	VBP
38536	385	9	treated	treat	VBN
38536	385	10	exactly	exactly	RB
38536	385	11	the	the	DT
38536	385	12	same	same	JJ
38536	385	13	as	as	IN
38536	385	14	they	-PRON-	PRP
38536	385	15	are	be	VBP
38536	385	16	in	in	IN
38536	385	17	arithmetic	arithmetic	JJ
38536	385	18	.	.	.
38536	386	1	1	1	LS
38536	386	2	.	.	.
38536	387	1	Reduce	reduce	VB
38536	387	2	to	to	TO
38536	387	3	lowest	low	JJS
38536	387	4	terms	term	NNS
38536	387	5	:	:	:
38536	387	6	(	(	-LRB-
38536	387	7	_	_	NNP
38536	387	8	a	a	DT
38536	387	9	_	_	NNP
38536	387	10	)	)	-RRB-
38536	387	11	32/24	32/24	CD
38536	387	12	;	;	:
38536	387	13	(	(	-LRB-
38536	387	14	_	_	NNP
38536	387	15	b	b	NNP
38536	387	16	_	_	NNP
38536	387	17	)	)	-RRB-
38536	387	18	(	(	-LRB-
38536	387	19	a^6	a^6	CD
38536	387	20	-	-	HYPH
38536	387	21	x^6)/(a^4	x^6)/(a^4	NNP
38536	387	22	-	-	HYPH
38536	387	23	x^4	x^4	NNP
38536	387	24	)	)	-RRB-
38536	387	25	;	;	:
38536	387	26	(	(	-LRB-
38536	387	27	_	_	NNP
38536	387	28	c	c	NNP
38536	387	29	_	_	NNP
38536	387	30	)	)	-RRB-
38536	387	31	[	[	-LRB-
38536	387	32	(	(	-LRB-
38536	387	33	a	a	DT
38536	387	34	+	+	SYM
38536	387	35	b)^2	b)^2	NNP
38536	387	36	-	-	,
38536	387	37	(	(	-LRB-
38536	387	38	c	c	NN
38536	387	39	+	+	CC
38536	387	40	d)^2]/[(a	d)^2]/[(a	NNP
38536	387	41	+	+	SYM
38536	387	42	c)^2	c)^2	RB
38536	387	43	-	-	:
38536	387	44	(	(	-LRB-
38536	387	45	b	b	NN
38536	387	46	+	+	SYM
38536	387	47	d)^2	d)^2	NNP
38536	387	48	]	]	-RRB-
38536	387	49	.	.	.
38536	388	1	(	(	-LRB-
38536	388	2	_	_	NNP
38536	388	3	M.	M.	NNP
38536	389	1	I.	I.	NNP
38536	389	2	T.	T.	NNP
38536	389	3	_	_	NNP
38536	389	4	)	)	-RRB-
38536	389	5	2	2	CD
38536	389	6	.	.	.
38536	390	1	Reduce	reduce	VB
38536	390	2	to	to	IN
38536	390	3	a	a	DT
38536	390	4	mixed	mixed	JJ
38536	390	5	expression	expression	NN
38536	390	6	:	:	:
38536	390	7	(	(	-LRB-
38536	390	8	_	_	NNP
38536	390	9	a	a	DT
38536	390	10	_	_	NNP
38536	390	11	)	)	-RRB-
38536	390	12	756/11	756/11	NNS
38536	390	13	;	;	:
38536	390	14	(	(	-LRB-
38536	390	15	_	_	NNP
38536	390	16	b	b	NNP
38536	390	17	_	_	NNP
38536	390	18	)	)	-RRB-
38536	390	19	(	(	-LRB-
38536	390	20	a^3	a^3	NNP
38536	390	21	+	+	SYM
38536	390	22	b^3)/(a	b^3)/(a	NNP
38536	390	23	-	-	HYPH
38536	390	24	b	b	NN
38536	390	25	)	)	-RRB-
38536	390	26	.	.	.
38536	391	1	3	3	LS
38536	391	2	.	.	.
38536	392	1	Reduce	reduce	VB
38536	392	2	to	to	IN
38536	392	3	an	an	DT
38536	392	4	improper	improper	JJ
38536	392	5	fraction	fraction	NN
38536	392	6	:	:	:
38536	392	7	(	(	-LRB-
38536	392	8	_	_	NNP
38536	392	9	a	a	DT
38536	392	10	_	_	NNP
38536	392	11	)	)	-RRB-
38536	392	12	45	45	CD
38536	392	13	-	-	HYPH
38536	392	14	1/8	1/8	CD
38536	392	15	;	;	:
38536	392	16	(	(	-LRB-
38536	392	17	_	_	NNP
38536	392	18	b	b	NNP
38536	392	19	_	_	NNP
38536	392	20	)	)	-RRB-
38536	392	21	9	9	CD
38536	392	22	-	-	SYM
38536	392	23	11/12	11/12	CD
38536	392	24	qt	qt	NN
38536	392	25	.	.	.
38536	393	1	;	;	:
38536	393	2	(	(	-LRB-
38536	393	3	_	_	NNP
38536	393	4	c	c	NNP
38536	393	5	_	_	NNP
38536	393	6	)	)	-RRB-
38536	393	7	a^2	a^2	NNP
38536	393	8	-	-	HYPH
38536	393	9	ab	ab	NN
38536	393	10	+	+	DT
38536	393	11	b^2	b^2	NNP
38536	393	12	-	-	,
38536	393	13	(	(	-LRB-
38536	393	14	b^3)/(a	b^3)/(a	NNP
38536	393	15	+	+	SYM
38536	393	16	b	b	NN
38536	393	17	)	)	-RRB-
38536	393	18	.	.	.
38536	394	1	Add	add	NN
38536	394	2	:	:	:
38536	394	3	4	4	CD
38536	394	4	.	.	.
38536	395	1	5/18	5/18	CD
38536	395	2	+	+	SYM
38536	395	3	7/9	7/9	CD
38536	395	4	+	+	SYM
38536	395	5	11/16	11/16	CD
38536	395	6	+	+	SYM
38536	395	7	5/8	5/8	CD
38536	395	8	.	.	.
38536	396	1	5	5	CD
38536	396	2	.	.	.
38536	397	1	5/(1	5/(1	CD
38536	397	2	+	+	CC
38536	397	3	2x	2x	CD
38536	397	4	)	)	-RRB-
38536	397	5	-	-	:
38536	397	6	(	(	-LRB-
38536	397	7	3x)/(1	3x)/(1	NNP
38536	397	8	-	-	HYPH
38536	397	9	2x	2x	NNP
38536	397	10	)	)	-RRB-
38536	397	11	+	+	CC
38536	397	12	(	(	-LRB-
38536	397	13	4	4	CD
38536	397	14	-	-	CD
38536	397	15	13x)/(4x^2	13x)/(4x^2	CD
38536	397	16	-	-	HYPH
38536	397	17	1	1	CD
38536	397	18	)	)	-RRB-
38536	397	19	.	.	.
38536	398	1	6	6	CD
38536	398	2	.	.	.
38536	399	1	1/[x(x	1/[x(x	JJ
38536	399	2	-	-	HYPH
38536	399	3	a)(x	a)(x	NNP
38536	399	4	-	-	HYPH
38536	399	5	b	b	NNP
38536	399	6	)	)	-RRB-
38536	399	7	]	]	-RRB-
38536	399	8	+	+	CC
38536	399	9	1/[a(a	1/[a(a	CD
38536	399	10	-	-	HYPH
38536	399	11	x)(a	x)(a	NNP
38536	399	12	-	-	HYPH
38536	399	13	b	b	NNP
38536	399	14	)	)	-RRB-
38536	399	15	]	]	-RRB-
38536	399	16	+	+	CC
38536	399	17	1/[b(b	1/[b(b	CD
38536	399	18	-	-	HYPH
38536	399	19	x)(b	x)(b	NN
38536	399	20	-	-	HYPH
38536	399	21	a	a	NNP
38536	399	22	)	)	-RRB-
38536	399	23	]	]	-RRB-
38536	399	24	.	.	.
38536	400	1	Multiply	Multiply	NNP
38536	400	2	:	:	:
38536	400	3	7	7	CD
38536	400	4	.	.	.
38536	401	1	72/121	72/121	CD
38536	401	2	×	×	CD
38536	401	3	55/56	55/56	CD
38536	401	4	×	×	NNS
38536	401	5	77/90	77/90	CD
38536	401	6	.	.	.
38536	402	1	8	8	LS
38536	402	2	.	.	.
38536	403	1	(	(	-LRB-
38536	403	2	b	b	NNP
38536	403	3	-	-	HYPH
38536	403	4	y)/(a^3	y)/(a^3	NNP
38536	403	5	+	+	SYM
38536	403	6	y^3	y^3	NN
38536	403	7	)	)	-RRB-
38536	403	8	×	×	ADD
38536	403	9	(	(	-LRB-
38536	403	10	ca	ca	NN
38536	403	11	+	+	ADD
38536	403	12	cy)/(b^2	cy)/(b^2	.
38536	403	13	+	+	SYM
38536	403	14	by	by	IN
38536	403	15	)	)	-RRB-
38536	403	16	×	×	NN
38536	403	17	(	(	-LRB-
38536	403	18	b^6	b^6	NNS
38536	403	19	+	+	SYM
38536	403	20	y^6)/(b^2	y^6)/(b^2	NN
38536	403	21	+	+	SYM
38536	403	22	y^2	y^2	NN
38536	403	23	)	)	-RRB-
38536	403	24	×	×	NNP
38536	403	25	b	b	NNP
38536	403	26	/	/	SYM
38536	403	27	c	c	NN
38536	403	28	.	.	.
38536	404	1	Divide	divide	NN
38536	404	2	:	:	:
38536	404	3	9	9	CD
38536	404	4	.	.	.
38536	405	1	(	(	-LRB-
38536	405	2	12/25	12/25	CD
38536	405	3	)	)	-RRB-
38536	405	4	÷	÷	NNP
38536	405	5	(	(	-LRB-
38536	405	6	6/50	6/50	NNP
38536	405	7	)	)	-RRB-
38536	405	8	.	.	.
38536	406	1	10	10	CD
38536	406	2	.	.	.
38536	407	1	[	[	-LRB-
38536	407	2	1	1	CD
38536	407	3	-	-	,
38536	407	4	(	(	-LRB-
38536	407	5	ab)/(a^2	ab)/(a^2	NN
38536	407	6	-	-	HYPH
38536	407	7	ab	ab	NNP
38536	407	8	+	+	DT
38536	407	9	b^2	b^2	NNS
38536	407	10	)	)	-RRB-
38536	407	11	]	]	-RRB-
38536	407	12	[	[	-LRB-
38536	407	13	1	1	CD
38536	407	14	-	-	,
38536	407	15	(	(	-LRB-
38536	407	16	ab)/(a^2	ab)/(a^2	NN
38536	407	17	+	+	SYM
38536	407	18	2ab	2ab	NN
38536	407	19	+	+	SYM
38536	407	20	b^2	b^2	NNS
38536	407	21	)	)	-RRB-
38536	407	22	]	]	-RRB-
38536	407	23	÷	÷	NNP
38536	407	24	(	(	-LRB-
38536	407	25	a^3	a^3	NNP
38536	407	26	-	-	HYPH
38536	407	27	b^3)/(a^3	b^3)/(a^3	NN
38536	407	28	+	+	SYM
38536	407	29	b^3	b^3	NNP
38536	407	30	)	)	-RRB-
38536	407	31	.	.	.
38536	408	1	(	(	-LRB-
38536	408	2	_	_	NNP
38536	408	3	Yale	Yale	NNP
38536	408	4	.	.	.
38536	408	5	_	_	NNP
38536	408	6	)	)	-RRB-
38536	408	7	11	11	CD
38536	408	8	.	.	.
38536	409	1	[	[	-LRB-
38536	409	2	(	(	-LRB-
38536	409	3	x^4	x^4	NNP
38536	409	4	-	-	HYPH
38536	409	5	y^4)/(x^2	y^4)/(x^2	JJ
38536	409	6	-	-	HYPH
38536	409	7	y^2	y^2	NN
38536	409	8	)	)	-RRB-
38536	409	9	÷	÷	NN
38536	409	10	(	(	-LRB-
38536	409	11	x	x	NN
38536	409	12	+	+	SYM
38536	409	13	y)/(x^2	y)/(x^2	:
38536	409	14	-	-	:
38536	409	15	xy	xy	NN
38536	409	16	)	)	-RRB-
38536	409	17	]	]	-RRB-
38536	409	18	÷	÷	NNP
38536	409	19	[	[	-LRB-
38536	409	20	(	(	-LRB-
38536	409	21	x^2	x^2	NNS
38536	409	22	+	+	SYM
38536	409	23	y^2)/(x	y^2)/(x	NN
38536	409	24	-	-	HYPH
38536	409	25	y	y	NN
38536	409	26	)	)	-RRB-
38536	409	27	÷	÷	NNP
38536	409	28	(	(	-LRB-
38536	409	29	x	x	NNP
38536	409	30	+	+	SYM
38536	409	31	y)/(xy	y)/(xy	NN
38536	409	32	-	-	HYPH
38536	409	33	y^2	y^2	NNP
38536	409	34	)	)	-RRB-
38536	409	35	]	]	-RRB-
38536	409	36	.	.	.
38536	410	1	(	(	-LRB-
38536	410	2	_	_	NNP
38536	410	3	Sheffield	Sheffield	NNP
38536	410	4	.	.	.
38536	410	5	_	_	NNP
38536	410	6	)	)	-RRB-
38536	410	7	Simplify	Simplify	NNP
38536	410	8	:	:	:
38536	410	9	12	12	CD
38536	410	10	.	.	.
38536	411	1	[	[	-LRB-
38536	411	2	(	(	-LRB-
38536	411	3	4y)/x	4y)/x	CD
38536	411	4	-	-	HYPH
38536	411	5	(	(	-LRB-
38536	411	6	15y^2)/(x^2	15y^2)/(x^2	NN
38536	411	7	)	)	-RRB-
38536	411	8	+	+	SYM
38536	411	9	4	4	LS
38536	411	10	]	]	-RRB-
38536	411	11	÷	÷	NN
38536	411	12	[	[	-LRB-
38536	411	13	4	4	CD
38536	411	14	-	-	HYPH
38536	411	15	(	(	-LRB-
38536	411	16	16y)/x	16y)/x	CD
38536	411	17	+	+	SYM
38536	411	18	(	(	-LRB-
38536	411	19	15y^2)/(x^2	15y^2)/(x^2	CD
38536	411	20	)	)	-RRB-
38536	411	21	]	]	-RRB-
38536	411	22	×	×	ADD
38536	411	23	[	[	-LRB-
38536	411	24	3	3	CD
38536	411	25	-	-	HYPH
38536	411	26	(	(	-LRB-
38536	411	27	4x	4x	NNS
38536	411	28	+	+	SYM
38536	411	29	20y)/(2x	20y)/(2x	CD
38536	411	30	+	+	SYM
38536	411	31	5y	5y	NNS
38536	411	32	)	)	-RRB-
38536	411	33	]	]	-RRB-
38536	411	34	.	.	.
38536	412	1	~Reference:~	~reference:~	VB
38536	412	2	The	the	DT
38536	412	3	chapter	chapter	NN
38536	412	4	on	on	IN
38536	412	5	Fractions	fraction	NNS
38536	412	6	in	in	IN
38536	412	7	any	any	DT
38536	412	8	algebra	algebra	NN
38536	412	9	.	.	.
38536	413	1	COMPLEX	COMPLEX	NNP
38536	413	2	FRACTIONS	fraction	NNS
38536	413	3	AND	and	CC
38536	413	4	FRACTIONAL	fractional	JJ
38536	413	5	EQUATIONS	equations	NN
38536	413	6	Define	Define	NNP
38536	413	7	a	a	DT
38536	413	8	complex	complex	JJ
38536	413	9	fraction	fraction	NN
38536	413	10	.	.	.
38536	414	1	Simplify	simplify	NN
38536	414	2	:	:	:
38536	414	3	1	1	CD
38536	414	4	.	.	.
38536	415	1	(	(	-LRB-
38536	415	2	3/7	3/7	CD
38536	415	3	+	+	SYM
38536	415	4	4/5)/(2	4/5)/(2	CD
38536	415	5	-	-	HYPH
38536	415	6	3/7	3/7	CD
38536	415	7	·	·	NFP
38536	415	8	4/5	4/5	CD
38536	415	9	)	)	-RRB-
38536	415	10	.	.	.
38536	416	1	2	2	LS
38536	416	2	.	.	.
38536	417	1	(	(	-LRB-
38536	417	2	2	2	CD
38536	417	3	-	-	SYM
38536	417	4	3/2	3/2	CD
38536	417	5	+	+	CC
38536	417	6	2/3)/(5	2/3)/(5	CD
38536	417	7	-	-	HYPH
38536	417	8	2/3	2/3	CD
38536	417	9	+	+	SYM
38536	417	10	3/2	3/2	CD
38536	417	11	)	)	-RRB-
38536	417	12	.	.	.
38536	418	1	3	3	LS
38536	418	2	.	.	.
38536	419	1	2	2	LS
38536	419	2	-	-	HYPH
38536	419	3	2/(1	2/(1	NNP
38536	419	4	-	-	HYPH
38536	419	5	1/[1	1/[1	NNP
38536	419	6	-	-	HYPH
38536	419	7	1/(1	1/(1	NNP
38536	419	8	+	+	CD
38536	419	9	1/2	1/2	CD
38536	419	10	)	)	-RRB-
38536	419	11	]	]	-RRB-
38536	419	12	)	)	-RRB-
38536	419	13	.	.	.
38536	420	1	4	4	LS
38536	420	2	.	.	.
38536	420	3	a/(b^2	a/(b^2	NNP
38536	420	4	)	)	-RRB-
38536	420	5	-	-	:
38536	420	6	a/[b^2	a/[b^2	NN
38536	420	7	+	+	NFP
38536	420	8	(	(	-LRB-
38536	420	9	cb)/(a	cb)/(a	NFP
38536	420	10	-	-	HYPH
38536	420	11	c	c	NN
38536	420	12	/	/	SYM
38536	420	13	b	b	NNP
38536	420	14	)	)	-RRB-
38536	420	15	]	]	-RRB-
38536	420	16	.	.	.
38536	421	1	(	(	-LRB-
38536	421	2	_	_	NNP
38536	421	3	Harvard	Harvard	NNP
38536	421	4	.	.	.
38536	421	5	_	_	NNP
38536	421	6	)	)	-RRB-
38536	421	7	5	5	CD
38536	421	8	.	.	.
38536	422	1	If	if	IN
38536	422	2	m	m	NN
38536	422	3	=	=	SYM
38536	422	4	1/(a	1/(a	CD
38536	422	5	+	+	SYM
38536	422	6	1	1	CD
38536	422	7	)	)	-RRB-
38536	422	8	,	,	,
38536	422	9	n	n	CC
38536	422	10	=	=	SYM
38536	422	11	2/(a	2/(a	CD
38536	422	12	+	+	SYM
38536	422	13	2	2	CD
38536	422	14	)	)	-RRB-
38536	422	15	,	,	,
38536	422	16	p	p	NN
38536	422	17	=	=	SYM
38536	422	18	3/(a	3/(a	CD
38536	422	19	+	+	SYM
38536	422	20	3	3	CD
38536	422	21	)	)	-RRB-
38536	422	22	,	,	,
38536	422	23	what	what	WP
38536	422	24	is	be	VBZ
38536	422	25	the	the	DT
38536	422	26	value	value	NN
38536	422	27	of	of	IN
38536	422	28	m/(1	m/(1	NN
38536	422	29	-	-	HYPH
38536	422	30	m	m	NN
38536	422	31	)	)	-RRB-
38536	422	32	+	+	CC
38536	422	33	n/(1	n/(1	NNP
38536	422	34	-	-	HYPH
38536	422	35	n	n	NN
38536	422	36	)	)	-RRB-
38536	422	37	+	+	SYM
38536	422	38	p/(1	p/(1	NNP
38536	422	39	-	-	HYPH
38536	422	40	p	p	NNP
38536	422	41	)	)	-RRB-
38536	422	42	?	?	.
38536	423	1	(	(	-LRB-
38536	423	2	_	_	NNP
38536	423	3	Univ	Univ	NNP
38536	423	4	.	.	.
38536	424	1	of	of	IN
38536	424	2	Penn	Penn	NNP
38536	424	3	.	.	.
38536	424	4	_	_	NNP
38536	424	5	)	)	-RRB-
38536	424	6	6	6	CD
38536	424	7	.	.	.
38536	425	1	Simplify	simplify	VB
38536	425	2	the	the	DT
38536	425	3	expression	expression	NN
38536	425	4	{	{	-LRB-
38536	425	5	x	x	SYM
38536	425	6	+	+	CC
38536	425	7	y	y	NNP
38536	425	8	-	-	HYPH
38536	425	9	1/[x	1/[x	NNP
38536	425	10	+	+	SYM
38536	425	11	y	y	NNP
38536	425	12	-	-	HYPH
38536	425	13	xy/(x	xy/(x	NNP
38536	425	14	+	+	CC
38536	425	15	y)]}(x^3	y)]}(x^3	NNP
38536	425	16	-	-	HYPH
38536	425	17	y^3)/(x^2	y^3)/(x^2	NN
38536	425	18	-	-	HYPH
38536	425	19	y^2	y^2	NNP
38536	425	20	)	)	-RRB-
38536	425	21	.	.	.
38536	426	1	(	(	-LRB-
38536	426	2	_	_	NNP
38536	426	3	Cornell	Cornell	NNP
38536	426	4	.	.	.
38536	426	5	_	_	NNP
38536	426	6	)	)	-RRB-
38536	426	7	7	7	CD
38536	426	8	.	.	.
38536	427	1	Simplify	Simplify	NNP
38536	427	2	[	[	-LRB-
38536	427	3	1	1	CD
38536	427	4	-	-	HYPH
38536	427	5	(	(	-LRB-
38536	427	6	2xy)/((x	2xy)/((x	CD
38536	427	7	+	+	SYM
38536	427	8	y)^2)]/[1	y)^2)]/[1	NN
38536	427	9	+	+	NFP
38536	427	10	(	(	-LRB-
38536	427	11	2xy)/((x	2xy)/((x	CD
38536	427	12	-	-	HYPH
38536	427	13	y)^2	y)^2	NNP
38536	427	14	)	)	-RRB-
38536	427	15	]	]	-RRB-
38536	427	16	÷	÷	NNP
38536	427	17	{	{	-LRB-
38536	427	18	(	(	-LRB-
38536	427	19	1	1	CD
38536	427	20	-	-	HYPH
38536	427	21	y	y	NNP
38536	427	22	/	/	SYM
38536	427	23	x)/(1	x)/(1	NNP
38536	427	24	+	+	SYM
38536	427	25	y	y	NNP
38536	427	26	/	/	SYM
38536	427	27	x)}^2	x)}^2	NNP
38536	427	28	.	.	.
38536	428	1	8	8	LS
38536	428	2	.	.	.
38536	429	1	Solve	solve	VB
38536	429	2	(	(	-LRB-
38536	429	3	7y	7y	NN
38536	429	4	+	+	SYM
38536	429	5	9)/4	9)/4	CD
38536	429	6	-	-	HYPH
38536	429	7	[	[	-LRB-
38536	429	8	y	y	FW
38536	429	9	-	-	HYPH
38536	429	10	(	(	-LRB-
38536	429	11	2y	2y	CD
38536	429	12	-	-	HYPH
38536	429	13	1)/9	1)/9	CD
38536	429	14	]	]	-RRB-
38536	429	15	=	=	SYM
38536	429	16	7	7	CD
38536	429	17	.	.	.
38536	430	1	9	9	CD
38536	430	2	.	.	.
38536	431	1	Solve	solve	VB
38536	431	2	2	2	CD
38536	431	3	-	-	HYPH
38536	431	4	1/3	1/3	CD
38536	431	5	-	-	HYPH
38536	431	6	(	(	-LRB-
38536	431	7	2/5)(x^2	2/5)(x^2	CD
38536	431	8	+	+	SYM
38536	431	9	3	3	CD
38536	431	10	)	)	-RRB-
38536	431	11	=	=	NFP
38536	431	12	(	(	-LRB-
38536	431	13	10x)/3	10x)/3	CD
38536	431	14	+	+	SYM
38536	431	15	1	1	CD
38536	431	16	-	-	HYPH
38536	431	17	(	(	-LRB-
38536	431	18	2x^2)/5	2x^2)/5	CD
38536	431	19	.	.	.
38536	432	1	10	10	CD
38536	432	2	.	.	.
38536	433	1	How	how	WRB
38536	433	2	much	much	JJ
38536	433	3	water	water	NN
38536	433	4	must	must	MD
38536	433	5	be	be	VB
38536	433	6	added	add	VBN
38536	433	7	to	to	IN
38536	433	8	80	80	CD
38536	433	9	pounds	pound	NNS
38536	433	10	of	of	IN
38536	433	11	a	a	DT
38536	433	12	5	5	CD
38536	433	13	per	per	NN
38536	433	14	cent	cent	NN
38536	433	15	salt	salt	NN
38536	433	16	solution	solution	NN
38536	433	17	to	to	TO
38536	433	18	obtain	obtain	VB
38536	433	19	a	a	DT
38536	433	20	4	4	CD
38536	433	21	per	per	NN
38536	433	22	cent	cent	NN
38536	433	23	solution	solution	NN
38536	433	24	?	?	.
38536	434	1	(	(	-LRB-
38536	434	2	_	_	NNP
38536	434	3	Yale	Yale	NNP
38536	434	4	.	.	.
38536	434	5	_	_	NNP
38536	434	6	)	)	-RRB-
38536	434	7	~Reference:~	~Reference:~	NNP
38536	434	8	See	See	NNP
38536	434	9	Complex	Complex	NNP
38536	434	10	Fractions	Fractions	NNPS
38536	434	11	,	,	,
38536	434	12	and	and	CC
38536	434	13	the	the	DT
38536	434	14	first	first	JJ
38536	434	15	part	part	NN
38536	434	16	of	of	IN
38536	434	17	the	the	DT
38536	434	18	chapter	chapter	NN
38536	434	19	on	on	IN
38536	434	20	Fractional	Fractional	NNP
38536	434	21	Equations	Equations	NNP
38536	434	22	in	in	IN
38536	434	23	any	any	DT
38536	434	24	algebra	algebra	NN
38536	434	25	.	.	.
38536	435	1	FRACTIONAL	fractional	JJ
38536	435	2	EQUATIONS	EQUATIONS	NNP
38536	435	3	1	1	CD
38536	435	4	.	.	.
38536	436	1	Solve	solve	VB
38536	436	2	for	for	IN
38536	436	3	each	each	DT
38536	436	4	letter	letter	NN
38536	436	5	in	in	IN
38536	436	6	turn	turn	NN
38536	436	7	1	1	CD
38536	436	8	/	/	SYM
38536	436	9	b	b	NN
38536	436	10	=	=	SYM
38536	436	11	1	1	CD
38536	436	12	/	/	SYM
38536	436	13	p	p	NN
38536	436	14	+	+	SYM
38536	436	15	1	1	CD
38536	436	16	/	/	SYM
38536	436	17	q	q	NN
38536	436	18	.	.	.
38536	437	1	2	2	LS
38536	437	2	.	.	.
38536	438	1	Solve	solve	VB
38536	438	2	and	and	CC
38536	438	3	check	check	VB
38536	438	4	:	:	:
38536	438	5	(	(	-LRB-
38536	438	6	5x	5x	CD
38536	438	7	+	+	SYM
38536	438	8	2)/3	2)/3	CD
38536	438	9	-	-	:
38536	438	10	(	(	-LRB-
38536	438	11	3	3	CD
38536	438	12	-	-	HYPH
38536	438	13	(	(	-LRB-
38536	438	14	3x	3x	CD
38536	438	15	-	-	HYPH
38536	438	16	1)/2	1)/2	CD
38536	438	17	)	)	-RRB-
38536	438	18	=	=	NFP
38536	438	19	(	(	-LRB-
38536	438	20	3x	3x	CD
38536	438	21	+	+	SYM
38536	438	22	19)/2	19)/2	CD
38536	438	23	-	-	:
38536	438	24	(	(	-LRB-
38536	438	25	(	(	-LRB-
38536	438	26	x	x	NN
38536	438	27	+	+	SYM
38536	438	28	1)/6	1)/6	CD
38536	438	29	+	+	SYM
38536	438	30	3	3	CD
38536	438	31	)	)	-RRB-
38536	438	32	.	.	.
38536	439	1	3	3	LS
38536	439	2	.	.	.
38536	440	1	Solve	solve	VB
38536	440	2	and	and	CC
38536	440	3	check	check	VB
38536	440	4	:	:	:
38536	440	5	(	(	-LRB-
38536	440	6	1/2)(x	1/2)(x	CD
38536	440	7	-	-	HYPH
38536	440	8	a/3	a/3	NNP
38536	440	9	)	)	-RRB-
38536	440	10	-	-	:
38536	440	11	(	(	-LRB-
38536	440	12	1/3)(x	1/3)(x	CD
38536	440	13	-	-	HYPH
38536	440	14	a/4	a/4	NN
38536	440	15	)	)	-RRB-
38536	440	16	+	+	CC
38536	440	17	(	(	-LRB-
38536	440	18	1/4)(x	1/4)(x	CD
38536	440	19	-	-	HYPH
38536	440	20	a/5	a/5	NNP
38536	440	21	)	)	-RRB-
38536	440	22	=	=	SYM
38536	440	23	0	0	NFP
38536	440	24	.	.	.
38536	441	1	4	4	LS
38536	441	2	.	.	.
38536	442	1	Solve	solve	VB
38536	442	2	(	(	-LRB-
38536	442	3	after	after	IN
38536	442	4	looking	look	VBG
38536	442	5	up	up	RP
38536	442	6	the	the	DT
38536	442	7	special	special	JJ
38536	442	8	_	_	NNP
38536	442	9	short	short	JJ
38536	442	10	_	_	NNP
38536	442	11	method	method	NN
38536	442	12	)	)	-RRB-
38536	442	13	:	:	:
38536	442	14	(	(	-LRB-
38536	442	15	3x	3x	CD
38536	442	16	-	-	HYPH
38536	442	17	1)/30	1)/30	CD
38536	442	18	+	+	NFP
38536	442	19	(	(	-LRB-
38536	442	20	4x	4x	NNP
38536	442	21	-	-	HYPH
38536	442	22	7)/15	7)/15	CD
38536	442	23	=	=	SYM
38536	442	24	x/4	x/4	NN
38536	442	25	-	-	HYPH
38536	442	26	(	(	-LRB-
38536	442	27	2x	2x	JJ
38536	442	28	-	-	HYPH
38536	442	29	3)/(12x	3)/(12x	CD
38536	442	30	-	-	HYPH
38536	442	31	11	11	CD
38536	442	32	)	)	-RRB-
38536	442	33	+	+	NFP
38536	442	34	(	(	-LRB-
38536	442	35	7x	7x	CD
38536	442	36	-	-	HYPH
38536	442	37	15)/60	15)/60	CD
38536	442	38	.	.	.
38536	443	1	5	5	CD
38536	443	2	.	.	.
38536	444	1	Solve	solve	VB
38536	444	2	by	by	IN
38536	444	3	the	the	DT
38536	444	4	special	special	JJ
38536	444	5	_	_	NNP
38536	444	6	short	short	JJ
38536	444	7	_	_	NNP
38536	444	8	method	method	NN
38536	444	9	:	:	:
38536	444	10	1/(x	1/(x	CD
38536	444	11	-	-	SYM
38536	444	12	2	2	CD
38536	444	13	)	)	-RRB-
38536	444	14	-	-	HYPH
38536	444	15	1/(x	1/(x	CD
38536	444	16	-	-	HYPH
38536	444	17	3	3	CD
38536	444	18	)	)	-RRB-
38536	444	19	=	=	SYM
38536	444	20	1/(x	1/(x	CD
38536	444	21	-	-	SYM
38536	444	22	4	4	CD
38536	444	23	)	)	-RRB-
38536	444	24	-	-	HYPH
38536	444	25	1/(x	1/(x	CD
38536	444	26	-	-	HYPH
38536	444	27	5	5	CD
38536	444	28	)	)	-RRB-
38536	444	29	.	.	.
38536	445	1	6	6	CD
38536	445	2	.	.	.
38536	446	1	At	at	IN
38536	446	2	what	what	WDT
38536	446	3	time	time	NN
38536	446	4	between	between	IN
38536	446	5	8	8	CD
38536	446	6	and	and	CC
38536	446	7	9	9	CD
38536	446	8	o'clock	o'clock	NN
38536	446	9	are	be	VBP
38536	446	10	the	the	DT
38536	446	11	hands	hand	NNS
38536	446	12	of	of	IN
38536	446	13	a	a	DT
38536	446	14	watch	watch	NN
38536	446	15	(	(	-LRB-
38536	446	16	_	_	NNP
38536	446	17	a	a	DT
38536	446	18	_	_	NNP
38536	446	19	)	)	-RRB-
38536	446	20	opposite	opposite	IN
38536	446	21	each	each	DT
38536	446	22	other	other	JJ
38536	446	23	?	?	.
38536	447	1	(	(	-LRB-
38536	447	2	_	_	NNP
38536	447	3	b	b	NNP
38536	447	4	_	_	NNP
38536	447	5	)	)	-RRB-
38536	447	6	at	at	IN
38536	447	7	right	right	JJ
38536	447	8	angles	angle	NNS
38536	447	9	?	?	.
38536	448	1	(	(	-LRB-
38536	448	2	_	_	NNP
38536	448	3	c	c	NNP
38536	448	4	_	_	NNP
38536	448	5	)	)	-RRB-
38536	448	6	together	together	RB
38536	448	7	?	?	.
38536	449	1	Work	work	VB
38536	449	2	out	out	RP
38536	449	3	(	(	-LRB-
38536	449	4	_	_	NNP
38536	449	5	a	a	DT
38536	449	6	_	_	NNP
38536	449	7	)	)	-RRB-
38536	449	8	and	and	CC
38536	449	9	state	state	VB
38536	449	10	the	the	DT
38536	449	11	equations	equation	NNS
38536	449	12	for	for	IN
38536	449	13	(	(	-LRB-
38536	449	14	_	_	NNP
38536	449	15	b	b	NNP
38536	449	16	_	_	NNP
38536	449	17	)	)	-RRB-
38536	449	18	and	and	CC
38536	449	19	(	(	-LRB-
38536	449	20	_	_	NNP
38536	449	21	c	c	NNP
38536	449	22	_	_	NNP
38536	449	23	)	)	-RRB-
38536	449	24	.	.	.
38536	450	1	7	7	LS
38536	450	2	.	.	.
38536	451	1	The	the	DT
38536	451	2	formula	formula	NN
38536	451	3	for	for	IN
38536	451	4	converting	convert	VBG
38536	451	5	a	a	DT
38536	451	6	temperature	temperature	NN
38536	451	7	of	of	IN
38536	451	8	F	F	NNP
38536	451	9	degrees	degrees	NNPS
38536	451	10	Fahrenheit	Fahrenheit	NNP
38536	451	11	into	into	IN
38536	451	12	its	-PRON-	PRP$
38536	451	13	equivalent	equivalent	JJ
38536	451	14	temperature	temperature	NN
38536	451	15	of	of	IN
38536	451	16	C	C	NNP
38536	451	17	degrees	degree	NNS
38536	451	18	Centigrade	Centigrade	NNP
38536	451	19	is	be	VBZ
38536	451	20	C	C	NNP
38536	451	21	=	=	NN
38536	451	22	(	(	-LRB-
38536	451	23	5/9)(F	5/9)(f	CD
38536	451	24	-	-	SYM
38536	451	25	32	32	CD
38536	451	26	)	)	-RRB-
38536	451	27	.	.	.
38536	452	1	Express	Express	NNP
38536	452	2	F	F	NNP
38536	452	3	in	in	IN
38536	452	4	terms	term	NNS
38536	452	5	of	of	IN
38536	452	6	C	C	NNP
38536	452	7	,	,	,
38536	452	8	and	and	CC
38536	452	9	compute	compute	VB
38536	452	10	F	f	NN
38536	452	11	for	for	IN
38536	452	12	the	the	DT
38536	452	13	values	value	NNS
38536	452	14	C	C	NNP
38536	452	15	=	=	SYM
38536	452	16	30	30	CD
38536	452	17	and	and	CC
38536	452	18	C	C	NNP
38536	452	19	=	=	SYM
38536	452	20	28	28	CD
38536	452	21	.	.	.
38536	453	1	(	(	-LRB-
38536	453	2	_	_	NNP
38536	453	3	College	College	NNP
38536	453	4	Entrance	Entrance	NNP
38536	453	5	Exam	Exam	NNP
38536	453	6	.	.	.
38536	454	1	Board	Board	NNP
38536	454	2	.	.	.
38536	454	3	_	_	NNP
38536	454	4	)	)	-RRB-
38536	454	5	8	8	CD
38536	454	6	.	.	.
38536	455	1	What	what	WP
38536	455	2	is	be	VBZ
38536	455	3	the	the	DT
38536	455	4	price	price	NN
38536	455	5	of	of	IN
38536	455	6	eggs	egg	NNS
38536	455	7	when	when	WRB
38536	455	8	2	2	CD
38536	455	9	less	less	JJR
38536	455	10	for	for	IN
38536	455	11	24	24	CD
38536	455	12	cents	cent	NNS
38536	455	13	raises	raise	VBZ
38536	455	14	the	the	DT
38536	455	15	price	price	NN
38536	455	16	2	2	CD
38536	455	17	cents	cent	NNS
38536	455	18	a	a	DT
38536	455	19	dozen	dozen	NN
38536	455	20	?	?	.
38536	456	1	(	(	-LRB-
38536	456	2	_	_	NNP
38536	456	3	Yale	Yale	NNP
38536	456	4	.	.	.
38536	456	5	_	_	NNP
38536	456	6	)	)	-RRB-
38536	456	7	9	9	CD
38536	456	8	.	.	.
38536	457	1	Solve	solve	VB
38536	457	2	2/(x	2/(x	CD
38536	457	3	-	-	SYM
38536	457	4	2	2	CD
38536	457	5	)	)	-RRB-
38536	457	6	+	+	CC
38536	457	7	2/(4	2/(4	CD
38536	457	8	-	-	HYPH
38536	457	9	x^2	x^2	NNP
38536	457	10	)	)	-RRB-
38536	457	11	=	=	NFP
38536	457	12	5/(x	5/(x	CD
38536	457	13	+	+	SYM
38536	457	14	2	2	CD
38536	457	15	)	)	-RRB-
38536	457	16	.	.	.
38536	458	1	~Reference:~	~reference:~	VB
38536	458	2	The	the	DT
38536	458	3	Chapter	chapter	NN
38536	458	4	on	on	IN
38536	458	5	Fractional	Fractional	NNP
38536	458	6	Equations	Equations	NNP
38536	458	7	in	in	IN
38536	458	8	any	any	DT
38536	458	9	algebra	algebra	NN
38536	458	10	.	.	.
38536	459	1	Note	note	VB
38536	459	2	particularly	particularly	RB
38536	459	3	the	the	DT
38536	459	4	special	special	JJ
38536	459	5	_	_	NNP
38536	459	6	short	short	JJ
38536	459	7	_	_	NNP
38536	459	8	methods	method	NNS
38536	459	9	,	,	,
38536	459	10	usually	usually	RB
38536	459	11	given	give	VBN
38536	459	12	about	about	IN
38536	459	13	the	the	DT
38536	459	14	middle	middle	NN
38536	459	15	of	of	IN
38536	459	16	the	the	DT
38536	459	17	chapter	chapter	NN
38536	459	18	.	.	.
38536	460	1	SIMULTANEOUS	simultaneou	VBG
38536	460	2	EQUATIONS	EQUATIONS	NNP
38536	460	3	NOTE	note	NN
38536	460	4	.	.	.
38536	461	1	Up	up	IN
38536	461	2	to	to	IN
38536	461	3	this	this	DT
38536	461	4	point	point	NN
38536	461	5	each	each	DT
38536	461	6	topic	topic	NN
38536	461	7	presented	present	VBD
38536	461	8	has	have	VBZ
38536	461	9	reviewed	review	VBN
38536	461	10	to	to	IN
38536	461	11	some	some	DT
38536	461	12	extent	extent	NN
38536	461	13	the	the	DT
38536	461	14	preceding	precede	VBG
38536	461	15	topics	topic	NNS
38536	461	16	.	.	.
38536	462	1	For	for	IN
38536	462	2	example	example	NN
38536	462	3	,	,	,
38536	462	4	factoring	factor	VBG
38536	462	5	reviews	review	NNS
38536	462	6	the	the	DT
38536	462	7	special	special	JJ
38536	462	8	rules	rule	NNS
38536	462	9	of	of	IN
38536	462	10	multiplication	multiplication	NN
38536	462	11	and	and	CC
38536	462	12	division	division	NN
38536	462	13	;	;	:
38536	462	14	H.	H.	NNP
38536	462	15	C.	C.	NNP
38536	462	16	F.	F.	NNP
38536	462	17	and	and	CC
38536	462	18	L.	L.	NNP
38536	462	19	C.	C.	NNP
38536	462	20	M.	M.	NNP
38536	462	21	review	review	NN
38536	462	22	factoring	factoring	NN
38536	462	23	;	;	:
38536	462	24	addition	addition	NN
38536	462	25	and	and	CC
38536	462	26	subtraction	subtraction	NN
38536	462	27	of	of	IN
38536	462	28	fractions	fraction	NNS
38536	462	29	and	and	CC
38536	462	30	fractional	fractional	JJ
38536	462	31	equations	equation	NNS
38536	462	32	review	review	NN
38536	462	33	H.	H.	NNP
38536	462	34	C.	C.	NNP
38536	462	35	F.	F.	NNP
38536	462	36	and	and	CC
38536	462	37	L.	L.	NNP
38536	462	38	C.	C.	NNP
38536	462	39	M.	M.	NNP
38536	462	40	,	,	,
38536	462	41	etc	etc	FW
38536	462	42	.	.	.
38536	463	1	From	from	IN
38536	463	2	this	this	DT
38536	463	3	point	point	NN
38536	463	4	on	on	RB
38536	463	5	,	,	,
38536	463	6	however	however	RB
38536	463	7	,	,	,
38536	463	8	the	the	DT
38536	463	9	interdependence	interdependence	NN
38536	463	10	is	be	VBZ
38536	463	11	not	not	RB
38536	463	12	so	so	RB
38536	463	13	marked	marked	JJ
38536	463	14	,	,	,
38536	463	15	and	and	CC
38536	463	16	miscellaneous	miscellaneous	JJ
38536	463	17	examples	example	NNS
38536	463	18	illustrating	illustrate	VBG
38536	463	19	the	the	DT
38536	463	20	work	work	NN
38536	463	21	already	already	RB
38536	463	22	covered	cover	VBN
38536	463	23	will	will	MD
38536	463	24	be	be	VB
38536	463	25	given	give	VBN
38536	463	26	very	very	RB
38536	463	27	frequently	frequently	RB
38536	463	28	in	in	IN
38536	463	29	order	order	NN
38536	463	30	to	to	TO
38536	463	31	keep	keep	VB
38536	463	32	the	the	DT
38536	463	33	whole	whole	JJ
38536	463	34	subject	subject	JJ
38536	463	35	fresh	fresh	JJ
38536	463	36	in	in	IN
38536	463	37	mind	mind	NN
38536	463	38	.	.	.
38536	464	1	1	1	LS
38536	464	2	.	.	.
38536	465	1	Solve	solve	VB
38536	465	2	by	by	IN
38536	465	3	three	three	CD
38536	465	4	methods	method	NNS
38536	465	5	--	--	:
38536	465	6	addition	addition	NN
38536	465	7	and	and	CC
38536	465	8	subtraction	subtraction	NN
38536	465	9	,	,	,
38536	465	10	substitution	substitution	NN
38536	465	11	,	,	,
38536	465	12	and	and	CC
38536	465	13	comparison	comparison	NN
38536	465	14	:	:	:
38536	465	15	{	{	-LRB-
38536	465	16	5x	5x	CD
38536	465	17	+	+	NNP
38536	465	18	y	y	NN
38536	465	19	=	=	SYM
38536	465	20	11	11	CD
38536	465	21	,	,	,
38536	465	22	{	{	-LRB-
38536	465	23	3x	3x	CD
38536	465	24	+	+	SYM
38536	465	25	2y	2y	CD
38536	465	26	=	=	SYM
38536	465	27	1	1	CD
38536	465	28	.	.	.
38536	466	1	Solve	solve	VB
38536	466	2	and	and	CC
38536	466	3	check	check	VB
38536	466	4	:	:	:
38536	466	5	2	2	CD
38536	466	6	.	.	.
38536	467	1	{	{	-LRB-
38536	467	2	12R_1	12r_1	CD
38536	467	3	-	-	HYPH
38536	467	4	11R_2	11r_2	CD
38536	467	5	=	=	SYM
38536	467	6	b	b	NN
38536	467	7	+	+	SYM
38536	467	8	12c	12c	NNS
38536	467	9	,	,	,
38536	467	10	{	{	-LRB-
38536	467	11	R_1	r_1	NN
38536	467	12	+	+	CC
38536	467	13	R_2	r_2	NN
38536	467	14	=	=	SYM
38536	467	15	2b	2b	NNP
38536	467	16	+	+	SYM
38536	467	17	c.	c.	NN
38536	467	18	3	3	CD
38536	467	19	.	.	.
38536	468	1	{	{	-LRB-
38536	468	2	(	(	-LRB-
38536	468	3	r	r	LS
38536	468	4	-	-	HYPH
38536	468	5	s)/2	s)/2	NNP
38536	468	6	=	=	SYM
38536	468	7	25/6	25/6	CD
38536	468	8	-	-	HYPH
38536	468	9	(	(	-LRB-
38536	468	10	r	r	NN
38536	468	11	+	+	CC
38536	468	12	s)/3	s)/3	NNS
38536	468	13	,	,	,
38536	468	14	{	{	-LRB-
38536	468	15	(	(	-LRB-
38536	468	16	r	r	NN
38536	468	17	+	+	CC
38536	468	18	s	s	NN
38536	468	19	-	-	HYPH
38536	468	20	9)/2	9)/2	CD
38536	468	21	-	-	HYPH
38536	468	22	(	(	-LRB-
38536	468	23	s	s	NNP
38536	468	24	-	-	HYPH
38536	468	25	r	r	NNP
38536	468	26	-	-	HYPH
38536	468	27	6)/3	6)/3	NNP
38536	468	28	=	=	SYM
38536	468	29	0	0	CD
38536	468	30	.	.	.
38536	469	1	4	4	LS
38536	469	2	.	.	.
38536	470	1	One	one	CD
38536	470	2	half	half	NN
38536	470	3	of	of	IN
38536	470	4	A	A	NNP
38536	470	5	's	's	POS
38536	470	6	marbles	marble	NNS
38536	470	7	exceeds	exceed	VBZ
38536	470	8	one	one	CD
38536	470	9	half	half	NN
38536	470	10	of	of	IN
38536	470	11	B	b	NN
38536	470	12	's	's	POS
38536	470	13	and	and	CC
38536	470	14	C	C	NNP
38536	470	15	's	be	VBZ
38536	470	16	together	together	RB
38536	470	17	by	by	IN
38536	470	18	2	2	CD
38536	470	19	;	;	:
38536	470	20	twice	twice	PDT
38536	470	21	B	B	NNP
38536	470	22	's	's	POS
38536	470	23	marbles	marble	NNS
38536	470	24	falls	fall	VBZ
38536	470	25	short	short	JJ
38536	470	26	of	of	IN
38536	470	27	A	A	NNP
38536	470	28	's	's	POS
38536	470	29	and	and	CC
38536	470	30	C	C	NNP
38536	470	31	's	be	VBZ
38536	470	32	together	together	RB
38536	470	33	by	by	IN
38536	470	34	16	16	CD
38536	470	35	;	;	:
38536	470	36	if	if	IN
38536	470	37	C	C	NNP
38536	470	38	had	have	VBD
38536	470	39	four	four	CD
38536	470	40	more	more	JJR
38536	470	41	marbles	marble	NNS
38536	470	42	,	,	,
38536	470	43	he	-PRON-	PRP
38536	470	44	would	would	MD
38536	470	45	have	have	VB
38536	470	46	one	one	CD
38536	470	47	fourth	fourth	JJ
38536	470	48	as	as	RB
38536	470	49	many	many	JJ
38536	470	50	as	as	IN
38536	470	51	A	a	NN
38536	470	52	and	and	CC
38536	470	53	B	b	NN
38536	470	54	together	together	RB
38536	470	55	.	.	.
38536	471	1	How	how	WRB
38536	471	2	many	many	JJ
38536	471	3	has	have	VBZ
38536	471	4	each	each	DT
38536	471	5	?	?	.
38536	472	1	(	(	-LRB-
38536	472	2	_	_	NNP
38536	472	3	College	College	NNP
38536	472	4	Entrance	Entrance	NNP
38536	472	5	Board	Board	NNP
38536	472	6	.	.	.
38536	472	7	_	_	NNP
38536	472	8	)	)	-RRB-
38536	472	9	5	5	CD
38536	472	10	.	.	.
38536	473	1	The	the	DT
38536	473	2	sides	side	NNS
38536	473	3	of	of	IN
38536	473	4	a	a	DT
38536	473	5	triangle	triangle	NN
38536	473	6	are	be	VBP
38536	473	7	a	a	DT
38536	473	8	,	,	,
38536	473	9	b	b	NN
38536	473	10	,	,	,
38536	473	11	c.	c.	NN
38536	473	12	Calculate	calculate	VB
38536	473	13	the	the	DT
38536	473	14	radii	radii	NN
38536	473	15	of	of	IN
38536	473	16	the	the	DT
38536	473	17	three	three	CD
38536	473	18	circles	circle	NNS
38536	473	19	having	have	VBG
38536	473	20	the	the	DT
38536	473	21	vertices	vertex	NNS
38536	473	22	as	as	IN
38536	473	23	centers	center	NNS
38536	473	24	,	,	,
38536	473	25	each	each	DT
38536	473	26	being	be	VBG
38536	473	27	tangent	tangent	NN
38536	473	28	externally	externally	RB
38536	473	29	to	to	IN
38536	473	30	the	the	DT
38536	473	31	other	other	JJ
38536	473	32	two	two	CD
38536	473	33	.	.	.
38536	474	1	(	(	-LRB-
38536	474	2	_	_	NNP
38536	474	3	Harvard	Harvard	NNP
38536	474	4	.	.	.
38536	474	5	_	_	NNP
38536	474	6	)	)	-RRB-
38536	474	7	6	6	CD
38536	474	8	.	.	.
38536	475	1	Solve	solve	VB
38536	475	2	{	{	-LRB-
38536	475	3	2x	2x	CD
38536	475	4	+	+	SYM
38536	475	5	3y	3y	NNS
38536	475	6	=	=	SYM
38536	475	7	7	7	CD
38536	475	8	,	,	,
38536	475	9	x	x	NNP
38536	475	10	-	-	NN
38536	475	11	y	y	NN
38536	475	12	=	=	NN
38536	475	13	1	1	CD
38536	475	14	}	}	-RRB-
38536	475	15	graphically	graphically	RB
38536	475	16	;	;	:
38536	475	17	then	then	RB
38536	475	18	solve	solve	VB
38536	475	19	algebraically	algebraically	RB
38536	475	20	and	and	CC
38536	475	21	compare	compare	VB
38536	475	22	results	result	NNS
38536	475	23	.	.	.
38536	476	1	(	(	-LRB-
38536	476	2	Use	use	VB
38536	476	3	coördinate	coördinate	NN
38536	476	4	or	or	CC
38536	476	5	squared	squared	JJ
38536	476	6	paper	paper	NN
38536	476	7	.	.	.
38536	476	8	)	)	-RRB-
38536	477	1	Factor	factor	NN
38536	477	2	:	:	:
38536	477	3	7	7	CD
38536	477	4	.	.	.
38536	477	5	x^4	x^4	NNP
38536	477	6	+	+	SYM
38536	477	7	4	4	CD
38536	477	8	.	.	.
38536	478	1	8	8	LS
38536	478	2	.	.	.
38536	479	1	2d^(10	2d^(10	CD
38536	479	2	)	)	-RRB-
38536	479	3	-	-	:
38536	479	4	1024d	1024d	CD
38536	479	5	.	.	.
38536	480	1	9	9	CD
38536	480	2	.	.	.
38536	481	1	2(x^3	2(x^3	NNP
38536	481	2	-	-	SYM
38536	481	3	1	1	CD
38536	481	4	)	)	-RRB-
38536	481	5	-	-	:
38536	481	6	7(x^2	7(x^2	NNP
38536	481	7	-	-	HYPH
38536	481	8	1	1	CD
38536	481	9	)	)	-RRB-
38536	481	10	.	.	.
38536	482	1	~References:~	~References:~	NNP
38536	482	2	The	the	DT
38536	482	3	chapters	chapter	NNS
38536	482	4	on	on	IN
38536	482	5	Simultaneous	Simultaneous	NNP
38536	482	6	Equations	Equations	NNPS
38536	482	7	and	and	CC
38536	482	8	Graphs	Graphs	NNPS
38536	482	9	in	in	IN
38536	482	10	any	any	DT
38536	482	11	algebra	algebra	NN
38536	482	12	.	.	.
38536	483	1	SIMULTANEOUS	SIMULTANEOUS	NNP
38536	483	2	EQUATIONS	EQUATIONS	NNP
38536	483	3	AND	and	CC
38536	483	4	INVOLUTION	involution	NN
38536	483	5	1	1	CD
38536	483	6	.	.	.
38536	484	1	Solve	solve	VB
38536	484	2	(	(	-LRB-
38536	484	3	3/4)x	3/4)x	NN
38536	484	4	-	-	,
38536	484	5	(	(	-LRB-
38536	484	6	5/3)y	5/3)y	CD
38536	484	7	=	=	SYM
38536	484	8	11	11	CD
38536	484	9	-	-	SYM
38536	484	10	1/2	1/2	CD
38536	484	11	,	,	,
38536	484	12	(	(	-LRB-
38536	484	13	5/8)x	5/8)x	CD
38536	484	14	-	-	HYPH
38536	484	15	(	(	-LRB-
38536	484	16	3/2)y	3/2)y	CD
38536	484	17	=	=	SYM
38536	484	18	10	10	CD
38536	484	19	-	-	SYM
38536	484	20	1/4	1/4	CD
38536	484	21	.	.	.
38536	485	1	Look	look	VB
38536	485	2	up	up	RP
38536	485	3	the	the	DT
38536	485	4	method	method	NN
38536	485	5	of	of	IN
38536	485	6	solving	solve	VBG
38536	485	7	when	when	WRB
38536	485	8	the	the	DT
38536	485	9	unknowns	unknown	NNS
38536	485	10	are	be	VBP
38536	485	11	in	in	IN
38536	485	12	the	the	DT
38536	485	13	denominator	denominator	NN
38536	485	14	.	.	.
38536	486	1	Should	Should	MD
38536	486	2	you	-PRON-	PRP
38536	486	3	clear	clear	JJ
38536	486	4	of	of	IN
38536	486	5	fractions	fraction	NNS
38536	486	6	?	?	.
38536	487	1	2	2	LS
38536	487	2	.	.	.
38536	488	1	Solve	solve	VB
38536	488	2	1	1	CD
38536	488	3	/	/	SYM
38536	488	4	x	x	SYM
38536	488	5	-	-	SYM
38536	488	6	1	1	CD
38536	488	7	/	/	SYM
38536	488	8	y	y	NNP
38536	488	9	-	-	HYPH
38536	488	10	1	1	NNP
38536	488	11	/	/	SYM
38536	488	12	z	z	NN
38536	488	13	=	=	SYM
38536	488	14	1	1	CD
38536	488	15	/	/	SYM
38536	488	16	a	a	DT
38536	488	17	,	,	,
38536	488	18	1	1	CD
38536	488	19	/	/	SYM
38536	488	20	y	y	NNP
38536	488	21	-	-	HYPH
38536	488	22	1	1	NNP
38536	488	23	/	/	SYM
38536	488	24	z	z	NNP
38536	488	25	-	-	HYPH
38536	488	26	1	1	CD
38536	488	27	/	/	SYM
38536	488	28	x	x	NNS
38536	488	29	=	=	SYM
38536	488	30	1	1	CD
38536	488	31	/	/	SYM
38536	488	32	b	b	NN
38536	488	33	,	,	,
38536	488	34	1	1	CD
38536	488	35	/	/	SYM
38536	488	36	z	z	NNP
38536	488	37	-	-	HYPH
38536	488	38	1	1	CD
38536	488	39	/	/	SYM
38536	488	40	x	x	SYM
38536	488	41	-	-	SYM
38536	488	42	1	1	CD
38536	488	43	/	/	SYM
38536	488	44	y	y	NNP
38536	488	45	=	=	SYM
38536	488	46	1	1	CD
38536	488	47	/	/	SYM
38536	488	48	c	c	NN
38536	488	49	.	.	.
38536	489	1	3	3	LS
38536	489	2	.	.	.
38536	490	1	Solve	solve	VB
38536	490	2	graphically	graphically	RB
38536	490	3	and	and	CC
38536	490	4	algebraically	algebraically	RB
38536	490	5	2x	2x	NNP
38536	490	6	-	-	HYPH
38536	490	7	y	y	NN
38536	490	8	=	=	SYM
38536	490	9	4	4	CD
38536	490	10	,	,	,
38536	490	11	2x	2x	CD
38536	490	12	+	+	SYM
38536	490	13	3y	3y	CD
38536	490	14	=	=	SYM
38536	490	15	12	12	CD
38536	490	16	.	.	.
38536	491	1	4	4	LS
38536	491	2	.	.	.
38536	492	1	Solve	solve	VB
38536	492	2	graphically	graphically	RB
38536	492	3	and	and	CC
38536	492	4	algebraically	algebraically	RB
38536	492	5	3x	3x	CD
38536	492	6	+	+	SYM
38536	492	7	7y	7y	NNS
38536	492	8	=	=	SYM
38536	492	9	5	5	CD
38536	492	10	,	,	,
38536	492	11	8x	8x	CD
38536	492	12	+	+	SYM
38536	492	13	3y	3y	NNS
38536	492	14	=	=	SYM
38536	492	15	-18	-18	NNP
38536	492	16	.	.	.
38536	492	17	Review	review	NN
38536	492	18	:	:	:
38536	492	19	5	5	CD
38536	492	20	.	.	.
38536	493	1	The	the	DT
38536	493	2	squares	square	NNS
38536	493	3	of	of	IN
38536	493	4	the	the	DT
38536	493	5	numbers	number	NNS
38536	493	6	from	from	IN
38536	493	7	1	1	CD
38536	493	8	to	to	IN
38536	493	9	25	25	CD
38536	493	10	.	.	.
38536	494	1	6	6	CD
38536	494	2	.	.	.
38536	495	1	The	the	DT
38536	495	2	cubes	cube	NNS
38536	495	3	of	of	IN
38536	495	4	the	the	DT
38536	495	5	numbers	number	NNS
38536	495	6	from	from	IN
38536	495	7	1	1	CD
38536	495	8	to	to	IN
38536	495	9	12	12	CD
38536	495	10	.	.	.
38536	496	1	7	7	LS
38536	496	2	.	.	.
38536	497	1	The	the	DT
38536	497	2	fourth	fourth	JJ
38536	497	3	powers	power	NNS
38536	497	4	of	of	IN
38536	497	5	the	the	DT
38536	497	6	numbers	number	NNS
38536	497	7	from	from	IN
38536	497	8	1	1	CD
38536	497	9	to	to	IN
38536	497	10	5	5	CD
38536	497	11	.	.	.
38536	498	1	8	8	LS
38536	498	2	.	.	.
38536	499	1	The	the	DT
38536	499	2	fifth	fifth	JJ
38536	499	3	powers	power	NNS
38536	499	4	of	of	IN
38536	499	5	the	the	DT
38536	499	6	numbers	number	NNS
38536	499	7	from	from	IN
38536	499	8	1	1	CD
38536	499	9	to	to	IN
38536	499	10	3	3	CD
38536	499	11	.	.	.
38536	500	1	9	9	CD
38536	500	2	.	.	.
38536	501	1	The	the	DT
38536	501	2	binomial	binomial	JJ
38536	501	3	theorem	theorem	NN
38536	501	4	laws	law	NNS
38536	501	5	.	.	.
38536	502	1	(	(	-LRB-
38536	502	2	See	see	VB
38536	502	3	Involution	involution	NN
38536	502	4	.	.	.
38536	502	5	)	)	-RRB-
38536	503	1	Expand	expand	VB
38536	503	2	:	:	:
38536	503	3	(	(	-LRB-
38536	503	4	Indicate	indicate	VB
38536	503	5	first	first	RB
38536	503	6	,	,	,
38536	503	7	then	then	RB
38536	503	8	reduce	reduce	VB
38536	503	9	.	.	.
38536	503	10	)	)	-RRB-
38536	504	1	10	10	CD
38536	504	2	.	.	.
38536	505	1	(	(	-LRB-
38536	505	2	b	b	NN
38536	505	3	+	+	CD
38536	505	4	y)^7	y)^7	NNS
38536	505	5	.	.	.
38536	506	1	11	11	CD
38536	506	2	.	.	.
38536	507	1	[	[	-LRB-
38536	507	2	(	(	-LRB-
38536	507	3	2a)/3	2a)/3	CD
38536	507	4	-	-	SYM
38536	507	5	1]^5	1]^5	CD
38536	507	6	.	.	.
38536	508	1	12	12	CD
38536	508	2	.	.	.
38536	509	1	(	(	-LRB-
38536	509	2	x^2	x^2	NNS
38536	509	3	+	+	SYM
38536	509	4	2a)^5	2a)^5	CD
38536	509	5	.	.	.
38536	510	1	13	13	CD
38536	510	2	.	.	.
38536	511	1	(	(	-LRB-
38536	511	2	x	x	NNP
38536	511	3	-	-	NN
38536	511	4	y	y	NNP
38536	511	5	+	+	CC
38536	511	6	2z)^3	2z)^3	CD
38536	511	7	.	.	.
38536	512	1	14	14	CD
38536	512	2	.	.	.
38536	513	1	A	a	DT
38536	513	2	train	train	NN
38536	513	3	lost	lose	VBD
38536	513	4	one	one	CD
38536	513	5	sixth	sixth	NN
38536	513	6	of	of	IN
38536	513	7	its	-PRON-	PRP$
38536	513	8	passengers	passenger	NNS
38536	513	9	at	at	IN
38536	513	10	the	the	DT
38536	513	11	first	first	JJ
38536	513	12	stop	stop	NN
38536	513	13	,	,	,
38536	513	14	25	25	CD
38536	513	15	at	at	IN
38536	513	16	the	the	DT
38536	513	17	second	second	JJ
38536	513	18	stop	stop	NN
38536	513	19	,	,	,
38536	513	20	20	20	CD
38536	513	21	%	%	NN
38536	513	22	of	of	IN
38536	513	23	the	the	DT
38536	513	24	remainder	remainder	NN
38536	513	25	at	at	IN
38536	513	26	the	the	DT
38536	513	27	third	third	JJ
38536	513	28	stop	stop	NN
38536	513	29	,	,	,
38536	513	30	three	three	CD
38536	513	31	quarters	quarter	NNS
38536	513	32	of	of	IN
38536	513	33	the	the	DT
38536	513	34	remainder	remainder	NN
38536	513	35	at	at	IN
38536	513	36	the	the	DT
38536	513	37	fourth	fourth	JJ
38536	513	38	stop	stop	NN
38536	513	39	;	;	:
38536	513	40	25	25	CD
38536	513	41	remain	remain	VBP
38536	513	42	.	.	.
38536	514	1	What	what	WP
38536	514	2	was	be	VBD
38536	514	3	the	the	DT
38536	514	4	original	original	JJ
38536	514	5	number	number	NN
38536	514	6	?	?	.
38536	515	1	(	(	-LRB-
38536	515	2	_	_	NNP
38536	515	3	M.	M.	NNP
38536	516	1	I.	I.	NNP
38536	516	2	T.	T.	NNP
38536	516	3	_	_	NNP
38536	516	4	)	)	-RRB-
38536	516	5	~References:~	~References:~	NNP
38536	516	6	The	the	DT
38536	516	7	chapter	chapter	NN
38536	516	8	on	on	IN
38536	516	9	Involution	involution	NN
38536	516	10	in	in	IN
38536	516	11	any	any	DT
38536	516	12	algebra	algebra	NN
38536	516	13	.	.	.
38536	517	1	Also	also	RB
38536	517	2	the	the	DT
38536	517	3	references	reference	NNS
38536	517	4	on	on	IN
38536	517	5	the	the	DT
38536	517	6	preceding	precede	VBG
38536	517	7	page	page	NN
38536	517	8	.	.	.
38536	518	1	SQUARE	SQUARE	NNP
38536	518	2	ROOT	ROOT	NNS
38536	518	3	Find	find	VBP
38536	518	4	the	the	DT
38536	518	5	square	square	JJ
38536	518	6	root	root	NN
38536	518	7	of	of	IN
38536	518	8	:	:	:
38536	518	9	1	1	CD
38536	518	10	.	.	.
38536	519	1	1	1	CD
38536	519	2	+	+	SYM
38536	519	3	16m^6	16m^6	CD
38536	519	4	-	-	HYPH
38536	519	5	40m^4	40m^4	CD
38536	519	6	+	+	SYM
38536	519	7	10	10	CD
38536	519	8	m	m	CD
38536	519	9	-	-	HYPH
38536	519	10	8m^3	8m^3	CD
38536	519	11	+	+	SYM
38536	519	12	25m^2	25m^2	CD
38536	519	13	.	.	.
38536	520	1	2	2	LS
38536	520	2	.	.	.
38536	521	1	(	(	-LRB-
38536	521	2	a^2)/(x^2	a^2)/(x^2	NN
38536	521	3	)	)	-RRB-
38536	521	4	+	+	CC
38536	521	5	(	(	-LRB-
38536	521	6	6a)/x	6a)/x	CD
38536	521	7	+	+	SYM
38536	521	8	11	11	CD
38536	521	9	+	+	SYM
38536	521	10	(	(	-LRB-
38536	521	11	6x)/a	6x)/a	CD
38536	521	12	+	+	SYM
38536	521	13	(	(	-LRB-
38536	521	14	x^2)/(a^2	x^2)/(a^2	NNP
38536	521	15	)	)	-RRB-
38536	521	16	.	.	.
38536	522	1	3	3	LS
38536	522	2	.	.	.
38536	523	1	Find	find	VB
38536	523	2	the	the	DT
38536	523	3	square	square	JJ
38536	523	4	root	root	NN
38536	523	5	to	to	IN
38536	523	6	three	three	CD
38536	523	7	terms	term	NNS
38536	523	8	of	of	IN
38536	523	9	x^2	x^2	NNS
38536	523	10	+	+	SYM
38536	523	11	5	5	CD
38536	523	12	.	.	.
38536	524	1	4	4	LS
38536	524	2	.	.	.
38536	525	1	Find	find	VB
38536	525	2	the	the	DT
38536	525	3	square	square	JJ
38536	525	4	root	root	NN
38536	525	5	of	of	IN
38536	525	6	337,561	337,561	CD
38536	525	7	.	.	.
38536	526	1	5	5	CD
38536	526	2	.	.	.
38536	527	1	Find	find	VB
38536	527	2	the	the	DT
38536	527	3	square	square	JJ
38536	527	4	root	root	NN
38536	527	5	of	of	IN
38536	527	6	1823.29	1823.29	CD
38536	527	7	.	.	.
38536	528	1	6	6	CD
38536	528	2	.	.	.
38536	529	1	Find	find	VB
38536	529	2	to	to	TO
38536	529	3	four	four	CD
38536	529	4	decimal	decimal	JJ
38536	529	5	places	place	NNS
38536	529	6	the	the	DT
38536	529	7	square	square	JJ
38536	529	8	root	root	NN
38536	529	9	of	of	IN
38536	529	10	1.672	1.672	CD
38536	529	11	.	.	.
38536	530	1	(	(	-LRB-
38536	530	2	_	_	NNP
38536	530	3	Princeton	Princeton	NNP
38536	530	4	.	.	.
38536	530	5	_	_	NNP
38536	530	6	)	)	-RRB-
38536	530	7	7	7	CD
38536	530	8	.	.	.
38536	531	1	Add	Add	NNP
38536	531	2	2/[(x	2/[(x	CD
38536	531	3	-	-	:
38536	531	4	1)^3	1)^3	NN
38536	531	5	]	]	-RRB-
38536	531	6	+	+	CC
38536	531	7	1/[(1	1/[(1	CD
38536	531	8	-	-	HYPH
38536	531	9	x)^2	x)^2	NNP
38536	531	10	]	]	-RRB-
38536	531	11	-	-	:
38536	531	12	2/(1	2/(1	CD
38536	531	13	-	-	HYPH
38536	531	14	x	x	NNP
38536	531	15	)	)	-RRB-
38536	531	16	-	-	:
38536	531	17	1	1	CD
38536	531	18	/	/	SYM
38536	531	19	x	x	NNS
38536	531	20	.	.	.
38536	532	1	8	8	LS
38536	532	2	.	.	.
38536	533	1	Find	find	VB
38536	533	2	the	the	DT
38536	533	3	value	value	NN
38536	533	4	of	of	IN
38536	533	5	:	:	:
38536	533	6	(	(	-LRB-
38536	533	7	64^(1/3	64^(1/3	NN
38536	533	8	)	)	-RRB-
38536	533	9	·	·	NFP
38536	533	10	12)/24	12)/24	CD
38536	533	11	÷	÷	NN
38536	533	12	2	2	CD
38536	533	13	×	×	CD
38536	533	14	3	3	CD
38536	533	15	-	-	HYPH
38536	533	16	(	(	-LRB-
38536	533	17	2	2	CD
38536	533	18	·	·	SYM
38536	533	19	7	7	CD
38536	533	20	^	^	.
38536	533	21	2)/(14	2)/(14	NNP
38536	533	22	)	)	-RRB-
38536	533	23	÷	÷	NN
38536	533	24	7	7	CD
38536	533	25	×	×	NN
38536	533	26	1	1	CD
38536	533	27	+	+	SYM
38536	533	28	(	(	-LRB-
38536	533	29	1^(1/3	1^(1/3	CD
38536	533	30	)	)	-RRB-
38536	533	31	·	·	NFP
38536	533	32	1	1	CD
38536	533	33	^	^	NN
38536	533	34	7)/(1	7)/(1	NN
38536	533	35	·	·	NFP
38536	533	36	1	1	CD
38536	533	37	^	^	SYM
38536	533	38	2	2	CD
38536	533	39	)	)	-RRB-
38536	533	40	-	-	SYM
38536	533	41	4	4	CD
38536	533	42	·	·	NFP
38536	533	43	0	0	NFP
38536	533	44	.	.	.
38536	534	1	9	9	CD
38536	534	2	.	.	.
38536	535	1	Simplify	Simplify	NNP
38536	535	2	[	[	-LRB-
38536	535	3	(	(	-LRB-
38536	535	4	x	x	NN
38536	535	5	+	+	SYM
38536	535	6	y)^5	y)^5	XX
38536	535	7	+	+	SYM
38536	535	8	(	(	-LRB-
38536	535	9	x	x	NN
38536	535	10	-	-	NNS
38536	535	11	y)^5][(x	y)^5][(x	NN
38536	535	12	+	+	SYM
38536	535	13	y)^5	y)^5	NN
38536	535	14	-	-	,
38536	535	15	(	(	-LRB-
38536	535	16	x	x	NN
38536	535	17	-	-	SYM
38536	535	18	y)^5	y)^5	NNP
38536	535	19	]	]	-RRB-
38536	535	20	.	.	.
38536	536	1	10	10	CD
38536	536	2	.	.	.
38536	537	1	Solve	solve	VB
38536	537	2	by	by	IN
38536	537	3	the	the	DT
38536	537	4	short	short	JJ
38536	537	5	method	method	NN
38536	537	6	:	:	:
38536	537	7	5/(7	5/(7	CD
38536	537	8	-	-	HYPH
38536	537	9	x	x	NN
38536	537	10	)	)	-RRB-
38536	537	11	-	-	,
38536	537	12	[	[	-LRB-
38536	537	13	(	(	-LRB-
38536	537	14	2	2	CD
38536	537	15	-	-	HYPH
38536	537	16	1/4)x	1/4)x	CD
38536	537	17	-	-	HYPH
38536	537	18	3]/4	3]/4	CD
38536	537	19	-	-	HYPH
38536	537	20	(	(	-LRB-
38536	537	21	x	x	NNS
38536	537	22	+	+	SYM
38536	537	23	11)/8	11)/8	CD
38536	537	24	+	+	SYM
38536	537	25	(	(	-LRB-
38536	537	26	11x	11x	NNS
38536	537	27	+	+	SYM
38536	537	28	5)/16	5)/16	CD
38536	537	29	=	=	SYM
38536	537	30	0	0	NFP
38536	537	31	.	.	.
38536	538	1	11	11	CD
38536	538	2	.	.	.
38536	539	1	It	-PRON-	PRP
38536	539	2	takes	take	VBZ
38536	539	3	3/4	3/4	CD
38536	539	4	of	of	IN
38536	539	5	a	a	DT
38536	539	6	second	second	NN
38536	539	7	for	for	IN
38536	539	8	a	a	DT
38536	539	9	ball	ball	NN
38536	539	10	to	to	TO
38536	539	11	go	go	VB
38536	539	12	from	from	IN
38536	539	13	the	the	DT
38536	539	14	pitcher	pitcher	NN
38536	539	15	to	to	IN
38536	539	16	the	the	DT
38536	539	17	catcher	catcher	NN
38536	539	18	,	,	,
38536	539	19	and	and	CC
38536	539	20	1/2	1/2	CD
38536	539	21	of	of	IN
38536	539	22	a	a	DT
38536	539	23	second	second	JJ
38536	539	24	for	for	IN
38536	539	25	the	the	DT
38536	539	26	catcher	catcher	NN
38536	539	27	to	to	TO
38536	539	28	handle	handle	VB
38536	539	29	it	-PRON-	PRP
38536	539	30	and	and	CC
38536	539	31	get	get	VB
38536	539	32	off	off	RP
38536	539	33	a	a	DT
38536	539	34	throw	throw	NN
38536	539	35	to	to	IN
38536	539	36	second	second	JJ
38536	539	37	base	base	NN
38536	539	38	.	.	.
38536	540	1	It	-PRON-	PRP
38536	540	2	is	be	VBZ
38536	540	3	90	90	CD
38536	540	4	feet	foot	NNS
38536	540	5	from	from	IN
38536	540	6	first	first	JJ
38536	540	7	base	base	NN
38536	540	8	to	to	IN
38536	540	9	second	second	JJ
38536	540	10	,	,	,
38536	540	11	and	and	CC
38536	540	12	130	130	CD
38536	540	13	feet	foot	NNS
38536	540	14	from	from	IN
38536	540	15	the	the	DT
38536	540	16	catcher	catcher	NN
38536	540	17	's	's	POS
38536	540	18	position	position	NN
38536	540	19	to	to	IN
38536	540	20	second	second	JJ
38536	540	21	.	.	.
38536	541	1	A	a	DT
38536	541	2	runner	runner	NN
38536	541	3	stealing	steal	VBG
38536	541	4	second	second	JJ
38536	541	5	has	have	VBZ
38536	541	6	a	a	DT
38536	541	7	start	start	NN
38536	541	8	of	of	IN
38536	541	9	13	13	CD
38536	541	10	feet	foot	NNS
38536	541	11	when	when	WRB
38536	541	12	the	the	DT
38536	541	13	ball	ball	NN
38536	541	14	leaves	leave	VBZ
38536	541	15	the	the	DT
38536	541	16	pitcher	pitcher	NN
38536	541	17	's	's	POS
38536	541	18	hand	hand	NN
38536	541	19	,	,	,
38536	541	20	and	and	CC
38536	541	21	beats	beat	VBZ
38536	541	22	the	the	DT
38536	541	23	throw	throw	NN
38536	541	24	to	to	IN
38536	541	25	the	the	DT
38536	541	26	base	base	NN
38536	541	27	by	by	IN
38536	541	28	1/8	1/8	CD
38536	541	29	of	of	IN
38536	541	30	a	a	DT
38536	541	31	second	second	NN
38536	541	32	.	.	.
38536	542	1	The	the	DT
38536	542	2	next	next	JJ
38536	542	3	time	time	NN
38536	542	4	he	-PRON-	PRP
38536	542	5	tries	try	VBZ
38536	542	6	it	-PRON-	PRP
38536	542	7	,	,	,
38536	542	8	he	-PRON-	PRP
38536	542	9	gets	get	VBZ
38536	542	10	a	a	DT
38536	542	11	start	start	NN
38536	542	12	of	of	IN
38536	542	13	only	only	JJ
38536	542	14	3	3	CD
38536	542	15	-	-	SYM
38536	542	16	1/2	1/2	CD
38536	542	17	feet	foot	NNS
38536	542	18	,	,	,
38536	542	19	and	and	CC
38536	542	20	is	be	VBZ
38536	542	21	caught	catch	VBN
38536	542	22	by	by	IN
38536	542	23	6	6	CD
38536	542	24	feet	foot	NNS
38536	542	25	.	.	.
38536	543	1	What	what	WP
38536	543	2	is	be	VBZ
38536	543	3	his	-PRON-	PRP$
38536	543	4	rate	rate	NN
38536	543	5	of	of	IN
38536	543	6	running	running	NN
38536	543	7	,	,	,
38536	543	8	and	and	CC
38536	543	9	the	the	DT
38536	543	10	velocity	velocity	NN
38536	543	11	of	of	IN
38536	543	12	the	the	DT
38536	543	13	catcher	catcher	NN
38536	543	14	's	's	POS
38536	543	15	throw	throw	NN
38536	543	16	?	?	.
38536	544	1	(	(	-LRB-
38536	544	2	_	_	NNP
38536	544	3	Cornell	Cornell	NNP
38536	544	4	.	.	.
38536	544	5	_	_	NNP
38536	544	6	)	)	-RRB-
38536	544	7	~Reference:~	~Reference:~	NNP
38536	544	8	The	the	DT
38536	544	9	chapter	chapter	NN
38536	544	10	on	on	IN
38536	544	11	Square	Square	NNP
38536	544	12	Root	Root	NNP
38536	544	13	in	in	IN
38536	544	14	any	any	DT
38536	544	15	algebra	algebra	NN
38536	544	16	.	.	.
38536	545	1	THEORY	THEORY	NNP
38536	545	2	OF	of	IN
38536	545	3	EXPONENTS	exponent	NNS
38536	545	4	Review	review	VBP
38536	545	5	the	the	DT
38536	545	6	proofs	proof	NNS
38536	545	7	,	,	,
38536	545	8	for	for	IN
38536	545	9	positive	positive	JJ
38536	545	10	integral	integral	JJ
38536	545	11	exponents	exponent	NNS
38536	545	12	,	,	,
38536	545	13	of	of	IN
38536	545	14	:	:	:
38536	545	15	I.	I.	NNP
38536	545	16	a^m	a^m	NNP
38536	545	17	×	×	NN
38536	545	18	a^n	a^n	UH
38536	545	19	=	=	NN
38536	545	20	a^(m	a^(m	NN
38536	545	21	+	+	SYM
38536	545	22	n	n	NN
38536	545	23	)	)	-RRB-
38536	545	24	.	.	.
38536	546	1	II	ii	CD
38536	546	2	.	.	.
38536	547	1	(	(	-LRB-
38536	547	2	a^m)/(a^n	a^m)/(a^n	NN
38536	547	3	)	)	-RRB-
38536	547	4	=	=	SYM
38536	547	5	a^(m	a^(m	CD
38536	547	6	-	-	HYPH
38536	547	7	n	n	NN
38536	547	8	)	)	-RRB-
38536	547	9	.	.	.
38536	548	1	III	iii	CD
38536	548	2	.	.	.
38536	549	1	(	(	-LRB-
38536	549	2	a^m)^n	a^m)^n	NNP
38536	549	3	=	=	NFP
38536	549	4	a^(mn	a^(mn	NNP
38536	549	5	)	)	-RRB-
38536	549	6	.	.	.
38536	550	1	IV	IV	NNP
38536	550	2	.	.	.
38536	551	1	[	[	-LRB-
38536	551	2	a^(mn)]^(1	a^(mn)]^(1	NNP
38536	551	3	/	/	SYM
38536	551	4	n	n	NN
38536	551	5	)	)	-RRB-
38536	551	6	=	=	NFP
38536	551	7	a^m	a^m	NNP
38536	551	8	.	.	.
38536	552	1	V.	V.	NNP
38536	552	2	[	[	-LRB-
38536	552	3	a	a	DT
38536	552	4	/	/	SYM
38536	552	5	b]^n	b]^n	NN
38536	552	6	=	=	NFP
38536	552	7	(	(	-LRB-
38536	552	8	a^n)/(b^n	a^n)/(b^n	NN
38536	552	9	)	)	-RRB-
38536	552	10	.	.	.
38536	553	1	VI	VI	NNP
38536	553	2	.	.	.
38536	554	1	(	(	-LRB-
38536	554	2	abc)^n	abc)^n	ADD
38536	554	3	=	=	SYM
38536	554	4	a^n	a^n	NNP
38536	554	5	b^n	b^n	NN
38536	554	6	c^n	c^n	NNS
38536	554	7	.	.	.
38536	555	1	~To	~To	NFP
38536	555	2	find	find	VB
38536	555	3	the	the	DT
38536	555	4	meaning	meaning	NN
38536	555	5	of	of	IN
38536	555	6	a	a	DT
38536	555	7	fractional	fractional	JJ
38536	555	8	exponent.~	exponent.~	NNP
38536	555	9	Assume	assume	VB
38536	555	10	that	that	IN
38536	555	11	Law	Law	NNP
38536	555	12	I	-PRON-	PRP
38536	555	13	holds	hold	VBZ
38536	555	14	for	for	IN
38536	555	15	_	_	NNP
38536	555	16	all	all	DT
38536	555	17	_	_	NNP
38536	555	18	exponents	exponent	NNS
38536	555	19	.	.	.
38536	556	1	If	if	IN
38536	556	2	so	so	RB
38536	556	3	,	,	,
38536	556	4	a^(2/3	a^(2/3	NNP
38536	556	5	)	)	-RRB-
38536	556	6	·	·	NFP
38536	556	7	a^(2/3	a^(2/3	NNP
38536	556	8	)	)	-RRB-
38536	556	9	·	·	NFP
38536	556	10	a^(2/3	a^(2/3	NNP
38536	556	11	)	)	-RRB-
38536	556	12	=	=	SYM
38536	556	13	a^(6/3	a^(6/3	NNP
38536	556	14	)	)	-RRB-
38536	556	15	=	=	NFP
38536	556	16	a^2	a^2	ADD
38536	556	17	.	.	.
38536	557	1	Hence	hence	RB
38536	557	2	,	,	,
38536	557	3	a^(2/3	a^(2/3	NNP
38536	557	4	)	)	-RRB-
38536	557	5	is	be	VBZ
38536	557	6	_	_	NNP
38536	557	7	one	one	CD
38536	557	8	of	of	IN
38536	557	9	the	the	DT
38536	557	10	three	three	CD
38536	557	11	equal	equal	JJ
38536	557	12	factors	factor	NNS
38536	557	13	_	_	NNP
38536	557	14	(	(	-LRB-
38536	557	15	hence	hence	RB
38536	557	16	the	the	DT
38536	557	17	cube	cube	NNP
38536	557	18	root	root	NN
38536	557	19	)	)	-RRB-
38536	557	20	of	of	IN
38536	557	21	a^2	a^2	PRP
38536	557	22	.	.	.
38536	558	1	Therefore	therefore	RB
38536	558	2	a^(2/3	a^(2/3	NNP
38536	558	3	)	)	-RRB-
38536	558	4	=	=	NFP
38536	558	5	[	[	-LRB-
38536	558	6	a^2]^(1/3	a^2]^(1/3	CD
38536	558	7	)	)	-RRB-
38536	558	8	.	.	.
38536	559	1	In	in	IN
38536	559	2	the	the	DT
38536	559	3	same	same	JJ
38536	559	4	way	way	NN
38536	559	5	,	,	,
38536	559	6	a^(4/5	a^(4/5	NNP
38536	559	7	)	)	-RRB-
38536	559	8	·	·	NFP
38536	559	9	a^(4/5	a^(4/5	NNP
38536	559	10	)	)	-RRB-
38536	559	11	·	·	NFP
38536	559	12	a^(4/5	a^(4/5	NNP
38536	559	13	)	)	-RRB-
38536	559	14	·	·	NFP
38536	559	15	a^(4/5	a^(4/5	NNP
38536	559	16	)	)	-RRB-
38536	559	17	·	·	NFP
38536	559	18	a^(4/5	a^(4/5	NNP
38536	559	19	)	)	-RRB-
38536	559	20	=	=	SYM
38536	559	21	a^(20/5	a^(20/5	NNS
38536	559	22	)	)	-RRB-
38536	559	23	=	=	SYM
38536	559	24	a^4	a^4	XX
38536	559	25	.	.	.
38536	560	1	Hence	hence	RB
38536	560	2	,	,	,
38536	560	3	a^(4/5	a^(4/5	NNP
38536	560	4	)	)	-RRB-
38536	560	5	is	be	VBZ
38536	560	6	_	_	NNP
38536	560	7	one	one	CD
38536	560	8	of	of	IN
38536	560	9	the	the	DT
38536	560	10	five	five	CD
38536	560	11	equal	equal	JJ
38536	560	12	factors	factor	NNS
38536	560	13	_	_	NNP
38536	560	14	(	(	-LRB-
38536	560	15	hence	hence	RB
38536	560	16	the	the	DT
38536	560	17	fifth	fifth	JJ
38536	560	18	root	root	NN
38536	560	19	)	)	-RRB-
38536	560	20	of	of	IN
38536	560	21	a^4	a^4	NNS
38536	560	22	.	.	.
38536	561	1	Therefore	therefore	RB
38536	561	2	a^(4/5	a^(4/5	NNP
38536	561	3	)	)	-RRB-
38536	561	4	=	=	NFP
38536	561	5	[	[	-LRB-
38536	561	6	a^4]^(1/5	a^4]^(1/5	CD
38536	561	7	)	)	-RRB-
38536	561	8	.	.	.
38536	562	1	In	in	IN
38536	562	2	the	the	DT
38536	562	3	same	same	JJ
38536	562	4	way	way	NN
38536	562	5	,	,	,
38536	562	6	in	in	IN
38536	562	7	general	general	JJ
38536	562	8	,	,	,
38536	562	9	a^(p	a^(p	NNP
38536	562	10	/	/	SYM
38536	562	11	q	q	NNP
38536	562	12	)	)	-RRB-
38536	562	13	=	=	NFP
38536	562	14	[	[	-LRB-
38536	562	15	a^p]^(1	a^p]^(1	NNP
38536	562	16	/	/	SYM
38536	562	17	q	q	NNP
38536	562	18	)	)	-RRB-
38536	562	19	.	.	.
38536	563	1	Hence	hence	RB
38536	563	2	,	,	,
38536	563	3	_	_	NNP
38536	563	4	the	the	DT
38536	563	5	numerator	numerator	NN
38536	563	6	of	of	IN
38536	563	7	a	a	DT
38536	563	8	fractional	fractional	JJ
38536	563	9	exponent	exponent	NN
38536	563	10	indicates	indicate	VBZ
38536	563	11	the	the	DT
38536	563	12	power	power	NN
38536	563	13	,	,	,
38536	563	14	the	the	DT
38536	563	15	denominator	denominator	NN
38536	563	16	indicates	indicate	VBZ
38536	563	17	the	the	DT
38536	563	18	root	root	NN
38536	563	19	_	_	NNP
38536	563	20	.	.	.
38536	564	1	~To	~To	NFP
38536	564	2	find	find	VB
38536	564	3	the	the	DT
38536	564	4	meaning	meaning	NN
38536	564	5	of	of	IN
38536	564	6	a	a	DT
38536	564	7	zero	zero	CD
38536	564	8	exponent.~	exponent.~	NNP
38536	564	9	Assume	Assume	NNP
38536	564	10	that	that	IN
38536	564	11	Law	Law	NNP
38536	564	12	II	II	NNP
38536	564	13	holds	hold	VBZ
38536	564	14	for	for	IN
38536	564	15	_	_	NNP
38536	564	16	all	all	DT
38536	564	17	_	_	NNP
38536	564	18	exponents	exponent	NNS
38536	564	19	.	.	.
38536	565	1	If	if	IN
38536	565	2	so	so	RB
38536	565	3	,	,	,
38536	565	4	(	(	-LRB-
38536	565	5	a^m)/(a^m	a^m)/(a^m	NNP
38536	565	6	)	)	-RRB-
38536	565	7	=	=	SYM
38536	565	8	a^(m	a^(m	NNP
38536	565	9	-	-	HYPH
38536	565	10	m	m	NN
38536	565	11	)	)	-RRB-
38536	565	12	=	=	SYM
38536	565	13	a^0	a^0	ADD
38536	565	14	.	.	.
38536	566	1	But	but	CC
38536	566	2	by	by	IN
38536	566	3	division	division	NN
38536	566	4	,	,	,
38536	566	5	(	(	-LRB-
38536	566	6	a^m)/(a^m	a^m)/(a^m	NNP
38536	566	7	)	)	-RRB-
38536	566	8	=	=	SYM
38536	566	9	1	1	CD
38536	566	10	.	.	.
38536	567	1	Therefore	therefore	RB
38536	567	2	a^0	a^0	NNP
38536	567	3	=	=	SYM
38536	567	4	1	1	CD
38536	567	5	.	.	.
38536	568	1	Axiom	Axiom	NNP
38536	568	2	I.	i.	NN
38536	569	1	~To	~To	NFP
38536	569	2	find	find	VB
38536	569	3	the	the	DT
38536	569	4	meaning	meaning	NN
38536	569	5	of	of	IN
38536	569	6	a	a	DT
38536	569	7	negative	negative	JJ
38536	569	8	exponent.~	exponent.~	NNP
38536	569	9	Assume	assume	VB
38536	569	10	that	that	IN
38536	569	11	Law	Law	NNP
38536	569	12	I	-PRON-	PRP
38536	569	13	holds	hold	VBZ
38536	569	14	for	for	IN
38536	569	15	_	_	NNP
38536	569	16	all	all	DT
38536	569	17	_	_	NNP
38536	569	18	exponents	exponent	NNS
38536	569	19	.	.	.
38536	570	1	If	if	IN
38536	570	2	so	so	RB
38536	570	3	,	,	,
38536	570	4	a^m	a^m	NNP
38536	570	5	×	×	NN
38536	570	6	a^(-m	a^(-m	UH
38536	570	7	)	)	-RRB-
38536	570	8	=	=	NFP
38536	570	9	a^(m	a^(m	CD
38536	570	10	-	-	HYPH
38536	570	11	m	m	NN
38536	570	12	)	)	-RRB-
38536	570	13	=	=	NFP
38536	570	14	a^0	a^0	NNP
38536	570	15	=	=	SYM
38536	570	16	1	1	CD
38536	570	17	.	.	.
38536	571	1	Hence	hence	RB
38536	571	2	,	,	,
38536	571	3	a^m	a^m	NNP
38536	571	4	×	×	NN
38536	571	5	a^(-m	a^(-m	NN
38536	571	6	)	)	-RRB-
38536	571	7	=	=	NFP
38536	571	8	1	1	CD
38536	571	9	.	.	.
38536	572	1	Therefore	therefore	RB
38536	572	2	a^(-m	a^(-m	VBP
38536	572	3	)	)	-RRB-
38536	572	4	=	=	NFP
38536	572	5	1/(a^m	1/(a^m	NN
38536	572	6	)	)	-RRB-
38536	572	7	.	.	.
38536	573	1	Rules	rule	NNS
38536	573	2	:	:	:
38536	573	3	_	_	NNP
38536	573	4	To	to	TO
38536	573	5	multiply	multiply	VB
38536	573	6	quantities	quantity	NNS
38536	573	7	having	have	VBG
38536	573	8	the	the	DT
38536	573	9	same	same	JJ
38536	573	10	base	base	NN
38536	573	11	,	,	,
38536	573	12	add	add	VB
38536	573	13	exponents	exponent	NNS
38536	573	14	.	.	.
38536	573	15	_	_	NNP
38536	573	16	_	_	NNP
38536	573	17	To	to	TO
38536	573	18	divide	divide	VB
38536	573	19	quantities	quantity	NNS
38536	573	20	having	have	VBG
38536	573	21	the	the	DT
38536	573	22	same	same	JJ
38536	573	23	base	base	NN
38536	573	24	,	,	,
38536	573	25	subtract	subtract	NN
38536	573	26	exponents	exponent	NNS
38536	573	27	.	.	.
38536	573	28	_	_	NNP
38536	573	29	_	_	NNP
38536	573	30	To	to	TO
38536	573	31	raise	raise	VB
38536	573	32	a	a	DT
38536	573	33	quantity	quantity	NN
38536	573	34	to	to	IN
38536	573	35	a	a	DT
38536	573	36	power	power	NN
38536	573	37	,	,	,
38536	573	38	multiply	multiply	JJ
38536	573	39	exponents	exponent	NNS
38536	573	40	.	.	.
38536	573	41	_	_	NNP
38536	573	42	_	_	NNP
38536	573	43	To	to	TO
38536	573	44	extract	extract	VB
38536	573	45	a	a	DT
38536	573	46	root	root	NN
38536	573	47	,	,	,
38536	573	48	divide	divide	VB
38536	573	49	the	the	DT
38536	573	50	exponent	exponent	NN
38536	573	51	of	of	IN
38536	573	52	the	the	DT
38536	573	53	power	power	NN
38536	573	54	by	by	IN
38536	573	55	the	the	DT
38536	573	56	index	index	NN
38536	573	57	of	of	IN
38536	573	58	the	the	DT
38536	573	59	root	root	NN
38536	573	60	.	.	.
38536	573	61	_	_	NNP
38536	573	62	1	1	CD
38536	573	63	.	.	.
38536	574	1	Find	find	VB
38536	574	2	the	the	DT
38536	574	3	value	value	NN
38536	574	4	of	of	IN
38536	574	5	3	3	CD
38536	574	6	^	^	SYM
38536	574	7	2	2	CD
38536	574	8	-	-	SYM
38536	574	9	5	5	CD
38536	574	10	×	×	NN
38536	574	11	4	4	CD
38536	574	12	^	^	SYM
38536	574	13	0	0	CD
38536	574	14	+	+	CC
38536	574	15	8^(-2/3	8^(-2/3	CD
38536	574	16	)	)	-RRB-
38536	574	17	+	+	CC
38536	574	18	1^(2/5	1^(2/5	CD
38536	574	19	)	)	-RRB-
38536	574	20	.	.	.
38536	575	1	2	2	LS
38536	575	2	.	.	.
38536	576	1	Find	find	VB
38536	576	2	the	the	DT
38536	576	3	value	value	NN
38536	576	4	of	of	IN
38536	576	5	8^(-2/3	8^(-2/3	CD
38536	576	6	)	)	-RRB-
38536	576	7	+	+	CC
38536	576	8	9^(3/2	9^(3/2	CD
38536	576	9	)	)	-RRB-
38536	576	10	-	-	:
38536	576	11	2^(-2	2^(-2	LS
38536	576	12	)	)	-RRB-
38536	576	13	+	+	CC
38536	576	14	1^(-2/5	1^(-2/5	CD
38536	576	15	)	)	-RRB-
38536	576	16	-	-	SYM
38536	576	17	7	7	CD
38536	576	18	^	^	SYM
38536	576	19	0	0	CD
38536	576	20	.	.	.
38536	577	1	Give	give	VB
38536	577	2	the	the	DT
38536	577	3	value	value	NN
38536	577	4	of	of	IN
38536	577	5	each	each	DT
38536	577	6	of	of	IN
38536	577	7	the	the	DT
38536	577	8	following	following	NN
38536	577	9	:	:	:
38536	577	10	3	3	CD
38536	577	11	.	.	.
38536	578	1	(	(	-LRB-
38536	578	2	3	3	CD
38536	578	3	^	^	NN
38536	578	4	0)/5	0)/5	NN
38536	578	5	,	,	,
38536	578	6	3/(5	3/(5	CD
38536	578	7	^	^	FW
38536	578	8	0	0	NFP
38536	578	9	)	)	-RRB-
38536	578	10	,	,	,
38536	578	11	(	(	-LRB-
38536	578	12	3	3	CD
38536	578	13	^	^	SYM
38536	578	14	0)/(5	0)/(5	NN
38536	578	15	^	^	NN
38536	578	16	0	0	NFP
38536	578	17	)	)	-RRB-
38536	578	18	,	,	,
38536	578	19	3	3	CD
38536	578	20	^	^	NN
38536	578	21	0	0	CD
38536	578	22	×	×	NN
38536	578	23	5	5	CD
38536	578	24	,	,	,
38536	578	25	3	3	CD
38536	578	26	×	×	NN
38536	578	27	5	5	CD
38536	578	28	^	^	SYM
38536	578	29	0	0	CD
38536	578	30	,	,	,
38536	578	31	3	3	CD
38536	578	32	^	^	NN
38536	578	33	0	0	CD
38536	578	34	×	×	NN
38536	578	35	5	5	CD
38536	578	36	^	^	SYM
38536	578	37	0	0	CD
38536	578	38	,	,	,
38536	578	39	3	3	CD
38536	578	40	^	^	NN
38536	578	41	0	0	CD
38536	578	42	+	+	SYM
38536	578	43	5	5	CD
38536	578	44	^	^	SYM
38536	578	45	0	0	CD
38536	578	46	,	,	,
38536	578	47	3	3	CD
38536	578	48	^	^	SYM
38536	578	49	0	0	CD
38536	578	50	-	-	SYM
38536	578	51	5	5	CD
38536	578	52	^	^	SYM
38536	578	53	0	0	CD
38536	578	54	.	.	.
38536	579	1	4	4	LS
38536	579	2	.	.	.
38536	580	1	Express	Express	NNP
38536	580	2	7	7	CD
38536	580	3	^	^	SYM
38536	580	4	0	0	NFP
38536	580	5	as	as	IN
38536	580	6	some	some	DT
38536	580	7	power	power	NN
38536	580	8	of	of	IN
38536	580	9	7	7	CD
38536	580	10	divided	divide	VBN
38536	580	11	by	by	IN
38536	580	12	itself	-PRON-	PRP
38536	580	13	.	.	.
38536	581	1	Simplify	simplify	NN
38536	581	2	:	:	:
38536	581	3	5	5	CD
38536	581	4	.	.	.
38536	582	1	16^(1/3	16^(1/3	CD
38536	582	2	)	)	-RRB-
38536	582	3	·	·	NFP
38536	582	4	2^(1/2	2^(1/2	CD
38536	582	5	)	)	-RRB-
38536	582	6	·	·	NFP
38536	582	7	32^(5/6	32^(5/6	CD
38536	582	8	)	)	-RRB-
38536	582	9	.	.	.
38536	583	1	(	(	-LRB-
38536	583	2	Change	change	NN
38536	583	3	to	to	IN
38536	583	4	the	the	DT
38536	583	5	same	same	JJ
38536	583	6	base	base	NN
38536	583	7	first	first	RB
38536	583	8	.	.	.
38536	583	9	)	)	-RRB-
38536	584	1	6	6	CD
38536	584	2	.	.	.
38536	585	1	[	[	-LRB-
38536	585	2	2/(8^(-3))]^(1/5	2/(8^(-3))]^(1/5	CD
38536	585	3	)	)	-RRB-
38536	585	4	.	.	.
38536	586	1	7	7	LS
38536	586	2	.	.	.
38536	587	1	[	[	-LRB-
38536	587	2	(	(	-LRB-
38536	587	3	x^n)^(n	x^n)^(n	NN
38536	587	4	+	+	SYM
38536	587	5	2)]/[(x^(n	2)]/[(x^(n	CD
38536	587	6	+	+	SYM
38536	587	7	1))(x^(n	1))(x^(n	CD
38536	587	8	-	-	SYM
38536	587	9	1	1	CD
38536	587	10	)	)	-RRB-
38536	587	11	)	)	-RRB-
38536	587	12	]	]	-RRB-
38536	587	13	.	.	.
38536	588	1	8	8	LS
38536	588	2	.	.	.
38536	589	1	(	(	-LRB-
38536	589	2	x	x	NN
38536	589	3	+	+	SYM
38536	589	4	3x^(2/3	3x^(2/3	CD
38536	589	5	)	)	-RRB-
38536	589	6	-	-	:
38536	589	7	2x^(1/3))(3	2x^(1/3))(3	CD
38536	589	8	-	-	HYPH
38536	589	9	2x^(-1/3	2x^(-1/3	CD
38536	589	10	)	)	-RRB-
38536	589	11	+	+	CC
38536	589	12	4x^(-2/3	4x^(-2/3	CD
38536	589	13	)	)	-RRB-
38536	589	14	)	)	-RRB-
38536	589	15	.	.	.
38536	590	1	9	9	CD
38536	590	2	.	.	.
38536	591	1	[	[	-LRB-
38536	591	2	(	(	-LRB-
38536	591	3	a^2b)/(c^2d)]^(1/2	a^2b)/(c^2d)]^(1/2	NNP
38536	591	4	)	)	-RRB-
38536	591	5	×	×	NFP
38536	591	6	[	[	-LRB-
38536	591	7	(	(	-LRB-
38536	591	8	c^3d)/(ab^3)]^(1/3	c^3d)/(ab^3)]^(1/3	NN
38536	591	9	)	)	-RRB-
38536	591	10	×	×	ADD
38536	591	11	[	[	-LRB-
38536	591	12	(	(	-LRB-
38536	591	13	a^(1/3)c)/(b^(1/4)d^(5/12))]^2	a^(1/3)c)/(b^(1/4)d^(5/12))]^2	NNP
38536	591	14	.	.	.
38536	592	1	10	10	CD
38536	592	2	.	.	.
38536	593	1	[	[	-LRB-
38536	593	2	(	(	-LRB-
38536	593	3	a^(-4))/(b^(-2)c)]^(-3/4	a^(-4))/(b^(-2)c)]^(-3/4	VBN
38536	593	4	)	)	-RRB-
38536	593	5	×	×	NFP
38536	593	6	[	[	-LRB-
38536	593	7	(	(	-LRB-
38536	593	8	a^(-1)b[c^(-3)]^(1/2))/(ab^(-1))]^(1/2	a^(-1)b[c^(-3)]^(1/2))/(ab^(-1))]^(1/2	NNP
38536	593	9	)	)	-RRB-
38536	593	10	.	.	.
38536	594	1	11	11	CD
38536	594	2	.	.	.
38536	595	1	[	[	-LRB-
38536	595	2	(	(	-LRB-
38536	595	3	[	[	-LRB-
38536	595	4	a^2]^(1/3))/([b^(-1)]^(1/4	a^2]^(1/3))/([b^(-1)]^(1/4	NNP
38536	595	5	)	)	-RRB-
38536	595	6	)	)	-RRB-
38536	595	7	·	·	NFP
38536	595	8	(	(	-LRB-
38536	595	9	[	[	-LRB-
38536	595	10	c^(-3)]^(1/2))/(a^(1/3	c^(-3)]^(1/2))/(a^(1/3	NN
38536	595	11	)	)	-RRB-
38536	595	12	)	)	-RRB-
38536	595	13	·	·	NFP
38536	595	14	(	(	-LRB-
38536	595	15	b^(-1/4)a^(1/3))/(c^(-1))]^(-6	b^(-1/4)a^(1/3))/(c^(-1))]^(-6	UH
38536	595	16	)	)	-RRB-
38536	595	17	.	.	.
38536	596	1	~Reference:~	~reference:~	VB
38536	596	2	The	the	DT
38536	596	3	chapter	chapter	NN
38536	596	4	on	on	IN
38536	596	5	Theory	theory	NN
38536	596	6	of	of	IN
38536	596	7	Exponents	exponent	NNS
38536	596	8	in	in	IN
38536	596	9	any	any	DT
38536	596	10	algebra	algebra	NN
38536	596	11	.	.	.
38536	597	1	Solve	solve	VB
38536	597	2	for	for	IN
38536	597	3	x	x	NNS
38536	597	4	:	:	:
38536	597	5	1	1	CD
38536	597	6	.	.	.
38536	597	7	x^(2/3	x^(2/3	NNP
38536	597	8	)	)	-RRB-
38536	597	9	=	=	SYM
38536	597	10	4	4	CD
38536	597	11	.	.	.
38536	598	1	2	2	LS
38536	598	2	.	.	.
38536	598	3	x^(-3/4	x^(-3/4	LS
38536	598	4	)	)	-RRB-
38536	598	5	=	=	SYM
38536	598	6	8	8	CD
38536	598	7	.	.	.
38536	599	1	Factor	factor	NN
38536	599	2	:	:	:
38536	599	3	3	3	LS
38536	599	4	.	.	.
38536	599	5	x^(2/3	x^(2/3	NNP
38536	599	6	)	)	-RRB-
38536	599	7	-	-	:
38536	599	8	9	9	CD
38536	599	9	.	.	.
38536	600	1	4	4	LS
38536	600	2	.	.	.
38536	600	3	x^(3/5	x^(3/5	NNP
38536	600	4	)	)	-RRB-
38536	600	5	+	+	CC
38536	600	6	27	27	CD
38536	600	7	.	.	.
38536	601	1	5	5	LS
38536	601	2	.	.	.
38536	601	3	x^(2a	x^(2a	NNP
38536	601	4	)	)	-RRB-
38536	601	5	-	-	:
38536	601	6	y^(-6	y^(-6	NNP
38536	601	7	)	)	-RRB-
38536	601	8	.	.	.
38536	602	1	6	6	LS
38536	602	2	.	.	.
38536	602	3	a^(1/3	a^(1/3	NNP
38536	602	4	)	)	-RRB-
38536	602	5	x^(1/2	x^(1/2	NNP
38536	602	6	)	)	-RRB-
38536	602	7	-	-	:
38536	602	8	3a^(1/3	3a^(1/3	CD
38536	602	9	)	)	-RRB-
38536	602	10	+	+	CC
38536	602	11	5x^(1/2	5x^(1/2	CD
38536	602	12	)	)	-RRB-
38536	602	13	-	-	:
38536	602	14	15	15	CD
38536	602	15	.	.	.
38536	603	1	7	7	LS
38536	603	2	.	.	.
38536	604	1	Find	find	VB
38536	604	2	the	the	DT
38536	604	3	H.	H.	NNP
38536	604	4	C.	C.	NNP
38536	604	5	F.	F.	NNP
38536	604	6	and	and	CC
38536	604	7	L.	L.	NNP
38536	604	8	C.	C.	NNP
38536	604	9	M.	M.	NNP
38536	604	10	of	of	IN
38536	604	11	a^2	a^2	NNP
38536	604	12	+	+	SYM
38536	604	13	a^(3/2	a^(3/2	NNP
38536	604	14	)	)	-RRB-
38536	604	15	b^(1/2	b^(1/2	NN
38536	604	16	)	)	-RRB-
38536	604	17	+	+	NFP
38536	604	18	a^(1/2	a^(1/2	NNP
38536	604	19	)	)	-RRB-
38536	604	20	b^(3/2	b^(3/2	NNP
38536	604	21	)	)	-RRB-
38536	604	22	-	-	:
38536	604	23	b^2	b^2	NNS
38536	604	24	,	,	,
38536	604	25	a^2	a^2	NNP
38536	604	26	-	-	HYPH
38536	604	27	a^(3/2	a^(3/2	NNP
38536	604	28	)	)	-RRB-
38536	604	29	b^(1/2	b^(1/2	NNP
38536	604	30	)	)	-RRB-
38536	604	31	-	-	:
38536	604	32	a^(1/2	a^(1/2	NN
38536	604	33	)	)	-RRB-
38536	604	34	b^(3/2	b^(3/2	NNP
38536	604	35	)	)	-RRB-
38536	604	36	-	-	:
38536	604	37	b^2	b^2	NNP
38536	604	38	.	.	.
38536	605	1	8	8	LS
38536	605	2	.	.	.
38536	606	1	Simplify	simplify	VB
38536	606	2	the	the	DT
38536	606	3	product	product	NN
38536	606	4	of	of	IN
38536	606	5	:	:	:
38536	606	6	(	(	-LRB-
38536	606	7	ayx^(-1))^(1/2	ayx^(-1))^(1/2	NNP
38536	606	8	)	)	-RRB-
38536	606	9	,	,	,
38536	606	10	(	(	-LRB-
38536	606	11	bxy^(-2))^(1/3	bxy^(-2))^(1/3	NNP
38536	606	12	)	)	-RRB-
38536	606	13	,	,	,
38536	606	14	and	and	CC
38536	606	15	(	(	-LRB-
38536	606	16	y^2a^(-2)b^(-2))^(1/4	y^2a^(-2)b^(-2))^(1/4	NNP
38536	606	17	)	)	-RRB-
38536	606	18	.	.	.
38536	607	1	(	(	-LRB-
38536	607	2	_	_	NNP
38536	607	3	Princeton	Princeton	NNP
38536	607	4	.	.	.
38536	607	5	_	_	NNP
38536	607	6	)	)	-RRB-
38536	607	7	9	9	CD
38536	607	8	.	.	.
38536	608	1	Find	find	VB
38536	608	2	the	the	DT
38536	608	3	square	square	JJ
38536	608	4	root	root	NN
38536	608	5	of	of	IN
38536	608	6	:	:	:
38536	608	7	25a^(4/3)b^(-3	25a^(4/3)b^(-3	CD
38536	608	8	)	)	-RRB-
38536	608	9	-	-	SYM
38536	608	10	10a^(2/3)b^(-3/2	10a^(2/3)b^(-3/2	CD
38536	608	11	)	)	-RRB-
38536	608	12	-	-	:
38536	608	13	49	49	CD
38536	608	14	+	+	SYM
38536	608	15	10a^(-2/3)b^(3/2	10a^(-2/3)b^(3/2	CD
38536	608	16	)	)	-RRB-
38536	608	17	+	+	CC
38536	608	18	25a^(-4/3)b^3	25a^(-4/3)b^3	CD
38536	608	19	.	.	.
38536	609	1	10	10	CD
38536	609	2	.	.	.
38536	610	1	Simplify	Simplify	NNP
38536	610	2	[	[	-LRB-
38536	610	3	(	(	-LRB-
38536	610	4	2^(n	2^(n	CD
38536	610	5	+	+	SYM
38536	610	6	2))/(4^(-n	2))/(4^(-n	CD
38536	610	7	)	)	-RRB-
38536	610	8	)	)	-RRB-
38536	610	9	÷	÷	NNP
38536	610	10	(	(	-LRB-
38536	610	11	8^n)/(2	8^n)/(2	CD
38536	610	12	^	^	.
38536	610	13	3)]^(1/5	3)]^(1/5	CD
38536	610	14	)	)	-RRB-
38536	610	15	.	.	.
38536	611	1	11	11	CD
38536	611	2	.	.	.
38536	612	1	Find	find	VB
38536	612	2	the	the	DT
38536	612	3	value	value	NN
38536	612	4	of	of	IN
38536	612	5	(	(	-LRB-
38536	612	6	7	7	CD
38536	612	7	·	·	SYM
38536	612	8	13	13	CD
38536	612	9	^	^	SYM
38536	612	10	0	0	CD
38536	612	11	÷	÷	NNP
38536	612	12	7)/(21	7)/(21	NNP
38536	612	13	^	^	NN
38536	612	14	0	0	NFP
38536	612	15	)	)	-RRB-
38536	612	16	+	+	SYM
38536	612	17	3	3	CD
38536	612	18	^	^	NN
38536	612	19	0	0	CD
38536	612	20	×	×	NNS
38536	612	21	(	(	-LRB-
38536	612	22	4	4	CD
38536	612	23	^	^	NN
38536	612	24	0	0	CD
38536	612	25	·	·	SYM
38536	612	26	7	7	CD
38536	612	27	^	^	SYM
38536	612	28	0)/[(7a	0)/[(7a	NNP
38536	612	29	+	+	SYM
38536	612	30	b)^0	b)^0	NNP
38536	612	31	]	]	-RRB-
38536	612	32	+	+	CC
38536	612	33	8^(-2/3	8^(-2/3	CD
38536	612	34	)	)	-RRB-
38536	612	35	.	.	.
38536	613	1	12	12	CD
38536	613	2	.	.	.
38536	614	1	Express	express	VB
38536	614	2	as	as	IN
38536	614	3	a	a	DT
38536	614	4	power	power	NN
38536	614	5	of	of	IN
38536	614	6	2	2	CD
38536	614	7	:	:	SYM
38536	614	8	8	8	CD
38536	614	9	^	^	SYM
38536	614	10	3	3	CD
38536	614	11	;	;	SYM
38536	614	12	4	4	CD
38536	614	13	^	^	SYM
38536	614	14	5	5	CD
38536	614	15	;	;	SYM
38536	614	16	4	4	CD
38536	614	17	^	^	SYM
38536	614	18	3	3	CD
38536	614	19	·	·	NFP
38536	614	20	8^(2/3	8^(2/3	CD
38536	614	21	)	)	-RRB-
38536	614	22	·	·	NFP
38536	614	23	16^(3/4	16^(3/4	CD
38536	614	24	)	)	-RRB-
38536	614	25	.	.	.
38536	615	1	13	13	CD
38536	615	2	.	.	.
38536	616	1	Simplify	simplify	VB
38536	616	2	{	{	-LRB-
38536	616	3	[	[	-LRB-
38536	616	4	(	(	-LRB-
38536	616	5	x^(a	x^(a	NNP
38536	616	6	+	+	SYM
38536	616	7	1))/(x^(1	1))/(x^(1	CD
38536	616	8	-	-	:
38536	616	9	a))]^a	a))]^a	JJ
38536	616	10	÷	÷	NN
38536	616	11	[	[	-LRB-
38536	616	12	(	(	-LRB-
38536	616	13	x^a)/(x^(1	x^a)/(x^(1	NNP
38536	616	14	-	-	HYPH
38536	616	15	a))]^(a	a))]^(a	NNP
38536	616	16	-	-	HYPH
38536	616	17	1)}^(1/(3a	1)}^(1/(3a	CD
38536	616	18	-	-	HYPH
38536	616	19	1	1	CD
38536	616	20	)	)	-RRB-
38536	616	21	)	)	-RRB-
38536	616	22	.	.	.
38536	617	1	14	14	CD
38536	617	2	.	.	.
38536	618	1	Simplify	Simplify	NNP
38536	618	2	[	[	-LRB-
38536	618	3	(	(	-LRB-
38536	618	4	x^(5/2	x^(5/2	NNP
38536	618	5	)	)	-RRB-
38536	618	6	y^(4/3))/(z^(-5/4	y^(4/3))/(z^(-5/4	NNP
38536	618	7	)	)	-RRB-
38536	618	8	)	)	-RRB-
38536	618	9	·	·	NFP
38536	618	10	(	(	-LRB-
38536	618	11	z^4)/(x^(-3	z^4)/(x^(-3	CD
38536	618	12	)	)	-RRB-
38536	618	13	y^(-5/3	y^(-5/3	NN
38536	618	14	)	)	-RRB-
38536	618	15	)	)	-RRB-
38536	618	16	÷	÷	NNP
38536	618	17	(	(	-LRB-
38536	618	18	y^(-2	y^(-2	NNP
38536	618	19	)	)	-RRB-
38536	618	20	z^(1/4))/(x^(-1/2))]^(1/5	z^(1/4))/(x^(-1/2))]^(1/5	NNP
38536	618	21	)	)	-RRB-
38536	618	22	.	.	.
38536	619	1	15	15	CD
38536	619	2	.	.	.
38536	620	1	Expand	Expand	NNP
38536	620	2	(	(	-LRB-
38536	620	3	a^(1/2	a^(1/2	NNP
38536	620	4	)	)	-RRB-
38536	620	5	+	+	NFP
38536	620	6	b^(1/3))^4	b^(1/3))^4	NNP
38536	620	7	,	,	,
38536	620	8	writing	write	VBG
38536	620	9	the	the	DT
38536	620	10	result	result	NN
38536	620	11	with	with	IN
38536	620	12	fractional	fractional	JJ
38536	620	13	exponents	exponent	NNS
38536	620	14	.	.	.
38536	621	1	~Reference:~	~reference:~	VB
38536	621	2	The	the	DT
38536	621	3	chapter	chapter	NN
38536	621	4	on	on	IN
38536	621	5	Theory	theory	NN
38536	621	6	of	of	IN
38536	621	7	Exponents	exponent	NNS
38536	621	8	in	in	IN
38536	621	9	any	any	DT
38536	621	10	algebra	algebra	NN
38536	621	11	.	.	.
38536	622	1	RADICALS	RADICALS	NNP
38536	622	2	1	1	CD
38536	622	3	.	.	.
38536	622	4	Review	review	VB
38536	622	5	all	all	DT
38536	622	6	definitions	definition	NNS
38536	622	7	in	in	IN
38536	622	8	Radicals	radical	NNS
38536	622	9	,	,	,
38536	622	10	also	also	RB
38536	622	11	the	the	DT
38536	622	12	methods	method	NNS
38536	622	13	of	of	IN
38536	622	14	transforming	transforming	NN
38536	622	15	and	and	CC
38536	622	16	simplifying	simplifying	NN
38536	622	17	radicals	radical	NNS
38536	622	18	.	.	.
38536	623	1	When	when	WRB
38536	623	2	is	be	VBZ
38536	623	3	_	_	NNP
38536	623	4	a	a	DT
38536	623	5	radical	radical	NN
38536	623	6	in	in	IN
38536	623	7	its	-PRON-	PRP$
38536	623	8	simplest	simple	JJS
38536	623	9	form	form	NN
38536	623	10	_	_	NNP
38536	623	11	?	?	.
38536	624	1	2	2	LS
38536	624	2	.	.	.
38536	625	1	Simplify	Simplify	NNP
38536	625	2	(	(	-LRB-
38536	625	3	to	to	TO
38536	625	4	simplest	simple	JJS
38536	625	5	form	form	NN
38536	625	6	)	)	-RRB-
38536	625	7	:	:	:
38536	625	8	[	[	-LRB-
38536	625	9	2/3]^(1/2	2/3]^(1/2	NNP
38536	625	10	)	)	-RRB-
38536	625	11	;	;	:
38536	625	12	[	[	-LRB-
38536	625	13	1/11]^(1/2	1/11]^(1/2	CD
38536	625	14	)	)	-RRB-
38536	625	15	;	;	:
38536	625	16	[	[	-LRB-
38536	625	17	3/5]^(1/3	3/5]^(1/3	NNP
38536	625	18	)	)	-RRB-
38536	625	19	;	;	:
38536	625	20	3[5/6]^(1/2	3[5/6]^(1/2	NNP
38536	625	21	)	)	-RRB-
38536	625	22	;	;	:
38536	625	23	(	(	-LRB-
38536	625	24	2a	2a	NNP
38536	625	25	/	/	SYM
38536	625	26	b)[(8b^2)/(27a)]^(1/2	b)[(8b^2)/(27a)]^(1/2	NNP
38536	625	27	)	)	-RRB-
38536	625	28	;	;	:
38536	625	29	[	[	-LRB-
38536	625	30	5/(x^n)]^(1/2n	5/(x^n)]^(1/2n	NNP
38536	625	31	)	)	-RRB-
38536	625	32	;	;	:
38536	625	33	(	(	-LRB-
38536	625	34	a	a	DT
38536	625	35	+	+	CD
38536	625	36	b)^2	b)^2	NN
38536	625	37	[	[	-LRB-
38536	625	38	(	(	-LRB-
38536	625	39	-a^4)/((a	-a^4)/((a	NFP
38536	625	40	+	+	NFP
38536	625	41	b)^5)]^(1/3	b)^5)]^(1/3	NNP
38536	625	42	)	)	-RRB-
38536	625	43	;	;	:
38536	625	44	27^(1/2	27^(1/2	CD
38536	625	45	)	)	-RRB-
38536	625	46	;	;	:
38536	625	47	[	[	-LRB-
38536	625	48	54]^(1/3	54]^(1/3	CD
38536	625	49	)	)	-RRB-
38536	625	50	;	;	:
38536	625	51	-5[125^(1/2	-5[125^(1/2	NFP
38536	625	52	)	)	-RRB-
38536	625	53	]	]	-RRB-
38536	625	54	.	.	.
38536	626	1	3	3	LS
38536	626	2	.	.	.
38536	627	1	Reduce	reduce	VB
38536	627	2	to	to	IN
38536	627	3	entire	entire	JJ
38536	627	4	surds	surd	NNS
38536	627	5	:	:	:
38536	627	6	2[3^(1/2	2[3^(1/2	LS
38536	627	7	)	)	-RRB-
38536	627	8	]	]	-RRB-
38536	627	9	;	;	:
38536	627	10	2[3^(1/4	2[3^(1/4	LS
38536	627	11	)	)	-RRB-
38536	627	12	]	]	-RRB-
38536	627	13	;	;	:
38536	627	14	6[2^(1/3	6[2^(1/3	LS
38536	627	15	)	)	-RRB-
38536	627	16	]	]	-RRB-
38536	627	17	;	;	:
38536	627	18	a[[b^2]^(1	a[[b^2]^(1	NNP
38536	627	19	/	/	SYM
38536	627	20	n	n	NNP
38536	627	21	)	)	-RRB-
38536	627	22	]	]	-RRB-
38536	627	23	;	;	:
38536	627	24	-3[2^(1/3	-3[2^(1/3	NNP
38536	627	25	)	)	-RRB-
38536	627	26	]	]	-RRB-
38536	627	27	;	;	:
38536	627	28	3a[[(a	3a[[(a	NN
38536	627	29	+	+	SYM
38536	627	30	2)/(6a^2)]^(1/3	2)/(6a^2)]^(1/3	CD
38536	627	31	)	)	-RRB-
38536	627	32	]	]	-RRB-
38536	627	33	;	;	,
38536	627	34	(	(	-LRB-
38536	627	35	a	a	DT
38536	627	36	+	+	SYM
38536	627	37	2y)[(a	2y)[(a	CD
38536	627	38	-	-	HYPH
38536	627	39	2y)/(a	2y)/(a	CD
38536	627	40	+	+	SYM
38536	627	41	2y)]^(1/2	2y)]^(1/2	CD
38536	627	42	)	)	-RRB-
38536	627	43	.	.	.
38536	628	1	4	4	LS
38536	628	2	.	.	.
38536	629	1	Reduce	reduce	VB
38536	629	2	to	to	IN
38536	629	3	radicals	radical	NNS
38536	629	4	of	of	IN
38536	629	5	lower	low	JJR
38536	629	6	order	order	NN
38536	629	7	(	(	-LRB-
38536	629	8	or	or	CC
38536	629	9	simplify	simplify	VB
38536	629	10	indices	indices	NNP
38536	629	11	)	)	-RRB-
38536	629	12	:	:	:
38536	629	13	[	[	-LRB-
38536	629	14	a^2]^(1/4	a^2]^(1/4	CD
38536	629	15	)	)	-RRB-
38536	629	16	;	;	:
38536	629	17	[	[	-LRB-
38536	629	18	a^3]^(1/6	a^3]^(1/6	NN
38536	629	19	)	)	-RRB-
38536	629	20	;	;	:
38536	629	21	[	[	-LRB-
38536	629	22	27a^3]^(1/6	27a^3]^(1/6	NN
38536	629	23	)	)	-RRB-
38536	629	24	;	;	:
38536	629	25	[	[	-LRB-
38536	629	26	81	81	CD
38536	629	27	a^4	a^4	NNS
38536	629	28	x^8]^(1/12	x^8]^(1/12	NNP
38536	629	29	)	)	-RRB-
38536	629	30	;	;	:
38536	629	31	[	[	-LRB-
38536	629	32	9x^2	9x^2	CD
38536	629	33	y^4	y^4	NNP
38536	629	34	z^10]^(1/2n	z^10]^(1/2n	NNP
38536	629	35	)	)	-RRB-
38536	629	36	.	.	.
38536	630	1	5	5	CD
38536	630	2	.	.	.
38536	631	1	Reduce	reduce	VB
38536	631	2	to	to	IN
38536	631	3	radicals	radical	NNS
38536	631	4	of	of	IN
38536	631	5	the	the	DT
38536	631	6	same	same	JJ
38536	631	7	degree	degree	NN
38536	631	8	(	(	-LRB-
38536	631	9	order	order	NN
38536	631	10	,	,	,
38536	631	11	or	or	CC
38536	631	12	index	index	NN
38536	631	13	)	)	-RRB-
38536	631	14	:	:	:
38536	631	15	7^(1/2	7^(1/2	CD
38536	631	16	)	)	-RRB-
38536	631	17	and	and	CC
38536	631	18	[	[	-LRB-
38536	631	19	11]^(1/3	11]^(1/3	CD
38536	631	20	)	)	-RRB-
38536	631	21	;	;	:
38536	631	22	5^(1/3	5^(1/3	CD
38536	631	23	)	)	-RRB-
38536	631	24	and	and	CC
38536	631	25	3^(1/4	3^(1/4	CD
38536	631	26	)	)	-RRB-
38536	631	27	;	;	:
38536	631	28	7^(1/6	7^(1/6	CD
38536	631	29	)	)	-RRB-
38536	631	30	and	and	CC
38536	631	31	3^(1/2	3^(1/2	CD
38536	631	32	)	)	-RRB-
38536	631	33	;	;	:
38536	631	34	[	[	-LRB-
38536	631	35	x^m]^(1	x^m]^(1	NN
38536	631	36	/	/	SYM
38536	631	37	n	n	IN
38536	631	38	)	)	-RRB-
38536	631	39	and	and	CC
38536	631	40	[	[	-LRB-
38536	631	41	x^n]^(1	x^n]^(1	NNP
38536	631	42	/	/	SYM
38536	631	43	m	m	NNP
38536	631	44	)	)	-RRB-
38536	631	45	;	;	:
38536	631	46	[	[	-LRB-
38536	631	47	c^y]^(1	c^y]^(1	NNP
38536	631	48	/	/	SYM
38536	631	49	x	x	NNPS
38536	631	50	)	)	-RRB-
38536	631	51	,	,	,
38536	631	52	[	[	-LRB-
38536	631	53	c^z]^(1	c^z]^(1	NNP
38536	631	54	/	/	SYM
38536	631	55	y	y	NNP
38536	631	56	)	)	-RRB-
38536	631	57	,	,	,
38536	631	58	and	and	CC
38536	631	59	[	[	-LRB-
38536	631	60	c^x]^(1	c^x]^(1	NNP
38536	631	61	/	/	SYM
38536	631	62	z	z	NNP
38536	631	63	)	)	-RRB-
38536	631	64	.	.	.
38536	632	1	6	6	CD
38536	632	2	.	.	.
38536	633	1	Which	which	WDT
38536	633	2	is	be	VBZ
38536	633	3	greater	great	JJR
38536	633	4	,	,	,
38536	633	5	3^(1/2	3^(1/2	CD
38536	633	6	)	)	-RRB-
38536	633	7	or	or	CC
38536	633	8	4^(1/3	4^(1/3	CD
38536	633	9	)	)	-RRB-
38536	633	10	?	?	.
38536	634	1	[	[	-LRB-
38536	634	2	23]^(1/3	23]^(1/3	CD
38536	634	3	)	)	-RRB-
38536	634	4	or	or	CC
38536	634	5	2[2^(1/2	2[2^(1/2	CD
38536	634	6	)	)	-RRB-
38536	634	7	]	]	-RRB-
38536	634	8	?	?	.
38536	635	1	7	7	LS
38536	635	2	.	.	.
38536	636	1	Which	which	WDT
38536	636	2	is	be	VBZ
38536	636	3	greatest	great	JJS
38536	636	4	,	,	,
38536	636	5	3^(1/2	3^(1/2	CD
38536	636	6	)	)	-RRB-
38536	636	7	,	,	,
38536	636	8	5^(1/3	5^(1/3	CD
38536	636	9	)	)	-RRB-
38536	636	10	,	,	,
38536	636	11	or	or	CC
38536	636	12	7^(1/4	7^(1/4	LS
38536	636	13	)	)	-RRB-
38536	636	14	?	?	.
38536	637	1	Give	give	VB
38536	637	2	work	work	NN
38536	637	3	and	and	CC
38536	637	4	arrange	arrange	VB
38536	637	5	in	in	IN
38536	637	6	descending	descend	VBG
38536	637	7	order	order	NN
38536	637	8	of	of	IN
38536	637	9	magnitude	magnitude	NN
38536	637	10	.	.	.
38536	638	1	Collect	collect	VB
38536	638	2	:	:	:
38536	638	3	8	8	CD
38536	638	4	.	.	.
38536	639	1	128^(1/2	128^(1/2	LS
38536	639	2	)	)	-RRB-
38536	639	3	-	-	:
38536	639	4	2[50^(1/2	2[50^(1/2	CD
38536	639	5	)	)	-RRB-
38536	639	6	]	]	-RRB-
38536	639	7	+	+	NNP
38536	639	8	72^(1/2	72^(1/2	CD
38536	639	9	)	)	-RRB-
38536	639	10	-	-	:
38536	639	11	18^(1/2	18^(1/2	CD
38536	639	12	)	)	-RRB-
38536	639	13	.	.	.
38536	640	1	9	9	CD
38536	640	2	.	.	.
38536	641	1	2[5/3]^(1/2	2[5/3]^(1/2	LS
38536	641	2	)	)	-RRB-
38536	641	3	+	+	CC
38536	641	4	(	(	-LRB-
38536	641	5	1/6)60^(1/2	1/6)60^(1/2	CD
38536	641	6	)	)	-RRB-
38536	641	7	+	+	SYM
38536	641	8	15^(1/2	15^(1/2	NNS
38536	641	9	)	)	-RRB-
38536	641	10	+	+	CC
38536	641	11	[	[	-LRB-
38536	641	12	3/5]^(1/2	3/5]^(1/2	NNP
38536	641	13	)	)	-RRB-
38536	641	14	.	.	.
38536	642	1	10	10	CD
38536	642	2	.	.	.
38536	643	1	[	[	-LRB-
38536	643	2	(	(	-LRB-
38536	643	3	m	m	NNP
38536	643	4	-	-	HYPH
38536	643	5	n)^2a]^(1/2	n)^2a]^(1/2	NNP
38536	643	6	)	)	-RRB-
38536	643	7	+	+	CC
38536	643	8	[	[	-LRB-
38536	643	9	(	(	-LRB-
38536	643	10	m	m	NN
38536	643	11	+	+	SYM
38536	643	12	n)^2a]^(1/2	n)^2a]^(1/2	NNP
38536	643	13	)	)	-RRB-
38536	643	14	-	-	:
38536	643	15	[	[	-LRB-
38536	643	16	am^2]^(1/2	am^2]^(1/2	CD
38536	643	17	)	)	-RRB-
38536	643	18	+	+	CC
38536	643	19	[	[	-LRB-
38536	643	20	a(n	a(n	NN
38536	643	21	-	-	HYPH
38536	643	22	m)^2]^(1/2	m)^2]^(1/2	NNP
38536	643	23	)	)	-RRB-
38536	643	24	-	-	:
38536	643	25	a^(1/2	a^(1/2	NN
38536	643	26	)	)	-RRB-
38536	643	27	.	.	.
38536	644	1	11	11	CD
38536	644	2	.	.	.
38536	645	1	A	a	NN
38536	645	2	and	and	CC
38536	645	3	B	b	NN
38536	645	4	each	each	DT
38536	645	5	shoot	shoot	VBP
38536	645	6	thirty	thirty	CD
38536	645	7	arrows	arrow	NNS
38536	645	8	at	at	IN
38536	645	9	a	a	DT
38536	645	10	target	target	NN
38536	645	11	.	.	.
38536	646	1	B	b	NN
38536	646	2	makes	make	VBZ
38536	646	3	twice	twice	PDT
38536	646	4	as	as	RB
38536	646	5	many	many	JJ
38536	646	6	hits	hit	NNS
38536	646	7	as	as	IN
38536	646	8	A	a	NN
38536	646	9	,	,	,
38536	646	10	and	and	CC
38536	646	11	A	a	DT
38536	646	12	makes	make	VBZ
38536	646	13	three	three	CD
38536	646	14	times	time	NNS
38536	646	15	as	as	IN
38536	646	16	many	many	JJ
38536	646	17	misses	miss	NNS
38536	646	18	as	as	IN
38536	646	19	B.	B.	NNP
38536	647	1	Find	find	VB
38536	647	2	the	the	DT
38536	647	3	number	number	NN
38536	647	4	of	of	IN
38536	647	5	hits	hit	NNS
38536	647	6	and	and	CC
38536	647	7	misses	miss	NNS
38536	647	8	of	of	IN
38536	647	9	each	each	DT
38536	647	10	.	.	.
38536	648	1	(	(	-LRB-
38536	648	2	_	_	NNP
38536	648	3	Univ	Univ	NNP
38536	648	4	.	.	.
38536	649	1	of	of	IN
38536	649	2	Cal	Cal	NNP
38536	649	3	.	.	.
38536	649	4	_	_	NNP
38536	649	5	)	)	-RRB-
38536	649	6	~Reference:~	~Reference:~	NNP
38536	649	7	The	the	DT
38536	649	8	chapter	chapter	NN
38536	649	9	on	on	IN
38536	649	10	Radicals	radical	NNS
38536	649	11	in	in	IN
38536	649	12	any	any	DT
38536	649	13	algebra	algebra	NN
38536	649	14	(	(	-LRB-
38536	649	15	first	first	JJ
38536	649	16	part	part	NN
38536	649	17	of	of	IN
38536	649	18	the	the	DT
38536	649	19	chapter	chapter	NN
38536	649	20	)	)	-RRB-
38536	649	21	.	.	.
38536	650	1	The	the	DT
38536	650	2	most	most	RBS
38536	650	3	important	important	JJ
38536	650	4	principle	principle	NN
38536	650	5	in	in	IN
38536	650	6	Radicals	radical	NNS
38536	650	7	is	be	VBZ
38536	650	8	the	the	DT
38536	650	9	following	follow	VBG
38536	650	10	:	:	:
38536	650	11	(	(	-LRB-
38536	650	12	ab)^(1	ab)^(1	NNP
38536	650	13	/	/	SYM
38536	650	14	n	n	NNP
38536	650	15	)	)	-RRB-
38536	650	16	=	=	NFP
38536	650	17	a^(1	a^(1	ADD
38536	650	18	/	/	SYM
38536	650	19	n	n	LS
38536	650	20	)	)	-RRB-
38536	650	21	b^(1	b^(1	NNS
38536	650	22	/	/	SYM
38536	650	23	n	n	NN
38536	650	24	)	)	-RRB-
38536	650	25	.	.	.
38536	651	1	Hence	hence	RB
38536	651	2	[	[	-LRB-
38536	651	3	ab]^(1	ab]^(1	NNP
38536	651	4	/	/	SYM
38536	651	5	n	n	NN
38536	651	6	)	)	-RRB-
38536	651	7	=	=	NFP
38536	651	8	a^(1	a^(1	ADD
38536	651	9	/	/	SYM
38536	651	10	n	n	NN
38536	651	11	)	)	-RRB-
38536	651	12	·	·	NFP
38536	651	13	b^(1	b^(1	ADD
38536	651	14	/	/	SYM
38536	651	15	n	n	NN
38536	651	16	)	)	-RRB-
38536	651	17	.	.	.
38536	652	1	Or	or	CC
38536	652	2	,	,	,
38536	652	3	a^(1	a^(1	ADD
38536	652	4	/	/	SYM
38536	652	5	n	n	NN
38536	652	6	)	)	-RRB-
38536	652	7	·	·	NFP
38536	652	8	b^(1	b^(1	NNS
38536	652	9	/	/	SYM
38536	652	10	n	n	NN
38536	652	11	)	)	-RRB-
38536	652	12	=	=	NFP
38536	652	13	[	[	-LRB-
38536	652	14	ab]^(1	ab]^(1	NNP
38536	652	15	/	/	SYM
38536	652	16	n	n	NNP
38536	652	17	)	)	-RRB-
38536	652	18	.	.	.
38536	653	1	From	from	IN
38536	653	2	this	this	DT
38536	653	3	also	also	RB
38536	653	4	(	(	-LRB-
38536	653	5	[	[	-LRB-
38536	653	6	ab]^(1	ab]^(1	NNP
38536	653	7	/	/	SYM
38536	653	8	n))/(a^(1	n))/(a^(1	NNP
38536	653	9	/	/	SYM
38536	653	10	n	n	NNP
38536	653	11	)	)	-RRB-
38536	653	12	)	)	-RRB-
38536	653	13	=	=	NFP
38536	653	14	b^(1	b^(1	ADD
38536	653	15	/	/	SYM
38536	653	16	n	n	NN
38536	653	17	)	)	-RRB-
38536	653	18	.	.	.
38536	654	1	Multiply	Multiply	NNP
38536	654	2	:	:	:
38536	654	3	1	1	CD
38536	654	4	.	.	.
38536	655	1	2[4^(1/3	2[4^(1/3	LS
38536	655	2	)	)	-RRB-
38536	655	3	]	]	-RRB-
38536	655	4	by	by	IN
38536	655	5	3[6^(1/3	3[6^(1/3	CD
38536	655	6	)	)	-RRB-
38536	655	7	]	]	-RRB-
38536	655	8	.	.	.
38536	656	1	2	2	LS
38536	656	2	.	.	.
38536	657	1	2^(1/2	2^(1/2	LS
38536	657	2	)	)	-RRB-
38536	657	3	by	by	IN
38536	657	4	3^(1/3	3^(1/3	CD
38536	657	5	)	)	-RRB-
38536	657	6	.	.	.
38536	658	1	3	3	LS
38536	658	2	.	.	.
38536	659	1	2^(1/4	2^(1/4	LS
38536	659	2	)	)	-RRB-
38536	659	3	by	by	IN
38536	659	4	4^(1/6	4^(1/6	CD
38536	659	5	)	)	-RRB-
38536	659	6	.	.	.
38536	660	1	4	4	LS
38536	660	2	.	.	.
38536	661	1	[	[	-LRB-
38536	661	2	a	a	DT
38536	661	3	+	+	SYM
38536	661	4	x^(1/2)]^(1/2	x^(1/2)]^(1/2	NN
38536	661	5	)	)	-RRB-
38536	661	6	by	by	IN
38536	661	7	[	[	-LRB-
38536	661	8	a	a	DT
38536	661	9	-	-	HYPH
38536	661	10	x^(1/2)]^(1/2	x^(1/2)]^(1/2	NN
38536	661	11	)	)	-RRB-
38536	661	12	.	.	.
38536	662	1	5	5	CD
38536	662	2	.	.	.
38536	663	1	2^(1/2	2^(1/2	CD
38536	663	2	)	)	-RRB-
38536	663	3	+	+	CC
38536	663	4	3^(1/2	3^(1/2	CD
38536	663	5	)	)	-RRB-
38536	663	6	-	-	:
38536	663	7	5^(1/2	5^(1/2	CD
38536	663	8	)	)	-RRB-
38536	663	9	by	by	IN
38536	663	10	2^(1/2	2^(1/2	CD
38536	663	11	)	)	-RRB-
38536	663	12	-	-	:
38536	663	13	3^(1/2	3^(1/2	CD
38536	663	14	)	)	-RRB-
38536	663	15	+	+	CC
38536	663	16	5^(1/2	5^(1/2	CD
38536	663	17	)	)	-RRB-
38536	663	18	.	.	.
38536	664	1	6	6	CD
38536	664	2	.	.	.
38536	665	1	-p/2	-p/2	NFP
38536	665	2	+	+	NFP
38536	665	3	(	(	-LRB-
38536	665	4	[	[	-LRB-
38536	665	5	p^2	p^2	NN
38536	665	6	-	-	:
38536	665	7	4q]^(1/2))/2	4q]^(1/2))/2	CD
38536	665	8	by	by	IN
38536	665	9	-p/2	-p/2	HYPH
38536	665	10	-	-	:
38536	665	11	(	(	-LRB-
38536	665	12	[	[	-LRB-
38536	665	13	p^2	p^2	NN
38536	665	14	-	-	SYM
38536	665	15	4q]^(1/2))/2	4q]^(1/2))/2	CD
38536	665	16	.	.	.
38536	666	1	Divide	divide	NN
38536	666	2	:	:	:
38536	666	3	7	7	CD
38536	666	4	.	.	.
38536	667	1	27^(1/2	27^(1/2	LS
38536	667	2	)	)	-RRB-
38536	667	3	by	by	IN
38536	667	4	3^(1/2	3^(1/2	CD
38536	667	5	)	)	-RRB-
38536	667	6	.	.	.
38536	668	1	8	8	LS
38536	668	2	.	.	.
38536	669	1	4[18^(1/2	4[18^(1/2	LS
38536	669	2	)	)	-RRB-
38536	669	3	]	]	-RRB-
38536	669	4	by	by	IN
38536	669	5	5[32^(1/2	5[32^(1/2	CD
38536	669	6	)	)	-RRB-
38536	669	7	]	]	-RRB-
38536	669	8	.	.	.
38536	670	1	9	9	CD
38536	670	2	.	.	.
38536	671	1	3[12]^(1/3	3[12]^(1/3	LS
38536	671	2	)	)	-RRB-
38536	671	3	by	by	IN
38536	671	4	6^(1/2	6^(1/2	CD
38536	671	5	)	)	-RRB-
38536	671	6	.	.	.
38536	672	1	10	10	CD
38536	672	2	.	.	.
38536	673	1	3^(1/2	3^(1/2	LS
38536	673	2	)	)	-RRB-
38536	673	3	by	by	IN
38536	673	4	3^(1/4	3^(1/4	CD
38536	673	5	)	)	-RRB-
38536	673	6	.	.	.
38536	674	1	11	11	CD
38536	674	2	.	.	.
38536	675	1	6[105^(1/2	6[105^(1/2	CD
38536	675	2	)	)	-RRB-
38536	675	3	]	]	-RRB-
38536	675	4	+	+	CC
38536	675	5	18[40^(1/2	18[40^(1/2	CD
38536	675	6	)	)	-RRB-
38536	675	7	]	]	-RRB-
38536	675	8	-	-	SYM
38536	675	9	45[12^(1/2	45[12^(1/2	CD
38536	675	10	)	)	-RRB-
38536	675	11	]	]	-RRB-
38536	675	12	by	by	IN
38536	675	13	3[15^(1/2	3[15^(1/2	CD
38536	675	14	)	)	-RRB-
38536	675	15	]	]	-RRB-
38536	675	16	.	.	.
38536	676	1	(	(	-LRB-
38536	676	2	_	_	NNP
38536	676	3	Short	Short	NNP
38536	676	4	division	division	NN
38536	676	5	.	.	.
38536	676	6	_	_	NNP
38536	676	7	)	)	-RRB-
38536	676	8	12	12	CD
38536	676	9	.	.	.
38536	677	1	10[18]^(1/3	10[18]^(1/3	LS
38536	677	2	)	)	-RRB-
38536	677	3	-	-	:
38536	677	4	4[60]^(1/3	4[60]^(1/3	CD
38536	677	5	)	)	-RRB-
38536	677	6	+	+	CC
38536	677	7	5[100]^(1/3	5[100]^(1/3	XX
38536	677	8	)	)	-RRB-
38536	677	9	by	by	IN
38536	677	10	3[30]^(1/3	3[30]^(1/3	CD
38536	677	11	)	)	-RRB-
38536	677	12	.	.	.
38536	678	1	Rationalize	rationalize	VB
38536	678	2	the	the	DT
38536	678	3	denominator	denominator	NN
38536	678	4	:	:	:
38536	678	5	13	13	CD
38536	678	6	.	.	.
38536	679	1	2/(3^(1/2	2/(3^(1/2	CD
38536	679	2	)	)	-RRB-
38536	679	3	)	)	-RRB-
38536	679	4	;	;	:
38536	679	5	7/(7^(1/2	7/(7^(1/2	LS
38536	679	6	)	)	-RRB-
38536	679	7	)	)	-RRB-
38536	679	8	;	;	:
38536	679	9	5/(2[5^(1/2	5/(2[5^(1/2	NNP
38536	679	10	)	)	-RRB-
38536	679	11	]	]	-RRB-
38536	679	12	)	)	-RRB-
38536	679	13	;	;	:
38536	679	14	3/([a^2]^(1/5	3/([a^2]^(1/5	NNP
38536	679	15	)	)	-RRB-
38536	679	16	)	)	-RRB-
38536	679	17	;	;	:
38536	679	18	4/([a^3]^(1/7	4/([a^3]^(1/7	LS
38536	679	19	)	)	-RRB-
38536	679	20	)	)	-RRB-
38536	679	21	.	.	.
38536	680	1	14	14	CD
38536	680	2	.	.	.
38536	681	1	2/(2^(1/2	2/(2^(1/2	LS
38536	681	2	)	)	-RRB-
38536	681	3	)	)	-RRB-
38536	681	4	+	+	CC
38536	681	5	3^(1/2	3^(1/2	CD
38536	681	6	)	)	-RRB-
38536	681	7	)	)	-RRB-
38536	681	8	;	;	:
38536	681	9	(	(	-LRB-
38536	681	10	a^(1/2	a^(1/2	NNP
38536	681	11	)	)	-RRB-
38536	681	12	+	+	NNP
38536	681	13	b^(1/2))/(a^(1/2	b^(1/2))/(a^(1/2	ADD
38536	681	14	)	)	-RRB-
38536	681	15	-	-	:
38536	681	16	b^(1/2	b^(1/2	NN
38536	681	17	)	)	-RRB-
38536	681	18	)	)	-RRB-
38536	681	19	;	;	:
38536	681	20	3/(3	3/(3	CD
38536	681	21	-	-	HYPH
38536	681	22	3^(1/2	3^(1/2	CD
38536	681	23	)	)	-RRB-
38536	681	24	)	)	-RRB-
38536	681	25	.	.	.
38536	682	1	15	15	CD
38536	682	2	.	.	.
38536	683	1	[	[	-LRB-
38536	683	2	3^(1/2	3^(1/2	CD
38536	683	3	)	)	-RRB-
38536	683	4	+	+	CC
38536	683	5	2^(1/2)]/[6^(1/2	2^(1/2)]/[6^(1/2	NNS
38536	683	6	)	)	-RRB-
38536	683	7	+	+	CC
38536	683	8	3^(1/2	3^(1/2	CD
38536	683	9	)	)	-RRB-
38536	683	10	-	-	:
38536	683	11	2^(1/2	2^(1/2	CD
38536	683	12	)	)	-RRB-
38536	683	13	]	]	-RRB-
38536	683	14	.	.	.
38536	684	1	Review	review	VB
38536	684	2	the	the	DT
38536	684	3	method	method	NN
38536	684	4	of	of	IN
38536	684	5	finding	find	VBG
38536	684	6	the	the	DT
38536	684	7	square	square	JJ
38536	684	8	root	root	NN
38536	684	9	of	of	IN
38536	684	10	a	a	DT
38536	684	11	binomial	binomial	JJ
38536	684	12	surd	surd	NN
38536	684	13	.	.	.
38536	685	1	(	(	-LRB-
38536	685	2	By	by	IN
38536	685	3	inspection	inspection	NN
38536	685	4	preferably	preferably	RB
38536	685	5	.	.	.
38536	685	6	)	)	-RRB-
38536	686	1	Then	then	RB
38536	686	2	find	find	VB
38536	686	3	square	square	JJ
38536	686	4	root	root	NN
38536	686	5	of	of	IN
38536	686	6	:	:	:
38536	686	7	16	16	CD
38536	686	8	.	.	.
38536	687	1	5	5	CD
38536	687	2	+	+	SYM
38536	687	3	2[6^(1/2	2[6^(1/2	CD
38536	687	4	)	)	-RRB-
38536	687	5	]	]	-RRB-
38536	687	6	.	.	.
38536	688	1	17	17	CD
38536	688	2	.	.	.
38536	689	1	17	17	CD
38536	689	2	-	-	SYM
38536	689	3	12[2^(1/2	12[2^(1/2	CD
38536	689	4	)	)	-RRB-
38536	689	5	]	]	-RRB-
38536	689	6	.	.	.
38536	690	1	18	18	CD
38536	690	2	.	.	.
38536	691	1	7	7	LS
38536	691	2	-	-	SYM
38536	691	3	33^(1/2	33^(1/2	CD
38536	691	4	)	)	-RRB-
38536	691	5	.	.	.
38536	692	1	~Reference:~	~reference:~	VB
38536	692	2	The	the	DT
38536	692	3	chapter	chapter	NN
38536	692	4	on	on	IN
38536	692	5	Radicals	radical	NNS
38536	692	6	in	in	IN
38536	692	7	any	any	DT
38536	692	8	algebra	algebra	NN
38536	692	9	,	,	,
38536	692	10	beginning	begin	VBG
38536	692	11	at	at	IN
38536	692	12	Addition	Addition	NNP
38536	692	13	and	and	CC
38536	692	14	Subtraction	Subtraction	NNP
38536	692	15	of	of	IN
38536	692	16	Radicals	Radicals	NNPS
38536	692	17	.	.	.
38536	693	1	MISCELLANEOUS	MISCELLANEOUS	NNP
38536	693	2	EXAMPLES	EXAMPLES	NNP
38536	693	3	,	,	,
38536	693	4	ALGEBRA	algebra	VBP
38536	693	5	TO	to	IN
38536	693	6	QUADRATICS	QUADRATICS	NNP
38536	693	7	Results	result	NNS
38536	693	8	by	by	IN
38536	693	9	inspection	inspection	NN
38536	693	10	,	,	,
38536	693	11	examples	example	NNS
38536	693	12	1	1	CD
38536	693	13	-	-	SYM
38536	693	14	10	10	CD
38536	693	15	.	.	.
38536	694	1	Divide	divide	NN
38536	694	2	:	:	:
38536	694	3	1	1	CD
38536	694	4	.	.	.
38536	695	1	(	(	-LRB-
38536	695	2	x^(5/17	x^(5/17	NNP
38536	695	3	)	)	-RRB-
38536	695	4	+	+	NFP
38536	695	5	y^(5/17))/(x^(1/17	y^(5/17))/(x^(1/17	NN
38536	695	6	)	)	-RRB-
38536	695	7	+	+	NFP
38536	695	8	y^(1/17	y^(1/17	NNPS
38536	695	9	)	)	-RRB-
38536	695	10	)	)	-RRB-
38536	695	11	.	.	.
38536	696	1	2	2	LS
38536	696	2	.	.	.
38536	697	1	(	(	-LRB-
38536	697	2	x	x	LS
38536	697	3	-	-	SYM
38536	697	4	y)/(x^(1/3	y)/(x^(1/3	NNP
38536	697	5	)	)	-RRB-
38536	697	6	-	-	HYPH
38536	697	7	y^(1/3	y^(1/3	NNP
38536	697	8	)	)	-RRB-
38536	697	9	)	)	-RRB-
38536	697	10	.	.	.
38536	698	1	3	3	LS
38536	698	2	.	.	.
38536	699	1	(	(	-LRB-
38536	699	2	m^2	m^2	CD
38536	699	3	+	+	SYM
38536	699	4	n^2)/(m^(2/3	n^2)/(m^(2/3	NN
38536	699	5	)	)	-RRB-
38536	699	6	+	+	NFP
38536	699	7	n^(2/3	n^(2/3	NNP
38536	699	8	)	)	-RRB-
38536	699	9	)	)	-RRB-
38536	699	10	.	.	.
38536	700	1	4	4	LS
38536	700	2	.	.	.
38536	701	1	(	(	-LRB-
38536	701	2	x	x	NN
38536	701	3	-	-	NN
38536	701	4	y^2)/(x^(1/3	y^2)/(x^(1/3	NN
38536	701	5	)	)	-RRB-
38536	701	6	-	-	:
38536	701	7	[	[	-LRB-
38536	701	8	y^2]^(1/3	y^2]^(1/3	NNP
38536	701	9	)	)	-RRB-
38536	701	10	)	)	-RRB-
38536	701	11	.	.	.
38536	702	1	Multiply	Multiply	NNP
38536	702	2	:	:	:
38536	702	3	5	5	CD
38536	702	4	.	.	.
38536	703	1	[	[	-LRB-
38536	703	2	a^(-3/4	a^(-3/4	NN
38536	703	3	)	)	-RRB-
38536	703	4	+	+	NNP
38536	703	5	2/(m^(1/2))]^2	2/(m^(1/2))]^2	CD
38536	703	6	.	.	.
38536	704	1	6	6	CD
38536	704	2	.	.	.
38536	705	1	(	(	-LRB-
38536	705	2	K^(-2/7	K^(-2/7	NNP
38536	705	3	)	)	-RRB-
38536	705	4	-	-	:
38536	705	5	g^(-11/25))^2	g^(-11/25))^2	NNP
38536	705	6	.	.	.
38536	706	1	7	7	LS
38536	706	2	.	.	.
38536	707	1	(	(	-LRB-
38536	707	2	r^(2s	r^(2s	NN
38536	707	3	)	)	-RRB-
38536	707	4	+	+	CC
38536	707	5	l^(-3m))(r^(2s	l^(-3m))(r^(2s	JJ
38536	707	6	)	)	-RRB-
38536	707	7	-	-	:
38536	707	8	l^(-3	l^(-3	VBN
38536	707	9	m	m	NNP
38536	707	10	)	)	-RRB-
38536	707	11	)	)	-RRB-
38536	707	12	.	.	.
38536	708	1	8	8	LS
38536	708	2	.	.	.
38536	709	1	[	[	-LRB-
38536	709	2	a^(-2	a^(-2	NN
38536	709	3	)	)	-RRB-
38536	709	4	+	+	CC
38536	709	5	b^(-3	b^(-3	VB
38536	709	6	)	)	-RRB-
38536	709	7	-	-	:
38536	709	8	1/(c^2)]^2	1/(c^2)]^2	CD
38536	709	9	.	.	.
38536	710	1	9	9	CD
38536	710	2	.	.	.
38536	711	1	(	(	-LRB-
38536	711	2	3K^x	3k^x	CD
38536	711	3	+	+	NNP
38536	711	4	4t^(-3))(3K^x	4t^(-3))(3K^x	NNP
38536	711	5	-	-	SYM
38536	711	6	7t^(-3	7t^(-3	CD
38536	711	7	)	)	-RRB-
38536	711	8	)	)	-RRB-
38536	711	9	.	.	.
38536	712	1	10	10	CD
38536	712	2	.	.	.
38536	713	1	(	(	-LRB-
38536	713	2	2y^(2/7	2y^(2/7	CD
38536	713	3	)	)	-RRB-
38536	713	4	-	-	:
38536	713	5	40K^3)(3y^(2/7	40k^3)(3y^(2/7	CD
38536	713	6	)	)	-RRB-
38536	713	7	+	+	CC
38536	713	8	55K^3	55k^3	LS
38536	713	9	)	)	-RRB-
38536	713	10	.	.	.
38536	714	1	Factor	factor	NN
38536	714	2	:	:	:
38536	714	3	11	11	CD
38536	714	4	.	.	.
38536	714	5	x^(2/3	x^(2/3	NNP
38536	714	6	)	)	-RRB-
38536	714	7	-	-	:
38536	714	8	64	64	CD
38536	714	9	.	.	.
38536	715	1	12	12	CD
38536	715	2	.	.	.
38536	715	3	y^(3/5	y^(3/5	NNP
38536	715	4	)	)	-RRB-
38536	715	5	+	+	CC
38536	715	6	27	27	CD
38536	715	7	.	.	.
38536	716	1	13	13	CD
38536	716	2	.	.	.
38536	716	3	b^(3/2	b^(3/2	NNP
38536	716	4	)	)	-RRB-
38536	716	5	-	-	:
38536	716	6	8m^(-1	8m^(-1	NNP
38536	716	7	)	)	-RRB-
38536	716	8	.	.	.
38536	717	1	14	14	CD
38536	717	2	.	.	.
38536	718	1	3p	3p	LS
38536	718	2	-	-	HYPH
38536	718	3	8p^(1/2	8p^(1/2	CD
38536	718	4	)	)	-RRB-
38536	718	5	-	-	:
38536	718	6	35	35	CD
38536	718	7	.	.	.
38536	719	1	Factor	Factor	NNP
38536	719	2	,	,	,
38536	719	3	using	use	VBG
38536	719	4	radicals	radical	NNS
38536	719	5	instead	instead	RB
38536	719	6	of	of	IN
38536	719	7	exponents	exponent	NNS
38536	719	8	:	:	:
38536	719	9	15	15	CD
38536	719	10	.	.	.
38536	720	1	60	60	CD
38536	720	2	-	-	SYM
38536	720	3	7[3b^(1/2	7[3b^(1/2	CD
38536	720	4	)	)	-RRB-
38536	720	5	]	]	-RRB-
38536	720	6	-	-	:
38536	720	7	6b	6b	VBN
38536	720	8	.	.	.
38536	721	1	16	16	CD
38536	721	2	.	.	.
38536	722	1	15	15	CD
38536	722	2	m	m	NNP
38536	722	3	-	-	HYPH
38536	722	4	2[[mn]^(1/2	2[[mn]^(1/2	CD
38536	722	5	)	)	-RRB-
38536	722	6	]	]	-RRB-
38536	722	7	-	-	NNS
38536	722	8	24n	24n	NNS
38536	722	9	.	.	.
38536	723	1	17	17	CD
38536	723	2	.	.	.
38536	723	3	a	a	DT
38536	723	4	-	-	HYPH
38536	723	5	b	b	NNP
38536	723	6	(	(	-LRB-
38536	723	7	factor	factor	NN
38536	723	8	as	as	IN
38536	723	9	difference	difference	NN
38536	723	10	of	of	IN
38536	723	11	two	two	CD
38536	723	12	squares	square	NNS
38536	723	13	)	)	-RRB-
38536	723	14	.	.	.
38536	724	1	18	18	CD
38536	724	2	.	.	.
38536	724	3	a	a	LS
38536	724	4	-	-	HYPH
38536	724	5	b	b	NNP
38536	724	6	(	(	-LRB-
38536	724	7	factor	factor	NN
38536	724	8	as	as	IN
38536	724	9	difference	difference	NN
38536	724	10	of	of	IN
38536	724	11	two	two	CD
38536	724	12	cubes	cube	NNS
38536	724	13	)	)	-RRB-
38536	724	14	.	.	.
38536	725	1	19	19	CD
38536	725	2	.	.	.
38536	725	3	a	a	LS
38536	725	4	-	-	HYPH
38536	725	5	b	b	NNP
38536	725	6	(	(	-LRB-
38536	725	7	factor	factor	NN
38536	725	8	as	as	IN
38536	725	9	difference	difference	NN
38536	725	10	of	of	IN
38536	725	11	two	two	CD
38536	725	12	fourth	fourth	JJ
38536	725	13	powers	power	NNS
38536	725	14	)	)	-RRB-
38536	725	15	.	.	.
38536	726	1	20	20	CD
38536	726	2	.	.	.
38536	727	1	Find	find	VB
38536	727	2	the	the	DT
38536	727	3	H.	H.	NNP
38536	727	4	C.	C.	NNP
38536	727	5	F.	F.	NNP
38536	727	6	and	and	CC
38536	727	7	L.	L.	NNP
38536	727	8	C.	C.	NNP
38536	727	9	M.	M.	NNP
38536	727	10	of	of	IN
38536	727	11	x^2	x^2	NNP
38536	727	12	+	+	SYM
38536	727	13	xy^(1/2	xy^(1/2	XX
38536	727	14	)	)	-RRB-
38536	727	15	-	-	:
38536	727	16	2y	2y	CD
38536	727	17	,	,	,
38536	727	18	2x^2	2x^2	CD
38536	727	19	+	+	SYM
38536	727	20	5xy^(1/2	5xy^(1/2	CD
38536	727	21	)	)	-RRB-
38536	727	22	+	+	CC
38536	727	23	2y	2y	CD
38536	727	24	,	,	,
38536	727	25	2x^2	2x^2	CD
38536	727	26	-	-	HYPH
38536	727	27	xy^(1/2	xy^(1/2	FW
38536	727	28	)	)	-RRB-
38536	727	29	-	-	:
38536	727	30	y.	y.	NNP
38536	728	1	21	21	CD
38536	728	2	.	.	.
38536	729	1	Solve	solve	VB
38536	729	2	(	(	-LRB-
38536	729	3	short	short	JJ
38536	729	4	method	method	NN
38536	729	5	)	)	-RRB-
38536	729	6	(	(	-LRB-
38536	729	7	x	x	LS
38536	729	8	-	-	SYM
38536	729	9	7)/(x	7)/(x	JJ
38536	729	10	-	-	HYPH
38536	729	11	8)	8)	CD
38536	729	12	-	-	HYPH
38536	729	13	(	(	-LRB-
38536	729	14	x	x	LS
38536	729	15	-	-	NNP
38536	729	16	8)/(x	8)/(x	CD
38536	729	17	-	-	HYPH
38536	729	18	9	9	CD
38536	729	19	)	)	-RRB-
38536	729	20	=	=	NFP
38536	729	21	(	(	-LRB-
38536	729	22	x	x	SYM
38536	729	23	-	-	NNP
38536	729	24	4)/(x	4)/(x	CD
38536	729	25	-	-	HYPH
38536	729	26	5	5	CD
38536	729	27	)	)	-RRB-
38536	729	28	-	-	:
38536	729	29	(	(	-LRB-
38536	729	30	x	x	LS
38536	729	31	-	-	:
38536	729	32	5)/(x	5)/(x	CD
38536	729	33	-	-	HYPH
38536	729	34	6	6	CD
38536	729	35	)	)	-RRB-
38536	729	36	.	.	.
38536	730	1	22	22	CD
38536	730	2	.	.	.
38536	731	1	Simplify	Simplify	NNP
38536	731	2	(	(	-LRB-
38536	731	3	ab	ab	NNP
38536	731	4	/	/	SYM
38536	731	5	c	c	NNP
38536	731	6	+	+	CC
38536	731	7	bc	bc	NNP
38536	731	8	/	/	SYM
38536	731	9	a	a	NN
38536	731	10	+	+	NNP
38536	731	11	ca	ca	NN
38536	731	12	/	/	SYM
38536	731	13	b)/(a	b)/(a	NNP
38536	731	14	/	/	SYM
38536	731	15	bc	bc	NNP
38536	731	16	+	+	NNP
38536	731	17	b	b	NN
38536	731	18	/	/	SYM
38536	731	19	ca	ca	NN
38536	731	20	+	+	NNP
38536	731	21	c	c	NN
38536	731	22	/	/	SYM
38536	731	23	ab	ab	NNP
38536	731	24	)	)	-RRB-
38536	731	25	×	×	ADD
38536	731	26	[	[	-LRB-
38536	731	27	(	(	-LRB-
38536	731	28	(	(	-LRB-
38536	731	29	a	a	DT
38536	731	30	+	+	SYM
38536	731	31	b	b	NN
38536	731	32	+	+	CC
38536	731	33	c)^2)/(ab	c)^2)/(ab	NN
38536	731	34	+	+	SYM
38536	731	35	bc	bc	NNP
38536	731	36	+	+	CC
38536	731	37	ca	ca	NN
38536	731	38	)	)	-RRB-
38536	731	39	-	-	:
38536	731	40	2	2	CD
38536	731	41	]	]	-RRB-
38536	731	42	.	.	.
38536	732	1	(	(	-LRB-
38536	732	2	_	_	NNP
38536	732	3	Princeton	Princeton	NNP
38536	732	4	.	.	.
38536	732	5	_	_	NNP
38536	732	6	)	)	-RRB-
38536	732	7	1	1	CD
38536	732	8	.	.	.
38536	733	1	Solve	solve	VB
38536	733	2	for	for	IN
38536	733	3	p	p	NN
38536	733	4	:	:	:
38536	733	5	2^(p	2^(p	CD
38536	733	6	-	-	SYM
38536	733	7	3	3	CD
38536	733	8	)	)	-RRB-
38536	733	9	=	=	SYM
38536	733	10	128	128	CD
38536	733	11	.	.	.
38536	734	1	2	2	LS
38536	734	2	.	.	.
38536	735	1	Solve	solve	VB
38536	735	2	for	for	IN
38536	735	3	t	t	NN
38536	735	4	:	:	:
38536	735	5	t^(3/2	t^(3/2	NN
38536	735	6	)	)	-RRB-
38536	735	7	=	=	NFP
38536	735	8	-27	-27	NNP
38536	735	9	.	.	.
38536	736	1	3	3	LS
38536	736	2	.	.	.
38536	737	1	Find	find	VB
38536	737	2	the	the	DT
38536	737	3	square	square	JJ
38536	737	4	root	root	NN
38536	737	5	of	of	IN
38536	737	6	8114.4064	8114.4064	CD
38536	737	7	.	.	.
38536	738	1	What	what	WP
38536	738	2	,	,	,
38536	738	3	then	then	RB
38536	738	4	,	,	,
38536	738	5	is	be	VBZ
38536	738	6	the	the	DT
38536	738	7	square	square	JJ
38536	738	8	root	root	NN
38536	738	9	of	of	IN
38536	738	10	.0081144064	.0081144064	.
38536	738	11	?	?	.
38536	739	1	of	of	IN
38536	739	2	811440.64	811440.64	CD
38536	739	3	?	?	.
38536	740	1	From	from	IN
38536	740	2	any	any	DT
38536	740	3	of	of	IN
38536	740	4	the	the	DT
38536	740	5	above	above	RB
38536	740	6	can	can	MD
38536	740	7	you	-PRON-	PRP
38536	740	8	determine	determine	VB
38536	740	9	the	the	DT
38536	740	10	square	square	JJ
38536	740	11	root	root	NN
38536	740	12	of	of	IN
38536	740	13	.081144064	.081144064	.
38536	740	14	?	?	.
38536	741	1	4	4	LS
38536	741	2	.	.	.
38536	742	1	The	the	DT
38536	742	2	H.	H.	NNP
38536	742	3	C.	C.	NNP
38536	742	4	F.	F.	NNP
38536	742	5	of	of	IN
38536	742	6	two	two	CD
38536	742	7	expressions	expression	NNS
38536	742	8	is	be	VBZ
38536	742	9	a(a	a(a	NN
38536	742	10	-	-	HYPH
38536	742	11	b	b	NN
38536	742	12	)	)	-RRB-
38536	742	13	,	,	,
38536	742	14	and	and	CC
38536	742	15	their	-PRON-	PRP$
38536	742	16	L.	L.	NNP
38536	742	17	C.	C.	NNP
38536	742	18	M.	M.	NNP
38536	742	19	is	be	VBZ
38536	742	20	a^2b(a	a^2b(a	NN
38536	742	21	+	+	SYM
38536	742	22	b)(a	b)(a	NNP
38536	742	23	-	-	HYPH
38536	742	24	b	b	NN
38536	742	25	)	)	-RRB-
38536	742	26	.	.	.
38536	743	1	If	if	IN
38536	743	2	one	one	CD
38536	743	3	expression	expression	NN
38536	743	4	is	be	VBZ
38536	743	5	ab(a^2	ab(a^2	DT
38536	743	6	-	-	HYPH
38536	743	7	b^2	b^2	FW
38536	743	8	)	)	-RRB-
38536	743	9	,	,	,
38536	743	10	what	what	WP
38536	743	11	is	be	VBZ
38536	743	12	the	the	DT
38536	743	13	other	other	JJ
38536	743	14	?	?	.
38536	744	1	5	5	CD
38536	744	2	.	.	.
38536	745	1	Solve	solve	VB
38536	745	2	(	(	-LRB-
38536	745	3	short	short	JJ
38536	745	4	method	method	NN
38536	745	5	)	)	-RRB-
38536	745	6	:	:	:
38536	745	7	5/(7	5/(7	CD
38536	745	8	-	-	HYPH
38536	745	9	x	x	NN
38536	745	10	)	)	-RRB-
38536	745	11	-	-	,
38536	745	12	[	[	-LRB-
38536	745	13	(	(	-LRB-
38536	745	14	2	2	CD
38536	745	15	-	-	HYPH
38536	745	16	1/4)x	1/4)x	CD
38536	745	17	-	-	HYPH
38536	745	18	3]/4	3]/4	CD
38536	745	19	-	-	HYPH
38536	745	20	(	(	-LRB-
38536	745	21	x	x	NNS
38536	745	22	+	+	SYM
38536	745	23	11)/8	11)/8	CD
38536	745	24	+	+	SYM
38536	745	25	(	(	-LRB-
38536	745	26	11x	11x	NNS
38536	745	27	+	+	SYM
38536	745	28	5)/16	5)/16	CD
38536	745	29	=	=	SYM
38536	745	30	0	0	NFP
38536	745	31	.	.	.
38536	746	1	6	6	CD
38536	746	2	.	.	.
38536	747	1	Solve	solve	VB
38536	747	2	2	2	CD
38536	747	3	/	/	SYM
38536	747	4	m	m	NNP
38536	747	5	-	-	HYPH
38536	747	6	3	3	CD
38536	747	7	/	/	SYM
38536	747	8	n	n	CD
38536	747	9	+	+	SYM
38536	747	10	10	10	CD
38536	747	11	/	/	SYM
38536	747	12	p	p	NN
38536	747	13	=	=	SYM
38536	747	14	-3	-3	:
38536	747	15	,	,	,
38536	747	16	4	4	CD
38536	747	17	/	/	SYM
38536	747	18	m	m	NN
38536	747	19	+	+	SYM
38536	747	20	5	5	CD
38536	747	21	/	/	SYM
38536	747	22	p	p	NN
38536	747	23	+	+	SYM
38536	747	24	6	6	CD
38536	747	25	/	/	SYM
38536	747	26	n	n	NN
38536	747	27	=	=	SYM
38536	747	28	15	15	CD
38536	747	29	,	,	,
38536	747	30	1	1	CD
38536	747	31	/	/	SYM
38536	747	32	m	m	NNP
38536	747	33	-	-	HYPH
38536	747	34	1	1	CD
38536	747	35	/	/	SYM
38536	747	36	n	n	CD
38536	747	37	+	+	SYM
38536	747	38	5	5	CD
38536	747	39	/	/	SYM
38536	747	40	p	p	NN
38536	747	41	=	=	SYM
38536	747	42	-1/2	-1/2	NN
38536	747	43	.	.	.
38536	748	1	7	7	LS
38536	748	2	.	.	.
38536	749	1	Simplify	Simplify	NNP
38536	749	2	21[2/3]^(1/2	21[2/3]^(1/2	CD
38536	749	3	)	)	-RRB-
38536	749	4	-	-	:
38536	749	5	5[4/5]^(1/2	5[4/5]^(1/2	NN
38536	749	6	)	)	-RRB-
38536	749	7	+	+	CC
38536	749	8	6[4	6[4	CD
38536	749	9	-	-	HYPH
38536	749	10	1/6]^(1/2	1/6]^(1/2	CD
38536	749	11	)	)	-RRB-
38536	749	12	-	-	HYPH
38536	749	13	10[3	10[3	CD
38536	749	14	-	-	HYPH
38536	749	15	1/5]^(1/2	1/5]^(1/2	NNS
38536	749	16	)	)	-RRB-
38536	749	17	+	+	CC
38536	749	18	(	(	-LRB-
38536	749	19	40/3)[11	40/3)[11	CD
38536	749	20	-	-	SYM
38536	749	21	1/4]^(1/2	1/4]^(1/2	NNP
38536	749	22	)	)	-RRB-
38536	749	23	.	.	.
38536	750	1	8	8	LS
38536	750	2	.	.	.
38536	751	1	Does	do	VBZ
38536	751	2	[	[	-LRB-
38536	751	3	16	16	CD
38536	751	4	×	×	CD
38536	751	5	25]^(1/2	25]^(1/2	CD
38536	751	6	)	)	-RRB-
38536	751	7	=	=	SYM
38536	751	8	4	4	CD
38536	751	9	×	×	CD
38536	751	10	5	5	CD
38536	751	11	?	?	.
38536	752	1	Does	do	VBZ
38536	752	2	[	[	-LRB-
38536	752	3	16	16	CD
38536	752	4	+	+	SYM
38536	752	5	25]^(1/2	25]^(1/2	CD
38536	752	6	)	)	-RRB-
38536	752	7	=	=	SYM
38536	752	8	4	4	CD
38536	752	9	+	+	SYM
38536	752	10	5	5	CD
38536	752	11	?	?	.
38536	753	1	9	9	CD
38536	753	2	.	.	.
38536	754	1	Write	write	VB
38536	754	2	the	the	DT
38536	754	3	fraction	fraction	NN
38536	754	4	5/(4	5/(4	CD
38536	754	5	+	+	SYM
38536	754	6	2[3^(1/2	2[3^(1/2	CD
38536	754	7	)	)	-RRB-
38536	754	8	]	]	-RRB-
38536	754	9	)	)	-RRB-
38536	754	10	with	with	IN
38536	754	11	rational	rational	JJ
38536	754	12	denominator	denominator	NN
38536	754	13	,	,	,
38536	754	14	and	and	CC
38536	754	15	find	find	VB
38536	754	16	its	-PRON-	PRP$
38536	754	17	value	value	NN
38536	754	18	correct	correct	JJ
38536	754	19	to	to	IN
38536	754	20	two	two	CD
38536	754	21	decimal	decimal	JJ
38536	754	22	places	place	NNS
38536	754	23	.	.	.
38536	755	1	10	10	CD
38536	755	2	.	.	.
38536	756	1	Simplify	simplify	NN
38536	756	2	[	[	-LRB-
38536	756	3	{	{	-LRB-
38536	756	4	(	(	-LRB-
38536	756	5	[	[	-LRB-
38536	756	6	p	p	NN
38536	756	7	+	+	CC
38536	756	8	[	[	-LRB-
38536	756	9	p^2	p^2	NNS
38536	756	10	-	-	HYPH
38536	756	11	q]^(1/2)]/2)^(1/2	q]^(1/2)]/2)^(1/2	NN
38536	756	12	)	)	-RRB-
38536	756	13	+	+	NFP
38536	756	14	(	(	-LRB-
38536	756	15	[	[	-LRB-
38536	756	16	p	p	NN
38536	756	17	-	-	HYPH
38536	756	18	[	[	-LRB-
38536	756	19	p^2	p^2	NNP
38536	756	20	-	-	HYPH
38536	756	21	q]^(1/2)]/2)^(1/2)}^2]/[p	q]^(1/2)]/2)^(1/2)}^2]/[p	NNP
38536	756	22	+	+	SYM
38536	756	23	q^(1/2	q^(1/2	NN
38536	756	24	)	)	-RRB-
38536	756	25	]	]	-RRB-
38536	756	26	.	.	.
38536	757	1	(	(	-LRB-
38536	757	2	_	_	NNP
38536	757	3	Princeton	Princeton	NNP
38536	757	4	.	.	.
38536	757	5	_	_	NNP
38536	757	6	)	)	-RRB-
38536	757	7	1	1	CD
38536	757	8	.	.	.
38536	758	1	Rationalize	rationalize	VB
38536	758	2	the	the	DT
38536	758	3	denominator	denominator	NN
38536	758	4	of	of	IN
38536	758	5	{	{	-LRB-
38536	758	6	6^(1/2	6^(1/2	CD
38536	758	7	)	)	-RRB-
38536	758	8	+	+	CC
38536	758	9	3^(1/2	3^(1/2	CD
38536	758	10	)	)	-RRB-
38536	758	11	-	-	:
38536	758	12	3[2^(1/2)]}/{6^(1/2	3[2^(1/2)]}/{6^(1/2	CD
38536	758	13	)	)	-RRB-
38536	758	14	-	-	:
38536	758	15	3^(1/2	3^(1/2	CD
38536	758	16	)	)	-RRB-
38536	758	17	+	+	SYM
38536	758	18	3[2^(1/2	3[2^(1/2	CD
38536	758	19	)	)	-RRB-
38536	758	20	]	]	-RRB-
38536	758	21	}	}	-RRB-
38536	758	22	.	.	.
38536	759	1	(	(	-LRB-
38536	759	2	_	_	NNP
38536	759	3	Univ	Univ	NNP
38536	759	4	.	.	.
38536	760	1	of	of	IN
38536	760	2	Cal	Cal	NNP
38536	760	3	.	.	NNP
38536	760	4	_	_	NNP
38536	760	5	)	)	-RRB-
38536	760	6	2	2	CD
38536	760	7	.	.	.
38536	761	1	Simplify	simplify	NN
38536	761	2	[	[	-LRB-
38536	761	3	2^(n	2^(n	CD
38536	761	4	+	+	SYM
38536	761	5	4	4	CD
38536	761	6	)	)	-RRB-
38536	761	7	-	-	:
38536	761	8	2(2^n)]/[2(2^(n	2(2^n)]/[2(2^(n	CD
38536	761	9	+	+	SYM
38536	761	10	3	3	CD
38536	761	11	)	)	-RRB-
38536	761	12	)	)	-RRB-
38536	761	13	]	]	-RRB-
38536	761	14	.	.	.
38536	762	1	(	(	-LRB-
38536	762	2	_	_	NNP
38536	762	3	Univ	Univ	NNP
38536	762	4	.	.	.
38536	763	1	of	of	IN
38536	763	2	Penn	Penn	NNP
38536	763	3	.	.	.
38536	763	4	_	_	NNP
38536	763	5	)	)	-RRB-
38536	763	6	3	3	CD
38536	763	7	.	.	.
38536	764	1	Find	find	VB
38536	764	2	the	the	DT
38536	764	3	value	value	NN
38536	764	4	of	of	IN
38536	764	5	[	[	-LRB-
38536	764	6	1	1	CD
38536	764	7	+	+	SYM
38536	764	8	8^(-x/3)]/[(8x)^(1/2	8^(-x/3)]/[(8x)^(1/2	CD
38536	764	9	)	)	-RRB-
38536	764	10	+	+	CC
38536	764	11	10^(x	10^(x	CD
38536	764	12	-	-	HYPH
38536	764	13	2	2	CD
38536	764	14	)	)	-RRB-
38536	764	15	]	]	-RRB-
38536	764	16	,	,	,
38536	764	17	when	when	WRB
38536	764	18	x	x	NNP
38536	764	19	=	=	SYM
38536	764	20	2	2	CD
38536	764	21	.	.	.
38536	765	1	(	(	-LRB-
38536	765	2	_	_	NNP
38536	765	3	Cornell	Cornell	NNP
38536	765	4	.	.	.
38536	765	5	_	_	NNP
38536	765	6	)	)	-RRB-
38536	765	7	4	4	CD
38536	765	8	.	.	.
38536	766	1	Find	find	VB
38536	766	2	the	the	DT
38536	766	3	value	value	NN
38536	766	4	of	of	IN
38536	766	5	x	x	NN
38536	766	6	if	if	IN
38536	766	7	x^(6/5	x^(6/5	NNP
38536	766	8	)	)	-RRB-
38536	766	9	=	=	NFP
38536	766	10	y^4	y^4	XX
38536	766	11	,	,	,
38536	766	12	y^(2/3	y^(2/3	NNP
38536	766	13	)	)	-RRB-
38536	766	14	=	=	NFP
38536	766	15	9	9	CD
38536	766	16	.	.	.
38536	767	1	(	(	-LRB-
38536	767	2	_	_	NNP
38536	767	3	M.	M.	NNP
38536	768	1	I.	I.	NNP
38536	768	2	T.	T.	NNP
38536	768	3	_	_	NNP
38536	768	4	)	)	-RRB-
38536	768	5	5	5	CD
38536	768	6	.	.	.
38536	769	1	A	a	DT
38536	769	2	fisherman	fisherman	NN
38536	769	3	told	tell	VBD
38536	769	4	a	a	DT
38536	769	5	yarn	yarn	NN
38536	769	6	about	about	IN
38536	769	7	a	a	DT
38536	769	8	fish	fish	NN
38536	769	9	he	-PRON-	PRP
38536	769	10	had	have	VBD
38536	769	11	caught	catch	VBN
38536	769	12	.	.	.
38536	770	1	If	if	IN
38536	770	2	the	the	DT
38536	770	3	fish	fish	NN
38536	770	4	were	be	VBD
38536	770	5	half	half	RB
38536	770	6	as	as	RB
38536	770	7	long	long	RB
38536	770	8	as	as	IN
38536	770	9	he	-PRON-	PRP
38536	770	10	said	say	VBD
38536	770	11	it	-PRON-	PRP
38536	770	12	was	be	VBD
38536	770	13	,	,	,
38536	770	14	it	-PRON-	PRP
38536	770	15	would	would	MD
38536	770	16	be	be	VB
38536	770	17	10	10	CD
38536	770	18	inches	inch	NNS
38536	770	19	more	more	JJR
38536	770	20	than	than	IN
38536	770	21	twice	twice	RB
38536	770	22	as	as	RB
38536	770	23	long	long	RB
38536	770	24	as	as	IN
38536	770	25	it	-PRON-	PRP
38536	770	26	is	be	VBZ
38536	770	27	.	.	.
38536	771	1	If	if	IN
38536	771	2	it	-PRON-	PRP
38536	771	3	were	be	VBD
38536	771	4	4	4	CD
38536	771	5	inches	inch	NNS
38536	771	6	longer	long	JJR
38536	771	7	than	than	IN
38536	771	8	it	-PRON-	PRP
38536	771	9	is	be	VBZ
38536	771	10	,	,	,
38536	771	11	and	and	CC
38536	771	12	he	-PRON-	PRP
38536	771	13	had	have	VBD
38536	771	14	further	further	RB
38536	771	15	exaggerated	exaggerate	VBN
38536	771	16	its	-PRON-	PRP$
38536	771	17	length	length	NN
38536	771	18	by	by	IN
38536	771	19	adding	add	VBG
38536	771	20	4	4	CD
38536	771	21	inches	inch	NNS
38536	771	22	,	,	,
38536	771	23	it	-PRON-	PRP
38536	771	24	would	would	MD
38536	771	25	be	be	VB
38536	771	26	1/5	1/5	CD
38536	771	27	as	as	RB
38536	771	28	long	long	RB
38536	771	29	as	as	IN
38536	771	30	he	-PRON-	PRP
38536	771	31	now	now	RB
38536	771	32	said	say	VBD
38536	771	33	it	-PRON-	PRP
38536	771	34	was	be	VBD
38536	771	35	.	.	.
38536	772	1	How	how	WRB
38536	772	2	long	long	RB
38536	772	3	is	be	VBZ
38536	772	4	the	the	DT
38536	772	5	fish	fish	NN
38536	772	6	,	,	,
38536	772	7	and	and	CC
38536	772	8	how	how	WRB
38536	772	9	long	long	RB
38536	772	10	did	do	VBD
38536	772	11	he	-PRON-	PRP
38536	772	12	first	first	RB
38536	772	13	say	say	VB
38536	772	14	it	-PRON-	PRP
38536	772	15	was	be	VBD
38536	772	16	?	?	.
38536	773	1	(	(	-LRB-
38536	773	2	_	_	NNP
38536	773	3	M.	M.	NNP
38536	774	1	I.	I.	NNP
38536	774	2	T.	T.	NNP
38536	774	3	_	_	NNP
38536	774	4	)	)	-RRB-
38536	774	5	6	6	CD
38536	774	6	.	.	.
38536	775	1	The	the	DT
38536	775	2	force	force	NN
38536	775	3	_	_	NNP
38536	775	4	P	P	NNP
38536	775	5	_	_	NNP
38536	775	6	necessary	necessary	JJ
38536	775	7	to	to	TO
38536	775	8	lift	lift	VB
38536	775	9	a	a	DT
38536	775	10	weight	weight	NN
38536	775	11	_	_	NNP
38536	775	12	W	W	NNP
38536	775	13	_	_	NNP
38536	775	14	by	by	IN
38536	775	15	means	mean	NNS
38536	775	16	of	of	IN
38536	775	17	a	a	DT
38536	775	18	certain	certain	JJ
38536	775	19	machine	machine	NN
38536	775	20	is	be	VBZ
38536	775	21	given	give	VBN
38536	775	22	by	by	IN
38536	775	23	the	the	DT
38536	775	24	formula	formula	NN
38536	775	25	P	p	NN
38536	775	26	=	=	SYM
38536	775	27	a	a	NN
38536	775	28	+	+	SYM
38536	775	29	bW	bW	NNP
38536	775	30	,	,	,
38536	775	31	where	where	WRB
38536	775	32	_	_	NNP
38536	775	33	a	a	DT
38536	775	34	_	_	NNP
38536	775	35	and	and	CC
38536	775	36	_	_	NNP
38536	775	37	b	b	NNP
38536	775	38	_	_	NNP
38536	775	39	are	be	VBP
38536	775	40	constants	constant	NNS
38536	775	41	depending	depend	VBG
38536	775	42	on	on	IN
38536	775	43	the	the	DT
38536	775	44	amount	amount	NN
38536	775	45	of	of	IN
38536	775	46	friction	friction	NN
38536	775	47	in	in	IN
38536	775	48	the	the	DT
38536	775	49	machine	machine	NN
38536	775	50	.	.	.
38536	776	1	If	if	IN
38536	776	2	a	a	DT
38536	776	3	force	force	NN
38536	776	4	of	of	IN
38536	776	5	7	7	CD
38536	776	6	pounds	pound	NNS
38536	776	7	will	will	MD
38536	776	8	raise	raise	VB
38536	776	9	a	a	DT
38536	776	10	weight	weight	NN
38536	776	11	of	of	IN
38536	776	12	20	20	CD
38536	776	13	pounds	pound	NNS
38536	776	14	,	,	,
38536	776	15	and	and	CC
38536	776	16	a	a	DT
38536	776	17	force	force	NN
38536	776	18	of	of	IN
38536	776	19	13	13	CD
38536	776	20	pounds	pound	NNS
38536	776	21	will	will	MD
38536	776	22	raise	raise	VB
38536	776	23	a	a	DT
38536	776	24	weight	weight	NN
38536	776	25	of	of	IN
38536	776	26	50	50	CD
38536	776	27	pounds	pound	NNS
38536	776	28	,	,	,
38536	776	29	what	what	WP
38536	776	30	force	force	NN
38536	776	31	is	be	VBZ
38536	776	32	necessary	necessary	JJ
38536	776	33	to	to	TO
38536	776	34	raise	raise	VB
38536	776	35	a	a	DT
38536	776	36	weight	weight	NN
38536	776	37	of	of	IN
38536	776	38	40	40	CD
38536	776	39	pounds	pound	NNS
38536	776	40	?	?	.
38536	777	1	(	(	-LRB-
38536	777	2	First	first	RB
38536	777	3	determine	determine	VB
38536	777	4	the	the	DT
38536	777	5	constants	constant	NNS
38536	777	6	_	_	NNP
38536	777	7	a	a	DT
38536	777	8	_	_	NNP
38536	777	9	and	and	CC
38536	777	10	_	_	NNP
38536	777	11	b	b	NNP
38536	777	12	_	_	NNP
38536	777	13	.	.	.
38536	777	14	)	)	-RRB-
38536	778	1	(	(	-LRB-
38536	778	2	_	_	NNP
38536	778	3	Harvard	Harvard	NNP
38536	778	4	.	.	.
38536	778	5	_	_	NNP
38536	778	6	)	)	-RRB-
38536	778	7	7	7	CD
38536	778	8	.	.	.
38536	779	1	Reduce	reduce	VB
38536	779	2	to	to	IN
38536	779	3	the	the	DT
38536	779	4	simplest	simple	JJS
38536	779	5	form	form	NN
38536	779	6	:	:	:
38536	779	7	[	[	-LRB-
38536	779	8	[	[	-LRB-
38536	779	9	4/[2^(n	4/[2^(n	CD
38536	779	10	+	+	SYM
38536	779	11	2)]]^(1	2)]]^(1	CD
38536	779	12	/	/	SYM
38536	779	13	n	n	NN
38536	779	14	)	)	-RRB-
38536	779	15	;	;	:
38536	779	16	[	[	-LRB-
38536	779	17	ax(a^(-1)x	ax(a^(-1)x	JJ
38536	779	18	-	-	HYPH
38536	779	19	ax^(-1))]/[x^(2/3	ax^(-1))]/[x^(2/3	NNP
38536	779	20	)	)	-RRB-
38536	779	21	-	-	:
38536	779	22	a^(2/3	a^(2/3	NNP
38536	779	23	)	)	-RRB-
38536	779	24	]	]	-RRB-
38536	779	25	.	.	.
38536	780	1	8	8	LS
38536	780	2	.	.	.
38536	781	1	Determine	determine	VB
38536	781	2	the	the	DT
38536	781	3	H.	H.	NNP
38536	781	4	C.	C.	NNP
38536	781	5	F.	F.	NNP
38536	781	6	and	and	CC
38536	781	7	L.	L.	NNP
38536	781	8	C.	C.	NNP
38536	781	9	M.	M.	NNP
38536	781	10	of	of	IN
38536	781	11	(	(	-LRB-
38536	781	12	xy	xy	NNP
38536	781	13	-	-	HYPH
38536	781	14	y^2)^3	y^2)^3	NNP
38536	781	15	and	and	CC
38536	781	16	y^3	y^3	NNP
38536	781	17	-	-	HYPH
38536	781	18	x^2y	x^2y	NNP
38536	781	19	.	.	.
38536	782	1	(	(	-LRB-
38536	782	2	_	_	NNP
38536	782	3	College	College	NNP
38536	782	4	Entrance	Entrance	NNP
38536	782	5	Board	Board	NNP
38536	782	6	.	.	.
38536	782	7	_	_	NNP
38536	782	8	)	)	-RRB-
38536	782	9	1	1	CD
38536	782	10	.	.	.
38536	783	1	Simplify	Simplify	NNP
38536	783	2	(	(	-LRB-
38536	783	3	a	a	DT
38536	783	4	-	-	HYPH
38536	783	5	8m)/(a^(1/3	8m)/(a^(1/3	NNP
38536	783	6	)	)	-RRB-
38536	783	7	-	-	:
38536	783	8	2m^(1/3	2m^(1/3	CD
38536	783	9	)	)	-RRB-
38536	783	10	)	)	-RRB-
38536	783	11	-	-	:
38536	783	12	2a^(1/3)m^(1/3	2a^(1/3)m^(1/3	NN
38536	783	13	)	)	-RRB-
38536	783	14	.	.	.
38536	784	1	2	2	LS
38536	784	2	.	.	.
38536	785	1	Simplify	Simplify	NNP
38536	785	2	,	,	,
38536	785	3	writing	write	VBG
38536	785	4	the	the	DT
38536	785	5	result	result	NN
38536	785	6	with	with	IN
38536	785	7	rational	rational	JJ
38536	785	8	denominator	denominator	NN
38536	785	9	:	:	:
38536	785	10	(	(	-LRB-
38536	785	11	[	[	-LRB-
38536	785	12	a^(1/2	a^(1/2	NNP
38536	785	13	)	)	-RRB-
38536	785	14	+	+	CC
38536	785	15	(	(	-LRB-
38536	785	16	1)/(x^(-1/2))]^2	1)/(x^(-1/2))]^2	CD
38536	785	17	-	-	HYPH
38536	785	18	[	[	-LRB-
38536	785	19	(	(	-LRB-
38536	785	20	1)/(a^(-1/2	1)/(a^(-1/2	CD
38536	785	21	)	)	-RRB-
38536	785	22	)	)	-RRB-
38536	785	23	-	-	:
38536	785	24	x^(1/2)]^2	x^(1/2)]^2	NNP
38536	785	25	)	)	-RRB-
38536	785	26	/	/	NFP
38536	785	27	(	(	-LRB-
38536	785	28	x	x	SYM
38536	785	29	+	+	SYM
38536	785	30	[	[	-LRB-
38536	785	31	a^2	a^2	CD
38536	785	32	+	+	SYM
38536	785	33	x^2]^(1/2	x^2]^(1/2	CD
38536	785	34	)	)	-RRB-
38536	785	35	)	)	-RRB-
38536	785	36	.	.	.
38536	786	1	(	(	-LRB-
38536	786	2	_	_	NNP
38536	786	3	M.	M.	NNP
38536	787	1	I.	I.	NNP
38536	787	2	T.	T.	NNP
38536	787	3	_	_	NNP
38536	787	4	)	)	-RRB-
38536	787	5	3	3	CD
38536	787	6	.	.	.
38536	788	1	Find	find	VB
38536	788	2	[	[	-LRB-
38536	788	3	7	7	CD
38536	788	4	-	-	SYM
38536	788	5	48^(1/2)]^(1/2	48^(1/2)]^(1/2	CD
38536	788	6	)	)	-RRB-
38536	788	7	.	.	.
38536	789	1	4	4	LS
38536	789	2	.	.	.
38536	790	1	Expand	expand	VB
38536	790	2	(	(	-LRB-
38536	790	3	[	[	-LRB-
38536	790	4	a^3]^(1/2	a^3]^(1/2	CD
38536	790	5	)	)	-RRB-
38536	790	6	-	-	,
38536	790	7	[	[	-LRB-
38536	790	8	b^5]^(1/2))^5	b^5]^(1/2))^5	NN
38536	790	9	.	.	.
38536	791	1	5	5	CD
38536	791	2	.	.	.
38536	792	1	Expand	expand	VB
38536	792	2	and	and	CC
38536	792	3	simplify	simplify	VB
38536	792	4	(	(	-LRB-
38536	792	5	1	1	CD
38536	792	6	-	-	SYM
38536	792	7	2[3^(1/2	2[3^(1/2	CD
38536	792	8	)	)	-RRB-
38536	792	9	]	]	-RRB-
38536	792	10	+	+	CC
38536	792	11	3[2^(1/2)])^2	3[2^(1/2)])^2	CD
38536	792	12	.	.	.
38536	793	1	6	6	CD
38536	793	2	.	.	.
38536	794	1	Solve	solve	VB
38536	794	2	the	the	DT
38536	794	3	simultaneous	simultaneous	JJ
38536	794	4	equations	equation	NNS
38536	794	5	x	x	LS
38536	794	6	^(-1/2	^(-1/2	NNP
38536	794	7	)	)	-RRB-
38536	794	8	+	+	CC
38536	794	9	2y^(-1/2	2y^(-1/2	CD
38536	794	10	)	)	-RRB-
38536	794	11	=	=	SYM
38536	794	12	7/6	7/6	CD
38536	794	13	,	,	,
38536	794	14	2x^(-1/2	2x^(-1/2	CD
38536	794	15	)	)	-RRB-
38536	794	16	-	-	:
38536	794	17	y^(-1/2	y^(-1/2	NN
38536	794	18	)	)	-RRB-
38536	794	19	=	=	NFP
38536	794	20	2/3	2/3	CD
38536	794	21	.	.	.
38536	795	1	(	(	-LRB-
38536	795	2	_	_	NNP
38536	795	3	Yale	Yale	NNP
38536	795	4	.	.	.
38536	795	5	_	_	NNP
38536	795	6	)	)	-RRB-
38536	795	7	7	7	CD
38536	795	8	.	.	.
38536	796	1	Find	find	VB
38536	796	2	to	to	IN
38536	796	3	three	three	CD
38536	796	4	places	place	NNS
38536	796	5	of	of	IN
38536	796	6	decimals	decimal	NNS
38536	796	7	the	the	DT
38536	796	8	value	value	NN
38536	796	9	of	of	IN
38536	796	10	{	{	-LRB-
38536	796	11	[	[	-LRB-
38536	796	12	(	(	-LRB-
38536	796	13	a	a	DT
38536	796	14	+	+	SYM
38536	796	15	b)^(-1/3)]/[(11a	b)^(-1/3)]/[(11a	NN
38536	796	16	+	+	SYM
38536	796	17	b^2)^(1/6	b^2)^(1/6	NNP
38536	796	18	)	)	-RRB-
38536	796	19	]	]	-RRB-
38536	796	20	·	·	NFP
38536	796	21	[	[	-LRB-
38536	796	22	(	(	-LRB-
38536	796	23	{	{	-LRB-
38536	796	24	a^3	a^3	VBN
38536	796	25	-	-	HYPH
38536	796	26	b^3)^(-1/2)]/[(a	b^3)^(-1/2)]/[(a	NNP
38536	796	27	-	-	HYPH
38536	796	28	b)^(1/2)]}^(1/2	b)^(1/2)]}^(1/2	NN
38536	796	29	)	)	-RRB-
38536	796	30	,	,	,
38536	796	31	when	when	WRB
38536	796	32	a	a	DT
38536	796	33	=	=	NN
38536	796	34	5	5	CD
38536	796	35	and	and	CC
38536	796	36	b	b	NN
38536	796	37	=	=	SYM
38536	796	38	3	3	CD
38536	796	39	.	.	.
38536	797	1	(	(	-LRB-
38536	797	2	_	_	NNP
38536	797	3	Columbia	Columbia	NNP
38536	797	4	.	.	.
38536	797	5	_	_	NNP
38536	797	6	)	)	-RRB-
38536	797	7	8	8	CD
38536	797	8	.	.	.
38536	798	1	Show	show	VB
38536	798	2	that	that	DT
38536	798	3	(	(	-LRB-
38536	798	4	10	10	CD
38536	798	5	-	-	SYM
38536	798	6	4[5^(1/2)])/(5	4[5^(1/2)])/(5	CD
38536	798	7	+	+	SYM
38536	798	8	3[5^(1/2	3[5^(1/2	CD
38536	798	9	)	)	-RRB-
38536	798	10	]	]	-RRB-
38536	798	11	)	)	-RRB-
38536	798	12	is	be	VBZ
38536	798	13	the	the	DT
38536	798	14	negative	negative	JJ
38536	798	15	of	of	IN
38536	798	16	the	the	DT
38536	798	17	reciprocal	reciprocal	NN
38536	798	18	of	of	IN
38536	798	19	(	(	-LRB-
38536	798	20	10	10	CD
38536	798	21	+	+	CD
38536	798	22	4[5^(1/2)])/(5	4[5^(1/2)])/(5	CD
38536	798	23	-	-	SYM
38536	798	24	3[5^(1/2	3[5^(1/2	CD
38536	798	25	)	)	-RRB-
38536	798	26	]	]	-RRB-
38536	798	27	)	)	-RRB-
38536	798	28	.	.	.
38536	799	1	(	(	-LRB-
38536	799	2	_	_	NNP
38536	799	3	Columbia	Columbia	NNP
38536	799	4	.	.	.
38536	799	5	_	_	NNP
38536	799	6	)	)	-RRB-
38536	799	7	9	9	CD
38536	799	8	.	.	.
38536	800	1	Solve	solve	VB
38536	800	2	and	and	CC
38536	800	3	check	check	VB
38536	800	4	{	{	-LRB-
38536	800	5	5}/{[3x	5}/{[3x	CD
38536	800	6	+	+	SYM
38536	800	7	2]^(1/2	2]^(1/2	CD
38536	800	8	)	)	-RRB-
38536	800	9	}	}	-RRB-
38536	800	10	=	=	NFP
38536	800	11	[	[	-LRB-
38536	800	12	3x	3x	CD
38536	800	13	+	+	SYM
38536	800	14	2]^(1/2	2]^(1/2	CD
38536	800	15	)	)	-RRB-
38536	800	16	+	+	CC
38536	800	17	[	[	-LRB-
38536	800	18	3x	3x	CD
38536	800	19	-	-	HYPH
38536	800	20	1]^(1/2	1]^(1/2	NN
38536	800	21	)	)	-RRB-
38536	800	22	.	.	.
38536	801	1	10	10	CD
38536	801	2	.	.	.
38536	802	1	Assuming	assume	VBG
38536	802	2	that	that	IN
38536	802	3	when	when	WRB
38536	802	4	an	an	DT
38536	802	5	apple	apple	NN
38536	802	6	falls	fall	VBZ
38536	802	7	from	from	IN
38536	802	8	a	a	DT
38536	802	9	tree	tree	NN
38536	802	10	the	the	DT
38536	802	11	distance	distance	NN
38536	802	12	(	(	-LRB-
38536	802	13	S	S	NNP
38536	802	14	meters	meter	NNS
38536	802	15	)	)	-RRB-
38536	802	16	through	through	IN
38536	802	17	which	which	WDT
38536	802	18	it	-PRON-	PRP
38536	802	19	falls	fall	VBZ
38536	802	20	in	in	IN
38536	802	21	any	any	DT
38536	802	22	time	time	NN
38536	802	23	(	(	-LRB-
38536	802	24	t	t	NN
38536	802	25	seconds	second	NNS
38536	802	26	)	)	-RRB-
38536	802	27	is	be	VBZ
38536	802	28	given	give	VBN
38536	802	29	by	by	IN
38536	802	30	the	the	DT
38536	802	31	formula	formula	NN
38536	802	32	S	s	NN
38536	802	33	=	=	NFP
38536	802	34	(	(	-LRB-
38536	802	35	1/2)gt^2	1/2)gt^2	CD
38536	802	36	(	(	-LRB-
38536	802	37	where	where	WRB
38536	802	38	g	g	NNP
38536	802	39	=	=	SYM
38536	802	40	9.8	9.8	CD
38536	802	41	)	)	-RRB-
38536	802	42	,	,	,
38536	802	43	find	find	VB
38536	802	44	to	to	IN
38536	802	45	two	two	CD
38536	802	46	decimal	decimal	JJ
38536	802	47	places	place	NNS
38536	802	48	the	the	DT
38536	802	49	time	time	NN
38536	802	50	taken	take	VBN
38536	802	51	by	by	IN
38536	802	52	an	an	DT
38536	802	53	apple	apple	NN
38536	802	54	in	in	IN
38536	802	55	falling	fall	VBG
38536	802	56	15	15	CD
38536	802	57	meters	meter	NNS
38536	802	58	.	.	.
38536	803	1	(	(	-LRB-
38536	803	2	_	_	NNP
38536	803	3	College	College	NNP
38536	803	4	Entrance	Entrance	NNP
38536	803	5	Board	Board	NNP
38536	803	6	.	.	.
38536	803	7	_	_	NNP
38536	803	8	)	)	-RRB-
38536	803	9	Excellent	excellent	JJ
38536	803	10	practice	practice	NN
38536	803	11	may	may	MD
38536	803	12	be	be	VB
38536	803	13	obtained	obtain	VBN
38536	803	14	by	by	IN
38536	803	15	solving	solve	VBG
38536	803	16	the	the	DT
38536	803	17	ordinary	ordinary	JJ
38536	803	18	formulas	formula	NNS
38536	803	19	used	use	VBN
38536	803	20	in	in	IN
38536	803	21	arithmetic	arithmetic	JJ
38536	803	22	,	,	,
38536	803	23	geometry	geometry	NN
38536	803	24	,	,	,
38536	803	25	and	and	CC
38536	803	26	physics	physics	NN
38536	803	27	_	_	NNP
38536	803	28	orally	orally	RB
38536	803	29	,	,	,
38536	803	30	for	for	IN
38536	803	31	each	each	DT
38536	803	32	letter	letter	NN
38536	803	33	in	in	IN
38536	803	34	turn	turn	NN
38536	803	35	_	_	NNP
38536	803	36	.	.	.
38536	804	1	ARITHMETIC	ARITHMETIC	NNP
38536	804	2	p	p	NNP
38536	804	3	=	=	NN
38536	804	4	br	br	UH
38536	804	5	i	i	PRP
38536	804	6	=	=	SYM
38536	804	7	prt	prt	NNP
38536	804	8	a	a	NN
38536	804	9	=	=	SYM
38536	804	10	p	p	NN
38536	804	11	+	+	CC
38536	804	12	prt	prt	NN
38536	804	13	GEOMETRY	GEOMETRY	NNP
38536	804	14	K	K	NNP
38536	804	15	=	=	NN
38536	804	16	(	(	-LRB-
38536	804	17	1/2	1/2	CD
38536	804	18	)	)	-RRB-
38536	804	19	bh	bh	NNP
38536	804	20	K	K	NNP
38536	804	21	=	=	SYM
38536	804	22	bh	bh	NNP
38536	804	23	K	K	NNP
38536	804	24	=	=	NFP
38536	804	25	(	(	-LRB-
38536	804	26	a^2)/4	a^2)/4	IN
38536	804	27	3^(1/2	3^(1/2	CD
38536	804	28	)	)	-RRB-
38536	804	29	K	K	NNP
38536	804	30	=	=	NFP
38536	804	31	(	(	-LRB-
38536	804	32	1/2	1/2	CD
38536	804	33	)	)	-RRB-
38536	804	34	(	(	-LRB-
38536	804	35	b	b	NN
38536	804	36	+	+	SYM
38536	804	37	b	b	NN
38536	804	38	'	'	''
38536	804	39	)	)	-RRB-
38536	804	40	h	h	NN
38536	804	41	K	k	NN
38536	804	42	=	=	NFP
38536	804	43	[	[	-LRB-
38536	804	44	pi	pi	NN
38536	804	45	]	]	-RRB-
38536	804	46	R^2	R^2	NNP
38536	804	47	C	C	NNP
38536	804	48	=	=	SYM
38536	804	49	2	2	CD
38536	804	50	[	[	-LRB-
38536	804	51	pi	pi	NN
38536	804	52	]	]	-RRB-
38536	804	53	R	r	NN
38536	804	54	K	k	NN
38536	804	55	=	=	NFP
38536	804	56	[	[	-LRB-
38536	804	57	pi	pi	NN
38536	804	58	]	]	-RRB-
38536	804	59	R	R	NNP
38536	804	60	L	l	NN
38536	804	61	S	s	NN
38536	804	62	=	=	SYM
38536	804	63	4	4	CD
38536	804	64	[	[	-LRB-
38536	804	65	pi	pi	NN
38536	804	66	]	]	-RRB-
38536	804	67	R^2	R^2	NNP
38536	804	68	V	v	NN
38536	804	69	=	=	NFP
38536	804	70	[	[	-LRB-
38536	804	71	pi	pi	NN
38536	804	72	]	]	-RRB-
38536	804	73	R^2	R^2	NNP
38536	804	74	H	h	NN
38536	804	75	V	v	NN
38536	804	76	=	=	NNS
38536	804	77	(	(	-LRB-
38536	804	78	1/3	1/3	LS
38536	804	79	)	)	-RRB-
38536	804	80	[	[	-LRB-
38536	804	81	pi	pi	NN
38536	804	82	]	]	-RRB-
38536	804	83	R^2	R^2	NNP
38536	804	84	H	h	NN
38536	804	85	V	v	NN
38536	804	86	=	=	NNS
38536	804	87	(	(	-LRB-
38536	804	88	4/3	4/3	CD
38536	804	89	)	)	-RRB-
38536	804	90	[	[	-LRB-
38536	804	91	pi	pi	NN
38536	804	92	]	]	-RRB-
38536	804	93	R^3	R^3	NNP
38536	804	94	S	S	NNP
38536	804	95	=	=	NFP
38536	804	96	(	(	-LRB-
38536	804	97	[	[	-LRB-
38536	804	98	pi	pi	NN
38536	804	99	]	]	-RRB-
38536	804	100	R^2	R^2	NNP
38536	804	101	E)/(180	e)/(180	VB
38536	804	102	)	)	-RRB-
38536	804	103	C/(C	C/(C	NNP
38536	804	104	'	'	''
38536	804	105	)	)	-RRB-
38536	804	106	=	=	NFP
38536	804	107	R/(R	r/(r	NN
38536	804	108	'	'	''
38536	804	109	)	)	-RRB-
38536	804	110	K/(K	K/(K	NNP
38536	804	111	'	'	''
38536	804	112	)	)	-RRB-
38536	804	113	=	=	NFP
38536	804	114	(	(	-LRB-
38536	804	115	R^2)/(R'^2	R^2)/(R'^2	NNP
38536	804	116	)	)	-RRB-
38536	804	117	PHYSICS	PHYSICS	NNP
38536	804	118	v	v	NNP
38536	804	119	=	=	SYM
38536	804	120	gt	gt	NNP
38536	804	121	s	s	NNP
38536	804	122	=	=	NN
38536	804	123	(	(	-LRB-
38536	804	124	1/2	1/2	CD
38536	804	125	)	)	-RRB-
38536	804	126	gt^2	gt^2	NN
38536	804	127	s	s	NNPS
38536	804	128	=	=	FW
38536	804	129	(	(	-LRB-
38536	804	130	v^2)/(2	v^2)/(2	NN
38536	804	131	g	g	NN
38536	804	132	)	)	-RRB-
38536	804	133	C	C	NNP
38536	804	134	=	=	SYM
38536	804	135	E	e	NN
38536	804	136	/	/	SYM
38536	804	137	R	r	NN
38536	804	138	E	e	NN
38536	804	139	=	=	NN
38536	804	140	(	(	-LRB-
38536	804	141	wv^2)/(2	wv^2)/(2	NNP
38536	804	142	g	g	NNP
38536	804	143	)	)	-RRB-
38536	804	144	e	e	NNP
38536	804	145	=	=	NNS
38536	804	146	(	(	-LRB-
38536	804	147	4Pl^3)/(bh^3	4pl^3)/(bh^3	CD
38536	804	148	m	m	NN
38536	804	149	)	)	-RRB-
38536	804	150	E	e	NN
38536	804	151	=	=	NFP
38536	804	152	(	(	-LRB-
38536	804	153	mv^2)/(2	mv^2)/(2	NNP
38536	804	154	)	)	-RRB-
38536	804	155	t	t	NNP
38536	804	156	=	=	NFP
38536	804	157	[	[	-LRB-
38536	804	158	pi	pi	NN
38536	804	159	]	]	-RRB-
38536	804	160	[	[	-LRB-
38536	804	161	l	l	NN
38536	804	162	/	/	SYM
38536	804	163	g]^(1/2	g]^(1/2	NNP
38536	804	164	)	)	-RRB-
38536	804	165	F	F	NNP
38536	804	166	=	=	NFP
38536	804	167	(	(	-LRB-
38536	804	168	mV^2)/(r	mV^2)/(r	NNP
38536	804	169	)	)	-RRB-
38536	804	170	mh	mh	VB
38536	804	171	=	=	NFP
38536	804	172	(	(	-LRB-
38536	804	173	mv^2)/(2	mv^2)/(2	NNP
38536	804	174	g	g	NNP
38536	804	175	)	)	-RRB-
38536	804	176	R	r	NN
38536	804	177	=	=	SYM
38536	804	178	gs/(g	gs/(g	NN
38536	804	179	+	+	SYM
38536	804	180	s	s	NN
38536	804	181	)	)	-RRB-
38536	804	182	E	e	NN
38536	804	183	=	=	NNS
38536	804	184	(	(	-LRB-
38536	804	185	4n^2l^2w)/(g	4n^2l^2w)/(g	LS
38536	804	186	)	)	-RRB-
38536	804	187	C	C	NNP
38536	804	188	=	=	NNS
38536	804	189	(	(	-LRB-
38536	804	190	5/9)(F	5/9)(f	CD
38536	804	191	-	-	SYM
38536	804	192	32	32	CD
38536	804	193	)	)	-RRB-
38536	804	194	QUADRATIC	QUADRATIC	NNP
38536	804	195	EQUATIONS	EQUATIONS	NNP
38536	804	196	1	1	CD
38536	804	197	.	.	.
38536	805	1	Define	define	VB
38536	805	2	a	a	DT
38536	805	3	quadratic	quadratic	JJ
38536	805	4	equation	equation	NN
38536	805	5	;	;	:
38536	805	6	a	a	DT
38536	805	7	pure	pure	JJ
38536	805	8	quadratic	quadratic	NN
38536	805	9	;	;	:
38536	805	10	an	an	DT
38536	805	11	affected	affect	VBN
38536	805	12	(	(	-LRB-
38536	805	13	or	or	CC
38536	805	14	complete	complete	JJ
38536	805	15	)	)	-RRB-
38536	805	16	quadratic	quadratic	NN
38536	805	17	;	;	:
38536	805	18	an	an	DT
38536	805	19	equation	equation	NN
38536	805	20	in	in	IN
38536	805	21	the	the	DT
38536	805	22	quadratic	quadratic	JJ
38536	805	23	form	form	NN
38536	805	24	.	.	.
38536	806	1	2	2	LS
38536	806	2	.	.	.
38536	807	1	Solve	solve	VB
38536	807	2	the	the	DT
38536	807	3	pure	pure	JJ
38536	807	4	quadratic	quadratic	NN
38536	807	5	(	(	-LRB-
38536	807	6	7)/(3S^2	7)/(3S^2	NNP
38536	807	7	)	)	-RRB-
38536	807	8	-	-	:
38536	807	9	(	(	-LRB-
38536	807	10	11)/(9S^2	11)/(9s^2	LS
38536	807	11	)	)	-RRB-
38536	807	12	=	=	SYM
38536	807	13	5/6	5/6	XX
38536	807	14	.	.	.
38536	808	1	Review	review	VB
38536	808	2	the	the	DT
38536	808	3	first	first	JJ
38536	808	4	(	(	-LRB-
38536	808	5	or	or	CC
38536	808	6	usual	usual	JJ
38536	808	7	)	)	-RRB-
38536	808	8	method	method	NN
38536	808	9	of	of	IN
38536	808	10	completing	complete	VBG
38536	808	11	the	the	DT
38536	808	12	square	square	NN
38536	808	13	.	.	.
38536	809	1	Solve	solve	VB
38536	809	2	by	by	IN
38536	809	3	it	-PRON-	PRP
38536	809	4	the	the	DT
38536	809	5	following	follow	VBG
38536	809	6	:	:	:
38536	809	7	3	3	LS
38536	809	8	.	.	.
38536	809	9	x^2	x^2	NNS
38536	809	10	+	+	SYM
38536	809	11	10x	10x	NNS
38536	809	12	=	=	SYM
38536	809	13	24	24	CD
38536	809	14	.	.	.
38536	810	1	4	4	LS
38536	810	2	.	.	.
38536	811	1	2x^2	2x^2	CD
38536	811	2	-	-	HYPH
38536	811	3	5x	5x	CD
38536	811	4	=	=	SYM
38536	811	5	7	7	CD
38536	811	6	.	.	.
38536	812	1	5	5	CD
38536	812	2	.	.	.
38536	813	1	(	(	-LRB-
38536	813	2	x	x	NN
38536	813	3	-	-	SYM
38536	813	4	1)/2	1)/2	CD
38536	813	5	+	+	CD
38536	813	6	2/(x	2/(x	CD
38536	813	7	-	-	SYM
38536	813	8	1	1	CD
38536	813	9	)	)	-RRB-
38536	813	10	=	=	SYM
38536	813	11	2	2	CD
38536	813	12	-	-	SYM
38536	813	13	1/2	1/2	CD
38536	813	14	.	.	.
38536	814	1	6	6	LS
38536	814	2	.	.	.
38536	814	3	ax^2	ax^2	CD
38536	814	4	+	+	SYM
38536	814	5	bx	bx	NN
38536	814	6	+	+	SYM
38536	814	7	c	c	NN
38536	814	8	=	=	SYM
38536	814	9	0	0	NFP
38536	814	10	.	.	.
38536	814	11	Review	review	VB
38536	814	12	the	the	DT
38536	814	13	solution	solution	NN
38536	814	14	by	by	IN
38536	814	15	factoring	factor	VBG
38536	814	16	.	.	.
38536	815	1	Solve	solve	VB
38536	815	2	by	by	IN
38536	815	3	it	-PRON-	PRP
38536	815	4	the	the	DT
38536	815	5	following	follow	VBG
38536	815	6	:	:	:
38536	815	7	7	7	LS
38536	815	8	.	.	.
38536	815	9	x^2	x^2	NNS
38536	815	10	+	+	SYM
38536	815	11	8x	8x	CD
38536	815	12	+	+	SYM
38536	815	13	7	7	CD
38536	815	14	=	=	SYM
38536	815	15	0	0	CD
38536	815	16	.	.	.
38536	816	1	8	8	LS
38536	816	2	.	.	.
38536	817	1	24x^2	24x^2	CD
38536	817	2	=	=	SYM
38536	817	3	2x	2x	CD
38536	817	4	+	+	SYM
38536	817	5	15	15	CD
38536	817	6	.	.	.
38536	818	1	9	9	CD
38536	818	2	.	.	.
38536	819	1	3	3	CD
38536	819	2	=	=	SYM
38536	819	3	10x	10x	NNS
38536	819	4	-	-	SYM
38536	819	5	3x^2	3x^2	CD
38536	819	6	.	.	.
38536	820	1	10	10	CD
38536	820	2	.	.	.
38536	821	1	-7	-7	:
38536	821	2	=	=	NFP
38536	821	3	6x	6x	NN
38536	821	4	-	-	HYPH
38536	821	5	x^2	x^2	NNP
38536	821	6	.	.	.
38536	822	1	Solve	solve	VB
38536	822	2	,	,	,
38536	822	3	by	by	IN
38536	822	4	factoring	factor	VBG
38536	822	5	,	,	,
38536	822	6	these	these	DT
38536	822	7	equations	equation	NNS
38536	822	8	,	,	,
38536	822	9	which	which	WDT
38536	822	10	are	be	VBP
38536	822	11	not	not	RB
38536	822	12	quadratics	quadratic	NNS
38536	822	13	:	:	:
38536	822	14	11	11	CD
38536	822	15	.	.	.
38536	822	16	x^4	x^4	NNP
38536	822	17	=	=	SYM
38536	822	18	16	16	CD
38536	822	19	.	.	.
38536	823	1	12	12	CD
38536	823	2	.	.	.
38536	823	3	x^3	x^3	NNP
38536	823	4	=	=	SYM
38536	823	5	8	8	CD
38536	823	6	.	.	.
38536	824	1	13	13	CD
38536	824	2	.	.	.
38536	824	3	x^3	x^3	NNP
38536	824	4	=	=	SYM
38536	824	5	x.	x.	NNP
38536	825	1	Review	review	VB
38536	825	2	the	the	DT
38536	825	3	solution	solution	NN
38536	825	4	by	by	IN
38536	825	5	formula	formula	NN
38536	825	6	.	.	.
38536	826	1	Solve	solve	VB
38536	826	2	by	by	IN
38536	826	3	it	-PRON-	PRP
38536	826	4	the	the	DT
38536	826	5	following	follow	VBG
38536	826	6	:	:	:
38536	826	7	14	14	CD
38536	826	8	.	.	.
38536	827	1	5x^2	5x^2	CD
38536	827	2	-	-	HYPH
38536	827	3	6x	6x	CD
38536	827	4	=	=	SYM
38536	827	5	8	8	CD
38536	827	6	.	.	.
38536	828	1	15	15	CD
38536	828	2	.	.	.
38536	829	1	(	(	-LRB-
38536	829	2	1/2)(x	1/2)(x	CD
38536	829	3	+	+	SYM
38536	829	4	1	1	CD
38536	829	5	)	)	-RRB-
38536	829	6	-	-	,
38536	829	7	(	(	-LRB-
38536	829	8	x/3)(2x	x/3)(2x	NNP
38536	829	9	-	-	HYPH
38536	829	10	1	1	NNP
38536	829	11	)	)	-RRB-
38536	829	12	=	=	NFP
38536	829	13	-12	-12	XX
38536	829	14	.	.	.
38536	830	1	16	16	CD
38536	830	2	.	.	.
38536	830	3	x^2	x^2	NNS
38536	830	4	+	+	CC
38536	830	5	4ax	4ax	NN
38536	830	6	=	=	SYM
38536	830	7	12a^2	12a^2	XX
38536	830	8	.	.	.
38536	831	1	17	17	CD
38536	831	2	.	.	.
38536	832	1	3x^2	3x^2	CD
38536	832	2	=	=	SYM
38536	832	3	2rx	2rx	JJ
38536	832	4	+	+	SYM
38536	832	5	2r^2	2r^2	CD
38536	832	6	.	.	.
38536	833	1	Solve	solve	VB
38536	833	2	graphically	graphically	RB
38536	833	3	:	:	:
38536	833	4	18	18	CD
38536	833	5	.	.	.
38536	833	6	x^2	x^2	NNP
38536	833	7	-	-	HYPH
38536	833	8	2x	2x	CD
38536	833	9	-	-	HYPH
38536	833	10	8	8	CD
38536	833	11	=	=	SYM
38536	833	12	0	0	CD
38536	833	13	.	.	.
38536	834	1	19	19	CD
38536	834	2	.	.	.
38536	834	3	x^2	x^2	NNP
38536	834	4	+	+	SYM
38536	834	5	x	x	SYM
38536	834	6	-	-	SYM
38536	834	7	2	2	CD
38536	834	8	=	=	SYM
38536	834	9	0	0	CD
38536	834	10	.	.	.
38536	835	1	~Reference:~	~reference:~	VB
38536	835	2	The	the	DT
38536	835	3	chapter	chapter	NN
38536	835	4	on	on	IN
38536	835	5	Quadratic	Quadratic	NNP
38536	835	6	Equations	Equations	NNPS
38536	835	7	in	in	IN
38536	835	8	any	any	DT
38536	835	9	algebra	algebra	NN
38536	835	10	(	(	-LRB-
38536	835	11	first	first	JJ
38536	835	12	part	part	NN
38536	835	13	of	of	IN
38536	835	14	the	the	DT
38536	835	15	chapter	chapter	NN
38536	835	16	)	)	-RRB-
38536	835	17	.	.	.
38536	836	1	1	1	LS
38536	836	2	.	.	.
38536	837	1	Solve	solve	VB
38536	837	2	by	by	IN
38536	837	3	three	three	CD
38536	837	4	methods	method	NNS
38536	837	5	--	--	:
38536	837	6	formula	formula	NN
38536	837	7	,	,	,
38536	837	8	factoring	factoring	NN
38536	837	9	,	,	,
38536	837	10	and	and	CC
38536	837	11	completing	complete	VBG
38536	837	12	the	the	DT
38536	837	13	square	square	NN
38536	837	14	:	:	:
38536	837	15	x^2	x^2	NNS
38536	837	16	+	+	SYM
38536	837	17	10x	10x	NNS
38536	837	18	=	=	SYM
38536	837	19	24	24	CD
38536	837	20	.	.	.
38536	837	21	Review	review	NN
38536	837	22	equations	equation	NNS
38536	837	23	in	in	IN
38536	837	24	the	the	DT
38536	837	25	quadratic	quadratic	JJ
38536	837	26	form	form	NN
38536	837	27	and	and	CC
38536	837	28	solve	solve	VB
38536	837	29	:	:	:
38536	837	30	2	2	CD
38536	837	31	.	.	.
38536	837	32	x^4	x^4	NNP
38536	837	33	-	-	HYPH
38536	837	34	5x^2	5x^2	NNP
38536	837	35	=	=	SYM
38536	837	36	-4	-4	:
38536	837	37	.	.	.
38536	838	1	3	3	LS
38536	838	2	.	.	.
38536	839	1	2[x^(-2)]^(1/3	2[x^(-2)]^(1/3	LS
38536	839	2	)	)	-RRB-
38536	839	3	-	-	:
38536	839	4	3[x^(-1)]^(1/3	3[x^(-1)]^(1/3	CD
38536	839	5	)	)	-RRB-
38536	839	6	=	=	SYM
38536	839	7	2	2	CD
38536	839	8	.	.	.
38536	840	1	4	4	LS
38536	840	2	.	.	.
38536	841	1	(	(	-LRB-
38536	841	2	x	x	SYM
38536	841	3	+	+	NNP
38536	841	4	3)/(x	3)/(x	CD
38536	841	5	-	-	HYPH
38536	841	6	3	3	CD
38536	841	7	)	)	-RRB-
38536	841	8	+	+	CC
38536	841	9	6	6	CD
38536	841	10	=	=	SYM
38536	841	11	5[(x	5[(x	CD
38536	841	12	+	+	SYM
38536	841	13	3)/(x	3)/(x	CD
38536	841	14	-	-	HYPH
38536	841	15	3)]^(1/2	3)]^(1/2	NNP
38536	841	16	)	)	-RRB-
38536	841	17	.	.	.
38536	842	1	(	(	-LRB-
38536	842	2	Let	let	VB
38536	842	3	y	y	PRP
38536	842	4	=	=	NFP
38536	842	5	[	[	-LRB-
38536	842	6	(	(	-LRB-
38536	842	7	x	x	SYM
38536	842	8	+	+	NNP
38536	842	9	3)/(x	3)/(x	NNP
38536	842	10	-	-	HYPH
38536	842	11	3)]^(1/2	3)]^(1/2	NNP
38536	842	12	)	)	-RRB-
38536	842	13	and	and	CC
38536	842	14	substitute	substitute	NN
38536	842	15	.	.	.
38536	842	16	)	)	-RRB-
38536	843	1	5	5	CD
38536	843	2	.	.	.
38536	844	1	3x^2	3x^2	LS
38536	844	2	-	-	HYPH
38536	844	3	4x	4x	NNS
38536	844	4	+	+	SYM
38536	844	5	2[3x^2	2[3x^2	CD
38536	844	6	-	-	HYPH
38536	844	7	4x	4x	NN
38536	844	8	-	-	HYPH
38536	844	9	6]^(1/2	6]^(1/2	NN
38536	844	10	)	)	-RRB-
38536	844	11	=	=	SYM
38536	844	12	21	21	CD
38536	844	13	.	.	.
38536	845	1	6	6	LS
38536	845	2	.	.	.
38536	845	3	x^2	x^2	NNP
38536	845	4	+	+	SYM
38536	845	5	5x	5x	CD
38536	845	6	-	-	HYPH
38536	845	7	5	5	CD
38536	845	8	=	=	SYM
38536	845	9	(	(	-LRB-
38536	845	10	6)/(x^2	6)/(x^2	CD
38536	845	11	+	+	SYM
38536	845	12	5x	5x	CD
38536	845	13	)	)	-RRB-
38536	845	14	.	.	.
38536	846	1	Solve	solve	VB
38536	846	2	and	and	CC
38536	846	3	check	check	VB
38536	846	4	:	:	:
38536	846	5	7	7	CD
38536	846	6	.	.	.
38536	847	1	[	[	-LRB-
38536	847	2	x	x	NN
38536	847	3	+	+	CD
38536	847	4	7]^(1/2	7]^(1/2	NN
38536	847	5	)	)	-RRB-
38536	847	6	+	+	CC
38536	847	7	[	[	-LRB-
38536	847	8	3x	3x	CD
38536	847	9	-	-	HYPH
38536	847	10	2]^(1/2	2]^(1/2	CD
38536	847	11	)	)	-RRB-
38536	847	12	=	=	NFP
38536	847	13	(	(	-LRB-
38536	847	14	4x	4x	NN
38536	847	15	+	+	SYM
38536	847	16	9)/([3x	9)/([3x	CD
38536	847	17	-	-	HYPH
38536	847	18	2]^(1/2	2]^(1/2	NNP
38536	847	19	)	)	-RRB-
38536	847	20	)	)	-RRB-
38536	847	21	.	.	.
38536	848	1	8	8	LS
38536	848	2	.	.	.
38536	849	1	[	[	-LRB-
38536	849	2	x^2	x^2	NNP
38536	849	3	-	-	HYPH
38536	849	4	5]^(1/2	5]^(1/2	NNP
38536	849	5	)	)	-RRB-
38536	849	6	+	+	CC
38536	849	7	6/[[x^2	6/[[x^2	CD
38536	849	8	-	-	HYPH
38536	849	9	5]^(1/2	5]^(1/2	CD
38536	849	10	)	)	-RRB-
38536	849	11	]	]	-RRB-
38536	849	12	=	=	SYM
38536	849	13	5	5	CD
38536	849	14	.	.	.
38536	850	1	9	9	CD
38536	850	2	.	.	.
38536	851	1	(	(	-LRB-
38536	851	2	10w)/([10w	10w)/([10w	JJ
38536	851	3	-	-	HYPH
38536	851	4	9]^(1/2	9]^(1/2	NN
38536	851	5	)	)	-RRB-
38536	851	6	)	)	-RRB-
38536	851	7	-	-	:
38536	851	8	[	[	-LRB-
38536	851	9	10w	10w	CD
38536	851	10	+	+	SYM
38536	851	11	2]^(1/2	2]^(1/2	CD
38536	851	12	)	)	-RRB-
38536	851	13	=	=	NFP
38536	851	14	2/([10w	2/([10w	CD
38536	851	15	-	-	SYM
38536	851	16	9]^(1/2	9]^(1/2	NN
38536	851	17	)	)	-RRB-
38536	851	18	)	)	-RRB-
38536	851	19	.	.	.
38536	852	1	Give	give	VB
38536	852	2	results	result	NNS
38536	852	3	by	by	IN
38536	852	4	inspection	inspection	NN
38536	852	5	:	:	:
38536	852	6	10	10	CD
38536	852	7	.	.	.
38536	853	1	(	(	-LRB-
38536	853	2	a^(1/2	a^(1/2	NNP
38536	853	3	)	)	-RRB-
38536	853	4	+	+	NFP
38536	853	5	b^(1/2))(a^(1/2	b^(1/2))(a^(1/2	ADD
38536	853	6	)	)	-RRB-
38536	853	7	-	-	:
38536	853	8	b^(1/2	b^(1/2	NN
38536	853	9	)	)	-RRB-
38536	853	10	)	)	-RRB-
38536	853	11	.	.	.
38536	854	1	11	11	CD
38536	854	2	.	.	.
38536	855	1	(	(	-LRB-
38536	855	2	[	[	-LRB-
38536	855	3	10	10	CD
38536	855	4	+	+	SYM
38536	855	5	19^(1/2)]^(1/2))([10	19^(1/2)]^(1/2))([10	CD
38536	855	6	-	-	SYM
38536	855	7	19^(1/2)]^(1/2	19^(1/2)]^(1/2	CD
38536	855	8	)	)	-RRB-
38536	855	9	)	)	-RRB-
38536	855	10	.	.	.
38536	856	1	12	12	CD
38536	856	2	.	.	.
38536	857	1	How	how	WRB
38536	857	2	many	many	JJ
38536	857	3	gallons	gallon	NNS
38536	857	4	each	each	DT
38536	857	5	of	of	IN
38536	857	6	cream	cream	NN
38536	857	7	containing	contain	VBG
38536	857	8	33	33	CD
38536	857	9	%	%	NN
38536	857	10	butter	butter	NN
38536	857	11	fat	fat	NN
38536	857	12	and	and	CC
38536	857	13	milk	milk	NN
38536	857	14	containing	contain	VBG
38536	857	15	6	6	CD
38536	857	16	%	%	NN
38536	857	17	butter	butter	NN
38536	857	18	fat	fat	NN
38536	857	19	must	must	MD
38536	857	20	be	be	VB
38536	857	21	mixed	mix	VBN
38536	857	22	to	to	TO
38536	857	23	produce	produce	VB
38536	857	24	10	10	CD
38536	857	25	gallons	gallon	NNS
38536	857	26	of	of	IN
38536	857	27	cream	cream	NN
38536	857	28	containing	contain	VBG
38536	857	29	25	25	CD
38536	857	30	%	%	NN
38536	857	31	butter	butter	NN
38536	857	32	fat	fat	NN
38536	857	33	?	?	.
38536	858	1	13	13	CD
38536	858	2	.	.	.
38536	859	1	I	-PRON-	PRP
38536	859	2	have	have	VBP
38536	859	3	$	$	$
38536	859	4	6	6	CD
38536	859	5	in	in	IN
38536	859	6	dimes	dime	NNS
38536	859	7	,	,	,
38536	859	8	quarters	quarter	NNS
38536	859	9	,	,	,
38536	859	10	and	and	CC
38536	859	11	half	half	NN
38536	859	12	-	-	HYPH
38536	859	13	dollars	dollar	NNS
38536	859	14	,	,	,
38536	859	15	there	there	EX
38536	859	16	being	be	VBG
38536	859	17	33	33	CD
38536	859	18	coins	coin	NNS
38536	859	19	in	in	IN
38536	859	20	all	all	DT
38536	859	21	.	.	.
38536	860	1	The	the	DT
38536	860	2	number	number	NN
38536	860	3	of	of	IN
38536	860	4	dimes	dime	NNS
38536	860	5	and	and	CC
38536	860	6	quarters	quarter	NNS
38536	860	7	together	together	RB
38536	860	8	is	be	VBZ
38536	860	9	ten	ten	CD
38536	860	10	times	time	NNS
38536	860	11	the	the	DT
38536	860	12	number	number	NN
38536	860	13	of	of	IN
38536	860	14	half	half	NN
38536	860	15	-	-	HYPH
38536	860	16	dollars	dollar	NNS
38536	860	17	.	.	.
38536	861	1	How	how	WRB
38536	861	2	many	many	JJ
38536	861	3	coins	coin	NNS
38536	861	4	of	of	IN
38536	861	5	each	each	DT
38536	861	6	kind	kind	NN
38536	861	7	are	be	VBP
38536	861	8	there	there	RB
38536	861	9	?	?	.
38536	862	1	(	(	-LRB-
38536	862	2	_	_	NNP
38536	862	3	College	College	NNP
38536	862	4	Entrance	Entrance	NNP
38536	862	5	Board	Board	NNP
38536	862	6	.	.	.
38536	862	7	_	_	NNP
38536	862	8	)	)	-RRB-
38536	862	9	~Reference:~	~Reference:~	NNP
38536	862	10	The	the	DT
38536	862	11	last	last	JJ
38536	862	12	part	part	NN
38536	862	13	of	of	IN
38536	862	14	the	the	DT
38536	862	15	chapter	chapter	NN
38536	862	16	on	on	IN
38536	862	17	Quadratic	Quadratic	NNP
38536	862	18	Equations	Equations	NNPS
38536	862	19	in	in	IN
38536	862	20	any	any	DT
38536	862	21	algebra	algebra	NN
38536	862	22	.	.	.
38536	863	1	THE	the	DT
38536	863	2	THEORY	theory	NN
38536	863	3	OF	of	IN
38536	863	4	QUADRATIC	quadratic	NN
38536	863	5	EQUATIONS	EQUATIONS	NNP
38536	863	6	~I.	~i.	NN
38536	864	1	To	to	TO
38536	864	2	find	find	VB
38536	864	3	the	the	DT
38536	864	4	sum	sum	NN
38536	864	5	and	and	CC
38536	864	6	the	the	DT
38536	864	7	product	product	NN
38536	864	8	of	of	IN
38536	864	9	the	the	DT
38536	864	10	roots.~	roots.~	NNP
38536	864	11	The	the	DT
38536	864	12	general	general	JJ
38536	864	13	quadratic	quadratic	JJ
38536	864	14	equation	equation	NN
38536	864	15	is	be	VBZ
38536	864	16	ax^2	ax^2	DT
38536	864	17	+	+	SYM
38536	864	18	bx	bx	NN
38536	864	19	+	+	NNS
38536	864	20	c	c	NN
38536	864	21	=	=	SYM
38536	864	22	0	0	NFP
38536	864	23	.	.	.
38536	865	1	(	(	-LRB-
38536	865	2	1	1	LS
38536	865	3	)	)	-RRB-
38536	865	4	Or	or	CC
38536	865	5	,	,	,
38536	865	6	x^2	x^2	NNP
38536	865	7	+	+	SYM
38536	865	8	(	(	-LRB-
38536	865	9	b	b	NNP
38536	865	10	/	/	SYM
38536	865	11	a)x	a)x	NNP
38536	865	12	+	+	SYM
38536	865	13	c	c	NN
38536	865	14	/	/	SYM
38536	865	15	a	a	NN
38536	865	16	=	=	NN
38536	865	17	0	0	CD
38536	865	18	.	.	.
38536	866	1	(	(	-LRB-
38536	866	2	2	2	LS
38536	866	3	)	)	-RRB-
38536	866	4	To	to	TO
38536	866	5	derive	derive	VB
38536	866	6	the	the	DT
38536	866	7	formula	formula	NN
38536	866	8	,	,	,
38536	866	9	we	-PRON-	PRP
38536	866	10	have	have	VBP
38536	866	11	by	by	IN
38536	866	12	transposing	transpose	VBG
38536	866	13	x^2	x^2	NNS
38536	866	14	+	+	NFP
38536	866	15	(	(	-LRB-
38536	866	16	b	b	NNP
38536	866	17	/	/	SYM
38536	866	18	a)x	a)x	NNP
38536	866	19	=	=	SYM
38536	866	20	-c	-c	:
38536	866	21	/	/	SYM
38536	866	22	a	a	NN
38536	866	23	.	.	.
38536	867	1	Completing	complete	VBG
38536	867	2	the	the	DT
38536	867	3	square	square	NN
38536	867	4	,	,	,
38536	867	5	x^2	x^2	NNP
38536	867	6	+	+	SYM
38536	867	7	(	(	-LRB-
38536	867	8	b	b	NNP
38536	867	9	/	/	SYM
38536	867	10	a)x	a)x	NN
38536	867	11	+	+	CC
38536	867	12	[	[	-LRB-
38536	867	13	b/2a]^2	b/2a]^2	JJ
38536	867	14	=	=	NFP
38536	867	15	(	(	-LRB-
38536	867	16	b^2)/(4a^2	b^2)/(4a^2	NNP
38536	867	17	)	)	-RRB-
38536	867	18	-	-	:
38536	867	19	c	c	NN
38536	867	20	/	/	SYM
38536	867	21	a	a	NN
38536	867	22	=	=	NN
38536	867	23	(	(	-LRB-
38536	867	24	b^2	b^2	NNP
38536	867	25	-	-	:
38536	867	26	4ac)/(4a^2	4ac)/(4a^2	NNP
38536	867	27	)	)	-RRB-
38536	867	28	.	.	.
38536	868	1	Extracting	extract	VBG
38536	868	2	square	square	JJ
38536	868	3	root	root	NN
38536	868	4	,	,	,
38536	868	5	x	x	SYM
38536	868	6	+	+	NFP
38536	868	7	b/2a	b/2a	FW
38536	868	8	=	=	NFP
38536	868	9	[	[	-LRB-
38536	868	10	±[b^2	±[b^2	NNP
38536	868	11	-	-	HYPH
38536	868	12	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	868	13	)	)	-RRB-
38536	868	14	.	.	.
38536	869	1	Transposing	transpose	VBG
38536	869	2	,	,	,
38536	869	3	x	x	NNP
38536	869	4	=	=	SYM
38536	869	5	-b/2a	-b/2a	:
38536	869	6	±	±	NNP
38536	869	7	[	[	-LRB-
38536	869	8	[	[	-LRB-
38536	869	9	b^2	b^2	NNS
38536	869	10	-	-	SYM
38536	869	11	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	869	12	)	)	-RRB-
38536	869	13	.	.	.
38536	870	1	Hence	hence	RB
38536	870	2	,	,	,
38536	870	3	x	x	NNS
38536	870	4	=	=	SYM
38536	870	5	[	[	-LRB-
38536	870	6	-b	-b	:
38536	870	7	±	±	CD
38536	870	8	[	[	-LRB-
38536	870	9	b^2	b^2	NNS
38536	870	10	-	-	SYM
38536	870	11	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	870	12	)	)	-RRB-
38536	870	13	.	.	.
38536	871	1	These	these	DT
38536	871	2	two	two	CD
38536	871	3	values	value	NNS
38536	871	4	of	of	IN
38536	871	5	x	x	NNS
38536	871	6	we	-PRON-	PRP
38536	871	7	call	call	VBP
38536	871	8	_	_	NNP
38536	871	9	roots	root	NNS
38536	871	10	_	_	NNP
38536	871	11	.	.	.
38536	872	1	For	for	IN
38536	872	2	convenience	convenience	NN
38536	872	3	represent	represent	VBP
38536	872	4	them	-PRON-	PRP
38536	872	5	by	by	IN
38536	872	6	r_1	r_1	NNP
38536	872	7	and	and	CC
38536	872	8	r_2	r_2	NNP
38536	872	9	.	.	.
38536	873	1	Hence	hence	RB
38536	873	2	,	,	,
38536	873	3	r_1	r_1	NNP
38536	873	4	=	=	SYM
38536	873	5	-b/2a	-b/2a	:
38536	873	6	+	+	SYM
38536	873	7	[	[	-LRB-
38536	873	8	[	[	-LRB-
38536	873	9	b^2	b^2	NNS
38536	873	10	-	-	SYM
38536	873	11	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	873	12	)	)	-RRB-
38536	873	13	.	.	.
38536	874	1	r_2	r_2	NN
38536	874	2	=	=	SYM
38536	874	3	-b/2a	-b/2a	:
38536	874	4	-	-	HYPH
38536	874	5	[	[	-LRB-
38536	874	6	[	[	-LRB-
38536	874	7	b^2	b^2	NNS
38536	874	8	-	-	SYM
38536	874	9	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	874	10	)	)	-RRB-
38536	874	11	.	.	.
38536	875	1	---------------------------------------------	---------------------------------------------	NFP
38536	875	2	Adding	Adding	NNP
38536	875	3	,	,	,
38536	875	4	r_1	r_1	NNP
38536	875	5	+	+	CC
38536	875	6	r_2	r_2	CD
38536	875	7	=	=	SYM
38536	875	8	-(2b)/(2a	-(2b)/(2a	NFP
38536	875	9	)	)	-RRB-
38536	875	10	=	=	NFP
38536	875	11	-b	-b	:
38536	875	12	/	/	SYM
38536	875	13	a	a	NN
38536	875	14	.	.	.
38536	876	1	(	(	-LRB-
38536	876	2	3	3	LS
38536	876	3	)	)	-RRB-
38536	876	4	Also	also	RB
38536	876	5	,	,	,
38536	876	6	r_1	r_1	NNP
38536	876	7	=	=	SYM
38536	876	8	-b/2a	-b/2a	:
38536	876	9	+	+	SYM
38536	876	10	[	[	-LRB-
38536	876	11	[	[	-LRB-
38536	876	12	b^2	b^2	NNS
38536	876	13	-	-	SYM
38536	876	14	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	876	15	)	)	-RRB-
38536	876	16	.	.	.
38536	877	1	r_2	r_2	NN
38536	877	2	=	=	SYM
38536	877	3	-b/2a	-b/2a	:
38536	877	4	-	-	HYPH
38536	877	5	[	[	-LRB-
38536	877	6	[	[	-LRB-
38536	877	7	b^2	b^2	NNS
38536	877	8	-	-	SYM
38536	877	9	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	877	10	)	)	-RRB-
38536	877	11	.	.	.
38536	878	1	-------------------------------------------	-------------------------------------------	NFP
38536	878	2	Multiplying	multiply	VBG
38536	878	3	,	,	,
38536	878	4	r_1	r_1	NNP
38536	878	5	r_2	r_2	CD
38536	878	6	=	=	NFP
38536	878	7	(	(	-LRB-
38536	878	8	b^2)/(4a^2	b^2)/(4a^2	NNP
38536	878	9	)	)	-RRB-
38536	878	10	-	-	:
38536	878	11	(	(	-LRB-
38536	878	12	b^2	b^2	NNP
38536	878	13	-	-	:
38536	878	14	4ac)/(4a^2	4ac)/(4a^2	NNP
38536	878	15	)	)	-RRB-
38536	878	16	=	=	NFP
38536	878	17	(	(	-LRB-
38536	878	18	b^2	b^2	NNS
38536	878	19	-	-	HYPH
38536	878	20	b^2	b^2	NNP
38536	878	21	+	+	NNP
38536	878	22	4ac)/(4a^2	4ac)/(4a^2	NNP
38536	878	23	)	)	-RRB-
38536	878	24	=	=	NFP
38536	878	25	(	(	-LRB-
38536	878	26	4ac)/(4a^2	4ac)/(4a^2	NNP
38536	878	27	)	)	-RRB-
38536	878	28	=	=	SYM
38536	878	29	c	c	NN
38536	878	30	/	/	SYM
38536	878	31	a	a	NN
38536	878	32	.	.	.
38536	879	1	(	(	-LRB-
38536	879	2	4	4	LS
38536	879	3	)	)	-RRB-
38536	879	4	Hence	hence	RB
38536	879	5	we	-PRON-	PRP
38536	879	6	have	have	VBP
38536	879	7	shown	show	VBN
38536	879	8	that	that	IN
38536	879	9	r_1	r_1	NNP
38536	879	10	+	+	CC
38536	879	11	r_2	r_2	CD
38536	879	12	=	=	SYM
38536	879	13	-b	-b	:
38536	879	14	/	/	SYM
38536	879	15	a	a	NN
38536	879	16	,	,	,
38536	879	17	and	and	CC
38536	879	18	r_1	r_1	NNP
38536	879	19	r_2	r_2	NNP
38536	879	20	=	=	SYM
38536	879	21	c	c	NN
38536	879	22	/	/	SYM
38536	879	23	a	a	NN
38536	879	24	.	.	.
38536	880	1	Or	or	CC
38536	880	2	,	,	,
38536	880	3	referring	refer	VBG
38536	880	4	to	to	IN
38536	880	5	equation	equation	NN
38536	880	6	(	(	-LRB-
38536	880	7	2	2	LS
38536	880	8	)	)	-RRB-
38536	880	9	above	above	RB
38536	880	10	,	,	,
38536	880	11	we	-PRON-	PRP
38536	880	12	have	have	VBP
38536	880	13	the	the	DT
38536	880	14	following	follow	VBG
38536	880	15	rule	rule	NN
38536	880	16	:	:	:
38536	880	17	_	_	NNP
38536	880	18	When	when	WRB
38536	880	19	the	the	DT
38536	880	20	coefficient	coefficient	NN
38536	880	21	of	of	IN
38536	880	22	x^2	x^2	NNP
38536	880	23	is	be	VBZ
38536	880	24	unity	unity	NN
38536	880	25	,	,	,
38536	880	26	the	the	DT
38536	880	27	sum	sum	NN
38536	880	28	of	of	IN
38536	880	29	the	the	DT
38536	880	30	roots	root	NNS
38536	880	31	is	be	VBZ
38536	880	32	the	the	DT
38536	880	33	coefficient	coefficient	NN
38536	880	34	of	of	IN
38536	880	35	x	x	NN
38536	880	36	with	with	IN
38536	880	37	the	the	DT
38536	880	38	sign	sign	NN
38536	880	39	changed	change	VBN
38536	880	40	;	;	:
38536	880	41	the	the	DT
38536	880	42	product	product	NN
38536	880	43	of	of	IN
38536	880	44	the	the	DT
38536	880	45	roots	root	NNS
38536	880	46	is	be	VBZ
38536	880	47	the	the	DT
38536	880	48	independent	independent	JJ
38536	880	49	term	term	NN
38536	880	50	.	.	.
38536	880	51	_	_	NNP
38536	880	52	EXAMPLES	EXAMPLES	NNP
38536	880	53	:	:	:
38536	880	54	1	1	LS
38536	880	55	.	.	.
38536	880	56	x^2	x^2	NNP
38536	880	57	-	-	HYPH
38536	880	58	9x	9x	CD
38536	880	59	+	+	SYM
38536	880	60	21	21	CD
38536	880	61	=	=	SYM
38536	880	62	0	0	CD
38536	880	63	.	.	.
38536	881	1	Sum	Sum	NNP
38536	881	2	of	of	IN
38536	881	3	the	the	DT
38536	881	4	roots	root	NNS
38536	881	5	=	=	SYM
38536	881	6	9	9	CD
38536	881	7	.	.	.
38536	882	1	Products	product	NNS
38536	882	2	of	of	IN
38536	882	3	the	the	DT
38536	882	4	roots	root	NNS
38536	882	5	=	=	SYM
38536	882	6	21	21	CD
38536	882	7	.	.	.
38536	883	1	2	2	LS
38536	883	2	.	.	.
38536	884	1	3x^2	3x^2	CD
38536	884	2	-	-	HYPH
38536	884	3	7x	7x	CD
38536	884	4	-	-	HYPH
38536	884	5	18	18	CD
38536	884	6	=	=	SYM
38536	884	7	0	0	CD
38536	884	8	.	.	.
38536	885	1	Sum	Sum	NNP
38536	885	2	of	of	IN
38536	885	3	the	the	DT
38536	885	4	roots	root	NNS
38536	885	5	=	=	SYM
38536	885	6	7/3	7/3	CD
38536	885	7	.	.	.
38536	886	1	Product	product	NN
38536	886	2	of	of	IN
38536	886	3	the	the	DT
38536	886	4	roots	root	NNS
38536	886	5	=	=	:
38536	886	6	-6	-6	:
38536	886	7	.	.	.
38536	887	1	3	3	LS
38536	887	2	.	.	.
38536	888	1	-21x	-21x	NFP
38536	888	2	=	=	SYM
38536	888	3	17	17	CD
38536	888	4	-	-	SYM
38536	888	5	4x^2	4x^2	CD
38536	888	6	.	.	.
38536	889	1	Sum	Sum	NNP
38536	889	2	of	of	IN
38536	889	3	the	the	DT
38536	889	4	roots	root	NNS
38536	889	5	=	=	SYM
38536	889	6	21/4	21/4	CD
38536	889	7	.	.	.
38536	890	1	Product	product	NN
38536	890	2	of	of	IN
38536	890	3	the	the	DT
38536	890	4	roots	root	NNS
38536	890	5	=	=	SYM
38536	890	6	-17/4	-17/4	:
38536	890	7	.	.	.
38536	891	1	~II	~II	NNP
38536	891	2	.	.	.
38536	892	1	To	to	TO
38536	892	2	find	find	VB
38536	892	3	the	the	DT
38536	892	4	nature	nature	NN
38536	892	5	or	or	CC
38536	892	6	character	character	NN
38536	892	7	of	of	IN
38536	892	8	the	the	DT
38536	892	9	roots.~	roots.~	NNP
38536	892	10	As	as	IN
38536	892	11	before	before	RB
38536	892	12	,	,	,
38536	892	13	r_1	r_1	NNP
38536	892	14	=	=	SYM
38536	892	15	-b/2a	-b/2a	:
38536	892	16	+	+	SYM
38536	892	17	[	[	-LRB-
38536	892	18	[	[	-LRB-
38536	892	19	b^2	b^2	NNS
38536	892	20	-	-	SYM
38536	892	21	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	892	22	)	)	-RRB-
38536	892	23	,	,	,
38536	892	24	r_2	r_2	NN
38536	892	25	=	=	SYM
38536	892	26	-b/2a	-b/2a	:
38536	892	27	-	-	HYPH
38536	892	28	[	[	-LRB-
38536	892	29	[	[	-LRB-
38536	892	30	b^2	b^2	NNS
38536	892	31	-	-	SYM
38536	892	32	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	892	33	)	)	-RRB-
38536	892	34	.	.	.
38536	893	1	The	the	DT
38536	893	2	[	[	-LRB-
38536	893	3	b^2	b^2	FW
38536	893	4	-	-	:
38536	893	5	4ac]^(1/2	4ac]^(1/2	CD
38536	893	6	)	)	-RRB-
38536	893	7	determines	determine	VBZ
38536	893	8	the	the	DT
38536	893	9	_	_	NNP
38536	893	10	nature	nature	NN
38536	893	11	_	_	NNP
38536	893	12	or	or	CC
38536	893	13	_	_	NNP
38536	893	14	character	character	NN
38536	893	15	_	_	NNP
38536	893	16	of	of	IN
38536	893	17	the	the	DT
38536	893	18	roots	root	NNS
38536	893	19	;	;	:
38536	893	20	hence	hence	RB
38536	893	21	it	-PRON-	PRP
38536	893	22	is	be	VBZ
38536	893	23	called	call	VBN
38536	893	24	the	the	DT
38536	893	25	_	_	NNP
38536	893	26	discriminant	discriminant	JJ
38536	893	27	_	_	NNP
38536	893	28	.	.	.
38536	894	1	~If	~If	NFP
38536	894	2	b^2	b^2	NNP
38536	894	3	-	-	HYPH
38536	894	4	4ac	4ac	NNP
38536	894	5	is	be	VBZ
38536	894	6	positive	positive	JJ
38536	894	7	,	,	,
38536	894	8	the	the	DT
38536	894	9	roots	root	NNS
38536	894	10	are	be	VBP
38536	894	11	real	real	JJ
38536	894	12	,	,	,
38536	894	13	unequal	unequal	JJ
38536	894	14	,	,	,
38536	894	15	and	and	CC
38536	894	16	either	either	CC
38536	894	17	rational	rational	JJ
38536	894	18	or	or	CC
38536	894	19	irrational.~	irrational.~	NNPS
38536	894	20	~If	~If	NFP
38536	894	21	b^2	b^2	NNP
38536	894	22	-	-	HYPH
38536	894	23	4ac	4ac	NN
38536	894	24	is	be	VBZ
38536	894	25	negative	negative	JJ
38536	894	26	,	,	,
38536	894	27	the	the	DT
38536	894	28	roots	root	NNS
38536	894	29	are	be	VBP
38536	894	30	imaginary	imaginary	JJ
38536	894	31	and	and	CC
38536	894	32	unequal.~	unequal.~	JJ
38536	894	33	~If	~If	NFP
38536	894	34	b^2	b^2	NNP
38536	894	35	-	-	HYPH
38536	894	36	4ac	4ac	NN
38536	894	37	is	be	VBZ
38536	894	38	zero	zero	CD
38536	894	39	,	,	,
38536	894	40	the	the	DT
38536	894	41	roots	root	NNS
38536	894	42	are	be	VBP
38536	894	43	real	real	JJ
38536	894	44	,	,	,
38536	894	45	equal	equal	JJ
38536	894	46	,	,	,
38536	894	47	and	and	CC
38536	894	48	rational.~	rational.~	NNP
38536	894	49	EXAMPLES	EXAMPLES	NNP
38536	894	50	:	:	:
38536	894	51	1	1	LS
38536	894	52	.	.	.
38536	894	53	x^2	x^2	NNP
38536	894	54	-	-	HYPH
38536	894	55	4x	4x	NNP
38536	894	56	+	+	SYM
38536	894	57	2	2	CD
38536	894	58	=	=	SYM
38536	894	59	0	0	CD
38536	894	60	.	.	.
38536	895	1	[	[	-LRB-
38536	895	2	b^2	b^2	NNP
38536	895	3	-	-	:
38536	895	4	4ac]^(1/2	4ac]^(1/2	CD
38536	895	5	)	)	-RRB-
38536	895	6	=	=	NFP
38536	895	7	[	[	-LRB-
38536	895	8	16	16	CD
38536	895	9	-	-	HYPH
38536	895	10	8]^(1/2	8]^(1/2	NN
38536	895	11	)	)	-RRB-
38536	895	12	=	=	NFP
38536	895	13	8^(1/2	8^(1/2	CD
38536	895	14	)	)	-RRB-
38536	895	15	.	.	.
38536	896	1	Therefore	therefore	RB
38536	896	2	:	:	:
38536	896	3	The	the	DT
38536	896	4	roots	root	NNS
38536	896	5	are	be	VBP
38536	896	6	real	real	JJ
38536	896	7	,	,	,
38536	896	8	unequal	unequal	JJ
38536	896	9	,	,	,
38536	896	10	and	and	CC
38536	896	11	irrational	irrational	JJ
38536	896	12	.	.	.
38536	897	1	2	2	LS
38536	897	2	.	.	.
38536	897	3	x^2	x^2	NNP
38536	897	4	-	-	HYPH
38536	897	5	4x	4x	NNS
38536	897	6	+	+	SYM
38536	897	7	6	6	CD
38536	897	8	=	=	SYM
38536	897	9	0	0	CD
38536	897	10	.	.	.
38536	898	1	[	[	-LRB-
38536	898	2	b^2	b^2	NNP
38536	898	3	-	-	:
38536	898	4	4ac]^(1/2	4ac]^(1/2	CD
38536	898	5	)	)	-RRB-
38536	898	6	=	=	NFP
38536	898	7	[	[	-LRB-
38536	898	8	16	16	CD
38536	898	9	-	-	SYM
38536	898	10	24]^(1/2	24]^(1/2	CD
38536	898	11	)	)	-RRB-
38536	898	12	=	=	NFP
38536	898	13	-8^(1/2	-8^(1/2	NFP
38536	898	14	)	)	-RRB-
38536	898	15	.	.	.
38536	899	1	Therefore	therefore	RB
38536	899	2	:	:	:
38536	899	3	The	the	DT
38536	899	4	roots	root	NNS
38536	899	5	are	be	VBP
38536	899	6	imaginary	imaginary	JJ
38536	899	7	and	and	CC
38536	899	8	unequal	unequal	JJ
38536	899	9	.	.	.
38536	900	1	3	3	LS
38536	900	2	.	.	.
38536	900	3	x^2	x^2	NNP
38536	900	4	-	-	HYPH
38536	900	5	4x	4x	NNS
38536	900	6	+	+	SYM
38536	900	7	4	4	CD
38536	900	8	=	=	SYM
38536	900	9	0	0	CD
38536	900	10	.	.	.
38536	901	1	[	[	-LRB-
38536	901	2	b^2	b^2	NNP
38536	901	3	-	-	:
38536	901	4	4ac]^(1/2	4ac]^(1/2	CD
38536	901	5	)	)	-RRB-
38536	901	6	=	=	NFP
38536	901	7	[	[	-LRB-
38536	901	8	16	16	CD
38536	901	9	-	-	SYM
38536	901	10	16]^(1/2	16]^(1/2	CD
38536	901	11	)	)	-RRB-
38536	901	12	=	=	SYM
38536	901	13	0^(1/2	0^(1/2	LS
38536	901	14	)	)	-RRB-
38536	901	15	.	.	.
38536	902	1	Therefore	therefore	RB
38536	902	2	:	:	:
38536	902	3	The	the	DT
38536	902	4	roots	root	NNS
38536	902	5	are	be	VBP
38536	902	6	real	real	JJ
38536	902	7	,	,	,
38536	902	8	equal	equal	JJ
38536	902	9	,	,	,
38536	902	10	and	and	CC
38536	902	11	rational	rational	JJ
38536	902	12	.	.	.
38536	903	1	~III	~III	NFP
38536	903	2	.	.	.
38536	904	1	To	to	TO
38536	904	2	form	form	VB
38536	904	3	the	the	DT
38536	904	4	quadratic	quadratic	JJ
38536	904	5	equation	equation	NN
38536	904	6	when	when	WRB
38536	904	7	the	the	DT
38536	904	8	roots	root	NNS
38536	904	9	are	be	VBP
38536	904	10	given.~	given.~	NNPS
38536	904	11	Suppose	suppose	VB
38536	904	12	the	the	DT
38536	904	13	roots	root	NNS
38536	904	14	are	be	VBP
38536	904	15	3	3	CD
38536	904	16	,	,	,
38536	904	17	-7	-7	:
38536	904	18	.	.	.
38536	905	1	Then	then	RB
38536	905	2	,	,	,
38536	905	3	x	x	NNS
38536	905	4	=	=	SYM
38536	905	5	3	3	CD
38536	905	6	,	,	,
38536	905	7	Or	or	CC
38536	905	8	,	,	,
38536	905	9	x	x	SYM
38536	905	10	-	-	SYM
38536	905	11	3	3	CD
38536	905	12	=	=	SYM
38536	905	13	0	0	CD
38536	905	14	,	,	,
38536	905	15	x	x	SYM
38536	905	16	=	=	SYM
38536	905	17	-7	-7	:
38536	905	18	.	.	.
38536	905	19	x	x	NNS
38536	905	20	+	+	SYM
38536	905	21	7	7	CD
38536	905	22	=	=	SYM
38536	905	23	0	0	CD
38536	905	24	.	.	.
38536	906	1	-------------------	-------------------	NFP
38536	906	2	Multiplying	multiply	VBG
38536	906	3	to	to	TO
38536	906	4	get	get	VB
38536	906	5	a	a	DT
38536	906	6	quadratic	quadratic	NN
38536	906	7	,	,	,
38536	906	8	(	(	-LRB-
38536	906	9	x	x	NNS
38536	906	10	-	-	SYM
38536	906	11	3)(x	3)(x	CD
38536	906	12	+	+	SYM
38536	906	13	7	7	CD
38536	906	14	)	)	-RRB-
38536	906	15	=	=	SYM
38536	906	16	0	0	NFP
38536	906	17	.	.	.
38536	907	1	Or	or	CC
38536	907	2	,	,	,
38536	907	3	x^2	x^2	NNS
38536	907	4	+	+	SYM
38536	907	5	4x	4x	NNP
38536	907	6	-	-	SYM
38536	907	7	21	21	CD
38536	907	8	=	=	SYM
38536	907	9	0	0	CD
38536	907	10	.	.	.
38536	908	1	_	_	NNP
38536	908	2	Or	or	CC
38536	908	3	_	_	NNP
38536	908	4	,	,	,
38536	908	5	use	use	VB
38536	908	6	the	the	DT
38536	908	7	sum	sum	NN
38536	908	8	and	and	CC
38536	908	9	product	product	NN
38536	908	10	idea	idea	NN
38536	908	11	developed	develop	VBN
38536	908	12	on	on	IN
38536	908	13	the	the	DT
38536	908	14	preceding	precede	VBG
38536	908	15	page	page	NN
38536	908	16	.	.	.
38536	909	1	The	the	DT
38536	909	2	coefficient	coefficient	NN
38536	909	3	of	of	IN
38536	909	4	x^2	x^2	NNS
38536	909	5	must	must	MD
38536	909	6	be	be	VB
38536	909	7	unity	unity	NN
38536	909	8	.	.	.
38536	910	1	Add	add	VB
38536	910	2	the	the	DT
38536	910	3	roots	root	NNS
38536	910	4	and	and	CC
38536	910	5	change	change	VB
38536	910	6	the	the	DT
38536	910	7	sign	sign	NN
38536	910	8	to	to	TO
38536	910	9	get	get	VB
38536	910	10	the	the	DT
38536	910	11	coefficient	coefficient	NN
38536	910	12	of	of	IN
38536	910	13	x.	x.	NNP
38536	911	1	Multiply	multiply	VB
38536	911	2	the	the	DT
38536	911	3	roots	root	NNS
38536	911	4	to	to	TO
38536	911	5	get	get	VB
38536	911	6	the	the	DT
38536	911	7	independent	independent	JJ
38536	911	8	term	term	NN
38536	911	9	.	.	.
38536	912	1	Therefore	therefore	RB
38536	912	2	:	:	:
38536	912	3	The	the	DT
38536	912	4	equation	equation	NN
38536	912	5	is	be	VBZ
38536	912	6	x^2	x^2	NNP
38536	912	7	+	+	SYM
38536	912	8	4x	4x	NNP
38536	912	9	-	-	SYM
38536	912	10	21	21	CD
38536	912	11	=	=	SYM
38536	912	12	0	0	CD
38536	912	13	.	.	.
38536	913	1	In	in	IN
38536	913	2	the	the	DT
38536	913	3	same	same	JJ
38536	913	4	way	way	NN
38536	913	5	,	,	,
38536	913	6	if	if	IN
38536	913	7	the	the	DT
38536	913	8	roots	root	NNS
38536	913	9	are	be	VBP
38536	913	10	[	[	-LRB-
38536	913	11	2	2	CD
38536	913	12	+	+	SYM
38536	913	13	3^(1/2)]/7	3^(1/2)]/7	CD
38536	913	14	,	,	,
38536	913	15	[	[	-LRB-
38536	913	16	2	2	CD
38536	913	17	-	-	SYM
38536	913	18	3^(1/2)]/7	3^(1/2)]/7	CD
38536	913	19	,	,	,
38536	913	20	the	the	DT
38536	913	21	equation	equation	NN
38536	913	22	is	be	VBZ
38536	913	23	x^2	x^2	NNP
38536	913	24	-	-	,
38536	913	25	(	(	-LRB-
38536	913	26	4/7)x	4/7)x	CD
38536	913	27	+	+	CC
38536	913	28	1/49	1/49	CD
38536	913	29	=	=	SYM
38536	913	30	0	0	CD
38536	913	31	.	.	.
38536	914	1	Find	find	VB
38536	914	2	the	the	DT
38536	914	3	sum	sum	NN
38536	914	4	,	,	,
38536	914	5	the	the	DT
38536	914	6	product	product	NN
38536	914	7	,	,	,
38536	914	8	and	and	CC
38536	914	9	the	the	DT
38536	914	10	nature	nature	NN
38536	914	11	or	or	CC
38536	914	12	character	character	NN
38536	914	13	of	of	IN
38536	914	14	the	the	DT
38536	914	15	roots	root	NNS
38536	914	16	of	of	IN
38536	914	17	the	the	DT
38536	914	18	following	follow	VBG
38536	914	19	:	:	:
38536	914	20	1	1	LS
38536	914	21	.	.	.
38536	914	22	x^2	x^2	NNP
38536	914	23	-	-	SYM
38536	914	24	7x	7x	NNP
38536	914	25	+	+	CC
38536	914	26	12	12	CD
38536	914	27	=	=	SYM
38536	914	28	0	0	CD
38536	914	29	.	.	.
38536	915	1	2	2	LS
38536	915	2	.	.	.
38536	916	1	9x^2	9x^2	CD
38536	916	2	-	-	HYPH
38536	916	3	6x	6x	CD
38536	916	4	+	+	SYM
38536	916	5	1	1	CD
38536	916	6	=	=	SYM
38536	916	7	0	0	CD
38536	916	8	.	.	.
38536	917	1	3	3	LS
38536	917	2	.	.	.
38536	917	3	x^2	x^2	NNS
38536	917	4	+	+	SYM
38536	917	5	2x	2x	CD
38536	917	6	+	+	SYM
38536	917	7	9734	9734	CD
38536	917	8	=	=	SYM
38536	917	9	0	0	CD
38536	917	10	.	.	.
38536	918	1	4	4	LS
38536	918	2	.	.	.
38536	919	1	16	16	CD
38536	919	2	+	+	SYM
38536	919	3	5	5	CD
38536	919	4	/	/	SYM
38536	919	5	x	x	NNS
38536	919	6	=	=	SYM
38536	919	7	17/(x^2	17/(x^2	CD
38536	919	8	)	)	-RRB-
38536	919	9	.	.	.
38536	920	1	5	5	CD
38536	920	2	.	.	.
38536	921	1	(	(	-LRB-
38536	921	2	x	x	LS
38536	921	3	-	-	NNP
38536	921	4	8)/(x	8)/(x	CD
38536	921	5	-	-	HYPH
38536	921	6	3	3	CD
38536	921	7	)	)	-RRB-
38536	921	8	=	=	NFP
38536	921	9	x.	x.	NNP
38536	922	1	6	6	CD
38536	922	2	.	.	.
38536	923	1	(	(	-LRB-
38536	923	2	x	x	SYM
38536	923	3	+	+	SYM
38536	923	4	7)(x	7)(x	CD
38536	923	5	-	-	SYM
38536	923	6	6	6	CD
38536	923	7	)	)	-RRB-
38536	923	8	=	=	SYM
38536	923	9	70	70	CD
38536	923	10	.	.	.
38536	924	1	7	7	LS
38536	924	2	.	.	.
38536	924	3	x^2	x^2	NNP
38536	924	4	-	-	HYPH
38536	924	5	x(2)^(1/2	x(2)^(1/2	NNP
38536	924	6	)	)	-RRB-
38536	924	7	=	=	SYM
38536	924	8	3	3	CD
38536	924	9	.	.	.
38536	925	1	8	8	LS
38536	925	2	.	.	.
38536	925	3	pr^2	pr^2	NFP
38536	925	4	+	+	$
38536	925	5	qr	qr	NN
38536	925	6	+	+	NNP
38536	925	7	s	s	NN
38536	925	8	=	=	SYM
38536	925	9	0	0	NFP
38536	925	10	.	.	.
38536	926	1	Form	form	VB
38536	926	2	the	the	DT
38536	926	3	equations	equation	NNS
38536	926	4	whose	whose	WP$
38536	926	5	roots	root	NNS
38536	926	6	are	be	VBP
38536	926	7	:	:	:
38536	926	8	9	9	CD
38536	926	9	.	.	.
38536	927	1	5	5	CD
38536	927	2	,	,	,
38536	927	3	-3	-3	.
38536	927	4	.	.	.
38536	928	1	10	10	CD
38536	928	2	.	.	.
38536	929	1	2/3	2/3	CD
38536	929	2	,	,	,
38536	929	3	5/3	5/3	CD
38536	929	4	.	.	.
38536	930	1	11	11	CD
38536	930	2	.	.	.
38536	930	3	c	c	NN
38536	930	4	+	+	SYM
38536	930	5	d	d	NN
38536	930	6	,	,	,
38536	930	7	c	c	NNP
38536	930	8	-	-	HYPH
38536	930	9	d.	d.	NNP
38536	930	10	12	12	CD
38536	930	11	.	.	.
38536	931	1	-3	-3	:
38536	931	2	,	,	,
38536	931	3	-5	-5	ADD
38536	931	4	.	.	.
38536	932	1	13	13	CD
38536	932	2	.	.	.
38536	933	1	[	[	-LRB-
38536	933	2	2	2	CD
38536	933	3	±	±	CD
38536	933	4	-3^(1/2)]/5	-3^(1/2)]/5	NN
38536	933	5	.	.	.
38536	934	1	14	14	CD
38536	934	2	.	.	.
38536	935	1	8/3	8/3	CD
38536	935	2	+	+	SYM
38536	935	3	(	(	-LRB-
38536	935	4	2/3)37^(1/2	2/3)37^(1/2	CD
38536	935	5	)	)	-RRB-
38536	935	6	,	,	,
38536	935	7	8/3	8/3	CD
38536	935	8	-	-	HYPH
38536	935	9	(	(	-LRB-
38536	935	10	2/3)37^(1/2	2/3)37^(1/2	CD
38536	935	11	)	)	-RRB-
38536	935	12	.	.	.
38536	936	1	15	15	CD
38536	936	2	.	.	.
38536	937	1	[	[	-LRB-
38536	937	2	-2	-2	:
38536	937	3	±	±	NNP
38536	937	4	-2^(1/2)]/2	-2^(1/2)]/2	CD
38536	937	5	.	.	.
38536	938	1	16	16	CD
38536	938	2	.	.	.
38536	939	1	Solve	solve	VB
38536	939	2	x^2	x^2	NNP
38536	939	3	-	-	HYPH
38536	939	4	3x	3x	CD
38536	939	5	+	+	SYM
38536	939	6	4	4	CD
38536	939	7	=	=	SYM
38536	939	8	0	0	CD
38536	939	9	.	.	.
38536	940	1	Check	check	VB
38536	940	2	by	by	IN
38536	940	3	substituting	substitute	VBG
38536	940	4	the	the	DT
38536	940	5	values	value	NNS
38536	940	6	of	of	IN
38536	940	7	x	x	NNS
38536	940	8	;	;	:
38536	940	9	then	then	RB
38536	940	10	check	check	VB
38536	940	11	by	by	IN
38536	940	12	finding	find	VBG
38536	940	13	the	the	DT
38536	940	14	sum	sum	NN
38536	940	15	and	and	CC
38536	940	16	the	the	DT
38536	940	17	product	product	NN
38536	940	18	of	of	IN
38536	940	19	the	the	DT
38536	940	20	roots	root	NNS
38536	940	21	.	.	.
38536	941	1	Compare	compare	VB
38536	941	2	the	the	DT
38536	941	3	amount	amount	NN
38536	941	4	of	of	IN
38536	941	5	labor	labor	NN
38536	941	6	required	require	VBN
38536	941	7	in	in	IN
38536	941	8	each	each	DT
38536	941	9	case	case	NN
38536	941	10	.	.	.
38536	942	1	17	17	CD
38536	942	2	.	.	.
38536	943	1	Solve	solve	VB
38536	943	2	(	(	-LRB-
38536	943	3	x	x	NNS
38536	943	4	-	-	SYM
38536	943	5	3)(x	3)(x	CD
38536	943	6	+	+	CC
38536	943	7	2)(x^2	2)(x^2	CD
38536	943	8	+	+	SYM
38536	943	9	3x	3x	CD
38536	943	10	-	-	SYM
38536	943	11	4	4	CD
38536	943	12	)	)	-RRB-
38536	943	13	=	=	SYM
38536	943	14	0	0	NFP
38536	943	15	.	.	.
38536	944	1	18	18	CD
38536	944	2	.	.	.
38536	945	1	Is	be	VBZ
38536	945	2	e^(4z	e^(4z	NN
38536	945	3	)	)	-RRB-
38536	945	4	+	+	NFP
38536	945	5	2e^(3z	2e^(3z	CD
38536	945	6	)	)	-RRB-
38536	945	7	+	+	CC
38536	945	8	e^(2z	e^(2z	NN
38536	945	9	)	)	-RRB-
38536	945	10	+	+	CC
38536	945	11	2e^z	2e^z	CD
38536	945	12	+	+	SYM
38536	945	13	2	2	CD
38536	945	14	+	+	CC
38536	945	15	e^(-2z	e^(-2z	NN
38536	945	16	)	)	-RRB-
38536	945	17	a	a	DT
38536	945	18	perfect	perfect	JJ
38536	945	19	square	square	NN
38536	945	20	?	?	.
38536	946	1	19	19	CD
38536	946	2	.	.	.
38536	947	1	Find	find	VB
38536	947	2	the	the	DT
38536	947	3	square	square	JJ
38536	947	4	root	root	NN
38536	947	5	(	(	-LRB-
38536	947	6	short	short	JJ
38536	947	7	method	method	NN
38536	947	8	)	)	-RRB-
38536	947	9	:	:	:
38536	947	10	(	(	-LRB-
38536	947	11	x^2	x^2	NNP
38536	947	12	-	-	HYPH
38536	947	13	1)(x^2	1)(x^2	CD
38536	947	14	-	-	HYPH
38536	947	15	3x	3x	CD
38536	947	16	+	+	SYM
38536	947	17	2)(x^2	2)(x^2	CD
38536	947	18	-	-	HYPH
38536	947	19	x	x	NN
38536	947	20	-	-	HYPH
38536	947	21	2	2	CD
38536	947	22	)	)	-RRB-
38536	947	23	.	.	.
38536	948	1	20	20	CD
38536	948	2	.	.	.
38536	949	1	Solve	solve	VB
38536	949	2	(	(	-LRB-
38536	949	3	1.2x	1.2x	CD
38536	949	4	-	-	SYM
38536	949	5	1.5)/(1.5	1.5)/(1.5	CD
38536	949	6	)	)	-RRB-
38536	949	7	+	+	NFP
38536	949	8	(	(	-LRB-
38536	949	9	.4x	.4x	NFP
38536	949	10	+	+	CC
38536	949	11	1)/(.2x	1)/(.2x	CD
38536	949	12	-	-	HYPH
38536	949	13	.2	.2	NN
38536	949	14	)	)	-RRB-
38536	949	15	=	=	NFP
38536	949	16	(	(	-LRB-
38536	949	17	.4x	.4x	NFP
38536	949	18	+	+	SYM
38536	949	19	1)/(.5	1)/(.5	CD
38536	949	20	)	)	-RRB-
38536	949	21	.	.	.
38536	950	1	21	21	CD
38536	950	2	.	.	.
38536	951	1	The	the	DT
38536	951	2	glass	glass	NN
38536	951	3	of	of	IN
38536	951	4	a	a	DT
38536	951	5	mirror	mirror	NN
38536	951	6	is	be	VBZ
38536	951	7	18	18	CD
38536	951	8	inches	inch	NNS
38536	951	9	by	by	IN
38536	951	10	12	12	CD
38536	951	11	inches	inch	NNS
38536	951	12	,	,	,
38536	951	13	and	and	CC
38536	951	14	it	-PRON-	PRP
38536	951	15	has	have	VBZ
38536	951	16	a	a	DT
38536	951	17	frame	frame	NN
38536	951	18	of	of	IN
38536	951	19	uniform	uniform	JJ
38536	951	20	width	width	NN
38536	951	21	whose	whose	WP$
38536	951	22	area	area	NN
38536	951	23	is	be	VBZ
38536	951	24	equal	equal	JJ
38536	951	25	to	to	IN
38536	951	26	that	that	DT
38536	951	27	of	of	IN
38536	951	28	the	the	DT
38536	951	29	glass	glass	NN
38536	951	30	.	.	.
38536	952	1	Find	find	VB
38536	952	2	the	the	DT
38536	952	3	width	width	NN
38536	952	4	of	of	IN
38536	952	5	the	the	DT
38536	952	6	frame	frame	NN
38536	952	7	.	.	.
38536	953	1	OUTLINE	OUTLINE	NNP
38536	953	2	OF	of	IN
38536	953	3	SIMULTANEOUS	SIMULTANEOUS	NNP
38536	953	4	QUADRATICS	QUADRATICS	NNP
38536	953	5	~Simultaneous	~Simultaneous	.
38536	953	6	Quadratics~	Quadratics~	JJR
38536	953	7	CASE	CASE	NNP
38536	953	8	I.	i.	NN
38536	954	1	One	one	CD
38536	954	2	equation	equation	NN
38536	954	3	linear	linear	NN
38536	954	4	.	.	.
38536	955	1	The	the	DT
38536	955	2	other	other	JJ
38536	955	3	quadratic	quadratic	NN
38536	955	4	.	.	.
38536	956	1	2x	2x	NNP
38536	956	2	+	+	SYM
38536	956	3	y	y	NN
38536	956	4	=	=	SYM
38536	956	5	7	7	CD
38536	956	6	,	,	,
38536	956	7	x^2	x^2	NNS
38536	956	8	+	+	SYM
38536	956	9	2y^2	2y^2	CD
38536	956	10	=	=	SYM
38536	956	11	22	22	CD
38536	956	12	.	.	.
38536	957	1	METHOD	METHOD	NNP
38536	957	2	:	:	:
38536	957	3	Solve	solve	VB
38536	957	4	for	for	IN
38536	957	5	x	x	NNS
38536	957	6	as	as	IN
38536	957	7	in	in	IN
38536	957	8	terms	term	NNS
38536	957	9	of	of	IN
38536	957	10	y	y	NNP
38536	957	11	,	,	,
38536	957	12	or	or	CC
38536	957	13	_	_	NNP
38536	957	14	vice	vice	NN
38536	957	15	versa	versa	RB
38536	957	16	_	_	NNP
38536	957	17	,	,	,
38536	957	18	in	in	IN
38536	957	19	the	the	DT
38536	957	20	linear	linear	NN
38536	957	21	and	and	CC
38536	957	22	substitute	substitute	NN
38536	957	23	in	in	IN
38536	957	24	the	the	DT
38536	957	25	quadratic	quadratic	NN
38536	957	26	.	.	.
38536	958	1	CASE	CASE	NNP
38536	958	2	II	II	NNP
38536	958	3	.	.	.
38536	959	1	Both	both	DT
38536	959	2	equations	equation	NNS
38536	959	3	homogeneous	homogeneous	JJ
38536	959	4	and	and	CC
38536	959	5	of	of	IN
38536	959	6	the	the	DT
38536	959	7	second	second	JJ
38536	959	8	degree	degree	NN
38536	959	9	.	.	.
38536	960	1	x^2	x^2	NNP
38536	960	2	-	-	HYPH
38536	960	3	xy	xy	NNP
38536	960	4	+	+	CC
38536	960	5	y^2	y^2	NN
38536	960	6	=	=	SYM
38536	960	7	39	39	CD
38536	960	8	,	,	,
38536	960	9	2x^2	2x^2	CD
38536	960	10	-	-	HYPH
38536	960	11	3xy	3xy	NN
38536	960	12	+	+	SYM
38536	960	13	2y^2	2y^2	CD
38536	960	14	=	=	SYM
38536	960	15	43	43	CD
38536	960	16	.	.	.
38536	961	1	METHOD	METHOD	NNP
38536	961	2	:	:	:
38536	961	3	Let	let	VB
38536	961	4	y	y	NNP
38536	961	5	=	=	SYM
38536	961	6	vx	vx	NNS
38536	961	7	,	,	,
38536	961	8	and	and	CC
38536	961	9	substitute	substitute	VB
38536	961	10	in	in	IN
38536	961	11	both	both	DT
38536	961	12	equations	equation	NNS
38536	961	13	.	.	.
38536	962	1	ALTERNATE	alternate	JJ
38536	962	2	METHOD	METHOD	NNP
38536	962	3	:	:	:
38536	962	4	Solve	solve	VB
38536	962	5	for	for	IN
38536	962	6	x	x	NNS
38536	962	7	in	in	IN
38536	962	8	terms	term	NNS
38536	962	9	of	of	IN
38536	962	10	y	y	NNP
38536	962	11	in	in	IN
38536	962	12	one	one	CD
38536	962	13	equation	equation	NN
38536	962	14	and	and	CC
38536	962	15	substitute	substitute	NN
38536	962	16	in	in	IN
38536	962	17	the	the	DT
38536	962	18	other	other	JJ
38536	962	19	.	.	.
38536	963	1	CASE	CASE	NNP
38536	963	2	III	III	NNP
38536	963	3	.	.	.
38536	964	1	Any	any	DT
38536	964	2	two	two	CD
38536	964	3	of	of	IN
38536	964	4	the	the	DT
38536	964	5	quantities	quantity	NNS
38536	964	6	x	x	SYM
38536	964	7	+	+	SYM
38536	964	8	y	y	NN
38536	964	9	x^2	x^2	NNS
38536	964	10	+	+	SYM
38536	964	11	y^2	y^2	PRP$
38536	964	12	xy	xy	NNP
38536	964	13	x	x	NNP
38536	964	14	-	-	NN
38536	964	15	y	y	NNP
38536	964	16	x^3	x^3	NNP
38536	964	17	+	+	SYM
38536	964	18	y^3	y^3	NNS
38536	964	19	x^3	x^3	NNS
38536	964	20	-	-	HYPH
38536	964	21	y^3	y^3	NNP
38536	964	22	x^2	x^2	NNP
38536	964	23	+	+	SYM
38536	964	24	xy	xy	NN
38536	964	25	+	+	SYM
38536	964	26	y^2	y^2	PRP$
38536	964	27	x^2	x^2	NNP
38536	964	28	-	-	HYPH
38536	964	29	xy	xy	NN
38536	964	30	+	+	SYM
38536	964	31	y^2	y^2	FW
38536	964	32	given	give	VBN
38536	964	33	.	.	.
38536	965	1	x	x	LS
38536	965	2	+	+	SYM
38536	965	3	y	y	NN
38536	965	4	=	=	SYM
38536	965	5	5	5	CD
38536	965	6	,	,	,
38536	965	7	x^2	x^2	NNP
38536	965	8	-	-	HYPH
38536	965	9	xy	xy	NN
38536	965	10	+	+	CC
38536	965	11	y^2	y^2	NN
38536	965	12	=	=	SYM
38536	965	13	7	7	CD
38536	965	14	.	.	.
38536	966	1	METHOD	METHOD	NNP
38536	966	2	:	:	:
38536	966	3	Solve	solve	VB
38536	966	4	for	for	IN
38536	966	5	x	x	NN
38536	966	6	+	+	SYM
38536	966	7	y	y	NN
38536	966	8	and	and	CC
38536	966	9	x	x	SYM
38536	966	10	-	-	NNP
38536	966	11	y	y	NNP
38536	966	12	;	;	:
38536	966	13	then	then	RB
38536	966	14	add	add	VB
38536	966	15	to	to	TO
38536	966	16	get	get	VB
38536	966	17	x	x	NNS
38536	966	18	,	,	,
38536	966	19	subtract	subtract	VB
38536	966	20	to	to	TO
38536	966	21	get	get	VB
38536	966	22	y.	y.	.
38536	967	1	CASE	CASE	NNP
38536	967	2	IV	IV	NNP
38536	967	3	.	.	.
38536	968	1	Both	both	DT
38536	968	2	equations	equation	NNS
38536	968	3	symmetrical	symmetrical	JJ
38536	968	4	or	or	CC
38536	968	5	symmetrical	symmetrical	JJ
38536	968	6	except	except	IN
38536	968	7	for	for	IN
38536	968	8	sign	sign	NN
38536	968	9	.	.	.
38536	969	1	Usually	usually	RB
38536	969	2	one	one	CD
38536	969	3	equation	equation	NN
38536	969	4	of	of	IN
38536	969	5	high	high	JJ
38536	969	6	degree	degree	NN
38536	969	7	,	,	,
38536	969	8	the	the	DT
38536	969	9	other	other	JJ
38536	969	10	of	of	IN
38536	969	11	the	the	DT
38536	969	12	first	first	JJ
38536	969	13	degree	degree	NN
38536	969	14	.	.	.
38536	970	1	x^5	x^5	NNP
38536	970	2	+	+	SYM
38536	970	3	y^5	y^5	NNP
38536	970	4	=	=	SYM
38536	970	5	242	242	CD
38536	970	6	,	,	,
38536	970	7	x	x	NNS
38536	970	8	+	+	SYM
38536	970	9	y	y	NN
38536	970	10	=	=	SYM
38536	970	11	2	2	CD
38536	970	12	.	.	.
38536	971	1	METHOD	METHOD	NNP
38536	971	2	:	:	:
38536	971	3	Let	let	VB
38536	971	4	x	x	NN
38536	971	5	=	=	SYM
38536	971	6	u	u	NN
38536	971	7	+	+	CC
38536	971	8	v	v	NN
38536	971	9	and	and	CC
38536	971	10	y	y	NNP
38536	971	11	=	=	SYM
38536	971	12	u	u	NNP
38536	971	13	-	-	HYPH
38536	971	14	v	v	NNP
38536	971	15	,	,	,
38536	971	16	and	and	CC
38536	971	17	substitute	substitute	NN
38536	971	18	in	in	IN
38536	971	19	both	both	DT
38536	971	20	equations	equation	NNS
38536	971	21	.	.	.
38536	972	1	~Special	~Special	NFP
38536	972	2	Devices~	Devices~	NNP
38536	972	3	I.	i.	NN
38536	973	1	Consider	consider	VB
38536	973	2	some	some	DT
38536	973	3	compound	compound	NN
38536	973	4	quantity	quantity	NN
38536	973	5	like	like	IN
38536	973	6	xy	xy	NNP
38536	973	7	,	,	,
38536	973	8	[	[	-LRB-
38536	973	9	x	x	NNP
38536	973	10	-	-	SYM
38536	973	11	y]^(1/2	y]^(1/2	NN
38536	973	12	)	)	-RRB-
38536	973	13	,	,	,
38536	973	14	[	[	-LRB-
38536	973	15	xy]^(1/2	xy]^(1/2	NNP
38536	973	16	)	)	-RRB-
38536	973	17	,	,	,
38536	973	18	x	x	NNP
38536	973	19	/	/	SYM
38536	973	20	y	y	NNP
38536	973	21	,	,	,
38536	973	22	etc	etc	FW
38536	973	23	.	.	FW
38536	973	24	,	,	,
38536	973	25	as	as	IN
38536	973	26	the	the	DT
38536	973	27	unknown	unknown	NN
38536	973	28	,	,	,
38536	973	29	at	at	IN
38536	973	30	first	first	RB
38536	973	31	.	.	.
38536	974	1	Solve	solve	VB
38536	974	2	for	for	IN
38536	974	3	the	the	DT
38536	974	4	compound	compound	NN
38536	974	5	unknown	unknown	NN
38536	974	6	,	,	,
38536	974	7	and	and	CC
38536	974	8	combine	combine	VB
38536	974	9	the	the	DT
38536	974	10	resulting	result	VBG
38536	974	11	equation	equation	NN
38536	974	12	with	with	IN
38536	974	13	the	the	DT
38536	974	14	simpler	simple	JJR
38536	974	15	original	original	JJ
38536	974	16	equation	equation	NN
38536	974	17	.	.	.
38536	975	1	x^2	x^2	NNP
38536	975	2	y^2	y^2	NNP
38536	975	3	+	+	SYM
38536	975	4	xy	xy	NN
38536	975	5	=	=	SYM
38536	975	6	6	6	CD
38536	975	7	,	,	,
38536	975	8	x	x	NNS
38536	975	9	+	+	SYM
38536	975	10	2y	2y	CD
38536	975	11	=	=	SYM
38536	975	12	-5	-5	ADD
38536	975	13	.	.	.
38536	976	1	II	ii	CD
38536	976	2	.	.	.
38536	977	1	Divide	divide	VB
38536	977	2	the	the	DT
38536	977	3	equations	equation	NNS
38536	977	4	member	member	NN
38536	977	5	by	by	IN
38536	977	6	member	member	NN
38536	977	7	.	.	.
38536	978	1	Then	then	RB
38536	978	2	solve	solve	VB
38536	978	3	by	by	IN
38536	978	4	Case	Case	NNP
38536	978	5	I	I	NNP
38536	978	6	,	,	,
38536	978	7	II	II	NNP
38536	978	8	,	,	,
38536	978	9	or	or	CC
38536	978	10	III	III	NNP
38536	978	11	.	.	.
38536	979	1	x^3	x^3	NNP
38536	979	2	-	-	:
38536	979	3	y^3	y^3	NNP
38536	979	4	=	=	SYM
38536	979	5	152	152	CD
38536	979	6	,	,	,
38536	979	7	x	x	NNP
38536	979	8	-	-	NN
38536	979	9	y	y	NN
38536	979	10	=	=	SYM
38536	979	11	2	2	CD
38536	979	12	.	.	.
38536	980	1	III	iii	CD
38536	980	2	.	.	.
38536	981	1	Eliminate	eliminate	VB
38536	981	2	the	the	DT
38536	981	3	quadratic	quadratic	JJ
38536	981	4	terms	term	NNS
38536	981	5	.	.	.
38536	982	1	Then	then	RB
38536	982	2	solve	solve	VB
38536	982	3	by	by	IN
38536	982	4	Case	Case	NNP
38536	982	5	I	I	NNP
38536	982	6	,	,	,
38536	982	7	II	II	NNP
38536	982	8	,	,	,
38536	982	9	or	or	CC
38536	982	10	III	III	NNP
38536	982	11	.	.	.
38536	983	1	xy	xy	NNP
38536	983	2	+	+	SYM
38536	983	3	x	x	SYM
38536	983	4	=	=	SYM
38536	983	5	15	15	CD
38536	983	6	,	,	,
38536	983	7	xy	xy	NNP
38536	983	8	+	+	NNP
38536	983	9	y	y	NN
38536	983	10	=	=	SYM
38536	983	11	16	16	CD
38536	983	12	.	.	.
38536	984	1	SIMULTANEOUS	SIMULTANEOUS	NNP
38536	984	2	QUADRATICS	QUADRATICS	NNP
38536	984	3	Solve	Solve	NNP
38536	984	4	:	:	:
38536	984	5	1	1	LS
38536	984	6	.	.	.
38536	984	7	x	x	SYM
38536	984	8	+	+	SYM
38536	984	9	y	y	NN
38536	984	10	=	=	SYM
38536	984	11	7	7	CD
38536	984	12	,	,	,
38536	984	13	x^2	x^2	NNS
38536	984	14	+	+	CC
38536	984	15	4xy	4xy	NN
38536	984	16	=	=	SYM
38536	984	17	57	57	CD
38536	984	18	.	.	.
38536	985	1	2	2	LS
38536	985	2	.	.	.
38536	986	1	2x^2	2x^2	CD
38536	986	2	=	=	SYM
38536	986	3	46	46	CD
38536	986	4	+	+	SYM
38536	986	5	y^2	y^2	UH
38536	986	6	,	,	,
38536	986	7	xy	xy	VBP
38536	986	8	+	+	NFP
38536	986	9	y^2	y^2	NNP
38536	986	10	=	=	SYM
38536	986	11	14	14	CD
38536	986	12	.	.	.
38536	987	1	3	3	LS
38536	987	2	.	.	.
38536	987	3	x^2	x^2	NNS
38536	987	4	+	+	SYM
38536	987	5	y^2	y^2	NN
38536	987	6	=	=	SYM
38536	987	7	25	25	CD
38536	987	8	,	,	,
38536	987	9	x	x	NN
38536	987	10	+	+	SYM
38536	987	11	y	y	NN
38536	987	12	=	=	SYM
38536	987	13	1	1	CD
38536	987	14	.	.	.
38536	988	1	4	4	LS
38536	988	2	.	.	.
38536	988	3	x^4	x^4	NNP
38536	988	4	+	+	SYM
38536	988	5	y^4	y^4	NNS
38536	988	6	=	=	SYM
38536	988	7	2	2	CD
38536	988	8	,	,	,
38536	988	9	x	x	NNP
38536	988	10	-	-	NN
38536	988	11	y	y	NNP
38536	988	12	=	=	SYM
38536	988	13	2	2	CD
38536	988	14	.	.	.
38536	989	1	5	5	CD
38536	989	2	.	.	.
38536	989	3	x^3	x^3	NNS
38536	989	4	+	+	SYM
38536	989	5	y^3	y^3	NNS
38536	989	6	=	=	SYM
38536	989	7	28	28	CD
38536	989	8	,	,	,
38536	989	9	x	x	NNP
38536	989	10	+	+	SYM
38536	989	11	y	y	NN
38536	989	12	=	=	SYM
38536	989	13	4	4	CD
38536	989	14	.	.	.
38536	990	1	6	6	LS
38536	990	2	.	.	.
38536	990	3	x^2	x^2	NNP
38536	990	4	y^2	y^2	NNP
38536	990	5	+	+	SYM
38536	990	6	xy	xy	NN
38536	990	7	-	-	HYPH
38536	990	8	12	12	CD
38536	990	9	=	=	SYM
38536	990	10	0	0	CD
38536	990	11	,	,	,
38536	990	12	x	x	NN
38536	990	13	+	+	SYM
38536	990	14	y	y	NN
38536	990	15	=	=	SYM
38536	990	16	4	4	CD
38536	990	17	.	.	.
38536	991	1	7	7	LS
38536	991	2	.	.	.
38536	992	1	2xy	2xy	LS
38536	992	2	-	-	HYPH
38536	992	3	x	x	NN
38536	992	4	+	+	CD
38536	992	5	2y	2y	CD
38536	992	6	=	=	SYM
38536	992	7	16	16	CD
38536	992	8	,	,	,
38536	992	9	3xy	3xy	NN
38536	992	10	+	+	SYM
38536	992	11	2x	2x	CD
38536	992	12	-	-	HYPH
38536	992	13	4y	4y	CD
38536	992	14	=	=	SYM
38536	992	15	10	10	CD
38536	992	16	.	.	.
38536	993	1	8	8	LS
38536	993	2	.	.	.
38536	994	1	(	(	-LRB-
38536	994	2	3x	3x	CD
38536	994	3	-	-	HYPH
38536	994	4	2y)(2x	2y)(2x	CD
38536	994	5	-	-	HYPH
38536	994	6	3y	3y	NNP
38536	994	7	)	)	-RRB-
38536	994	8	=	=	NFP
38536	994	9	26	26	CD
38536	994	10	,	,	,
38536	994	11	x	x	NNS
38536	994	12	+	+	SYM
38536	994	13	1	1	CD
38536	994	14	=	=	SYM
38536	994	15	2y	2y	CD
38536	994	16	.	.	.
38536	995	1	9	9	CD
38536	995	2	.	.	.
38536	996	1	4x^2	4x^2	CD
38536	996	2	+	+	SYM
38536	996	3	3xy	3xy	NN
38536	996	4	+	+	SYM
38536	996	5	2y^2	2y^2	CD
38536	996	6	=	=	SYM
38536	996	7	18	18	CD
38536	996	8	,	,	,
38536	996	9	3x^2	3x^2	CD
38536	996	10	+	+	CC
38536	996	11	2xy	2xy	JJ
38536	996	12	-	-	HYPH
38536	996	13	y^2	y^2	NNP
38536	996	14	=	=	SYM
38536	996	15	3	3	CD
38536	996	16	.	.	.
38536	997	1	10	10	CD
38536	997	2	.	.	.
38536	997	3	x^5	x^5	CD
38536	997	4	+	+	SYM
38536	997	5	y^5	y^5	NNP
38536	997	6	=	=	SYM
38536	997	7	242	242	CD
38536	997	8	,	,	,
38536	997	9	x	x	NNS
38536	997	10	+	+	SYM
38536	997	11	y	y	NN
38536	997	12	=	=	SYM
38536	997	13	2	2	CD
38536	997	14	.	.	.
38536	998	1	11	11	CD
38536	998	2	.	.	.
38536	998	3	x	x	LS
38536	998	4	-	-	NN
38536	998	5	y	y	NNP
38536	998	6	+	+	CC
38536	998	7	[	[	-LRB-
38536	998	8	x	x	NN
38536	998	9	-	-	SYM
38536	998	10	y]^(1/2	y]^(1/2	NN
38536	998	11	)	)	-RRB-
38536	998	12	=	=	SYM
38536	998	13	6	6	CD
38536	998	14	,	,	,
38536	998	15	xy	xy	NNP
38536	998	16	=	=	SYM
38536	998	17	5	5	CD
38536	998	18	.	.	.
38536	999	1	12	12	CD
38536	999	2	.	.	.
38536	1000	1	4x^2	4x^2	CD
38536	1000	2	-	-	HYPH
38536	1000	3	x	x	SYM
38536	1000	4	+	+	NN
38536	1000	5	y	y	NN
38536	1000	6	=	=	SYM
38536	1000	7	67	67	CD
38536	1000	8	,	,	,
38536	1000	9	3x^2	3x^2	CD
38536	1000	10	-	-	HYPH
38536	1000	11	3y	3y	CD
38536	1000	12	=	=	SYM
38536	1000	13	27	27	CD
38536	1000	14	.	.	.
38536	1001	1	13	13	CD
38536	1001	2	.	.	.
38536	1001	3	x	x	LS
38536	1001	4	-	-	HYPH
38536	1001	5	y	y	FW
38536	1001	6	-	-	HYPH
38536	1001	7	[	[	-LRB-
38536	1001	8	x	x	NN
38536	1001	9	-	-	SYM
38536	1001	10	y]^(1/2	y]^(1/2	NN
38536	1001	11	)	)	-RRB-
38536	1001	12	=	=	SYM
38536	1001	13	2	2	CD
38536	1001	14	,	,	,
38536	1001	15	x^3	x^3	NNP
38536	1001	16	-	-	HYPH
38536	1001	17	y^3	y^3	NNP
38536	1001	18	=	=	SYM
38536	1001	19	2044	2044	CD
38536	1001	20	.	.	.
38536	1002	1	(	(	-LRB-
38536	1002	2	_	_	NNP
38536	1002	3	Yale	Yale	NNP
38536	1002	4	.	.	.
38536	1002	5	_	_	NNP
38536	1002	6	)	)	-RRB-
38536	1002	7	14	14	CD
38536	1002	8	.	.	.
38536	1002	9	x^2	x^2	NNP
38536	1002	10	+	+	SYM
38536	1002	11	xy	xy	NN
38536	1002	12	+	+	SYM
38536	1002	13	x	x	NN
38536	1002	14	=	=	SYM
38536	1002	15	14	14	CD
38536	1002	16	,	,	,
38536	1002	17	y^2	y^2	NNP
38536	1002	18	+	+	SYM
38536	1002	19	xy	xy	NN
38536	1002	20	+	+	NNS
38536	1002	21	y	y	NN
38536	1002	22	=	=	SYM
38536	1002	23	28	28	CD
38536	1002	24	.	.	.
38536	1003	1	(	(	-LRB-
38536	1003	2	_	_	NNP
38536	1003	3	Princeton	Princeton	NNP
38536	1003	4	.	.	.
38536	1003	5	_	_	NNP
38536	1003	6	)	)	-RRB-
38536	1003	7	15	15	CD
38536	1003	8	.	.	.
38536	1003	9	x^2	x^2	NNS
38536	1003	10	+	+	SYM
38536	1003	11	y^2	y^2	NNS
38536	1003	12	=	=	SYM
38536	1003	13	13	13	CD
38536	1003	14	,	,	,
38536	1003	15	y^2	y^2	UH
38536	1003	16	=	=	SYM
38536	1003	17	4(x	4(x	CD
38536	1003	18	-	-	SYM
38536	1003	19	2	2	CD
38536	1003	20	)	)	-RRB-
38536	1003	21	.	.	.
38536	1004	1	Plot	plot	VB
38536	1004	2	the	the	DT
38536	1004	3	graph	graph	NN
38536	1004	4	of	of	IN
38536	1004	5	each	each	DT
38536	1004	6	equation	equation	NN
38536	1004	7	.	.	.
38536	1005	1	(	(	-LRB-
38536	1005	2	_	_	NNP
38536	1005	3	Cornell	Cornell	NNP
38536	1005	4	.	.	.
38536	1005	5	_	_	NNP
38536	1005	6	)	)	-RRB-
38536	1005	7	16	16	CD
38536	1005	8	.	.	.
38536	1005	9	x^2	x^2	NNS
38536	1005	10	+	+	SYM
38536	1005	11	y^2	y^2	UH
38536	1005	12	=	=	SYM
38536	1005	13	xy	xy	NN
38536	1005	14	+	+	CC
38536	1005	15	37	37	CD
38536	1005	16	,	,	,
38536	1005	17	x	x	NNS
38536	1005	18	+	+	SYM
38536	1005	19	y	y	NN
38536	1005	20	=	=	SYM
38536	1005	21	xy	xy	NNP
38536	1005	22	-	-	HYPH
38536	1005	23	17	17	CD
38536	1005	24	.	.	.
38536	1006	1	(	(	-LRB-
38536	1006	2	_	_	NNP
38536	1006	3	Columbia	Columbia	NNP
38536	1006	4	.	.	.
38536	1006	5	_	_	NNP
38536	1006	6	)	)	-RRB-
38536	1006	7	_	_	NNP
38536	1006	8	In	in	IN
38536	1006	9	grouping	group	VBG
38536	1006	10	the	the	DT
38536	1006	11	answers	answer	NNS
38536	1006	12	,	,	,
38536	1006	13	be	be	VB
38536	1006	14	sure	sure	JJ
38536	1006	15	to	to	TO
38536	1006	16	associate	associate	VB
38536	1006	17	each	each	DT
38536	1006	18	value	value	NN
38536	1006	19	of	of	IN
38536	1006	20	x	x	NN
38536	1006	21	with	with	IN
38536	1006	22	the	the	DT
38536	1006	23	corresponding	corresponding	JJ
38536	1006	24	value	value	NN
38536	1006	25	of	of	IN
38536	1006	26	y.	y.	NNP
38536	1006	27	_	_	NNP
38536	1006	28	17	17	CD
38536	1006	29	.	.	.
38536	1007	1	The	the	DT
38536	1007	2	course	course	NN
38536	1007	3	of	of	IN
38536	1007	4	a	a	DT
38536	1007	5	yacht	yacht	NN
38536	1007	6	is	be	VBZ
38536	1007	7	30	30	CD
38536	1007	8	miles	mile	NNS
38536	1007	9	in	in	IN
38536	1007	10	length	length	NN
38536	1007	11	and	and	CC
38536	1007	12	is	be	VBZ
38536	1007	13	in	in	IN
38536	1007	14	the	the	DT
38536	1007	15	shape	shape	NN
38536	1007	16	of	of	IN
38536	1007	17	a	a	DT
38536	1007	18	right	right	JJ
38536	1007	19	triangle	triangle	NN
38536	1007	20	one	one	CD
38536	1007	21	arm	arm	NN
38536	1007	22	of	of	IN
38536	1007	23	which	which	WDT
38536	1007	24	is	be	VBZ
38536	1007	25	2	2	CD
38536	1007	26	miles	mile	NNS
38536	1007	27	longer	long	RBR
38536	1007	28	than	than	IN
38536	1007	29	the	the	DT
38536	1007	30	other	other	JJ
38536	1007	31	.	.	.
38536	1008	1	What	what	WP
38536	1008	2	is	be	VBZ
38536	1008	3	the	the	DT
38536	1008	4	distance	distance	NN
38536	1008	5	along	along	IN
38536	1008	6	each	each	DT
38536	1008	7	side	side	NN
38536	1008	8	?	?	.
38536	1009	1	~Reference:~	~reference:~	VB
38536	1009	2	The	the	DT
38536	1009	3	chapter	chapter	NN
38536	1009	4	on	on	IN
38536	1009	5	Simultaneous	Simultaneous	NNP
38536	1009	6	Quadratics	Quadratics	NNP
38536	1009	7	in	in	IN
38536	1009	8	any	any	DT
38536	1009	9	algebra	algebra	NN
38536	1009	10	.	.	.
38536	1010	1	RATIO	ratio	NN
38536	1010	2	AND	and	CC
38536	1010	3	PROPORTION	proportion	NN
38536	1010	4	1	1	CD
38536	1010	5	.	.	.
38536	1011	1	Define	define	NN
38536	1011	2	ratio	ratio	NN
38536	1011	3	,	,	,
38536	1011	4	proportion	proportion	NN
38536	1011	5	,	,	,
38536	1011	6	mean	mean	VB
38536	1011	7	proportional	proportional	JJ
38536	1011	8	,	,	,
38536	1011	9	third	third	JJ
38536	1011	10	proportional	proportional	JJ
38536	1011	11	,	,	,
38536	1011	12	fourth	fourth	JJ
38536	1011	13	proportional	proportional	JJ
38536	1011	14	.	.	.
38536	1012	1	2	2	LS
38536	1012	2	.	.	.
38536	1013	1	Find	find	VB
38536	1013	2	a	a	DT
38536	1013	3	mean	mean	NN
38536	1013	4	proportional	proportional	NN
38536	1013	5	between	between	IN
38536	1013	6	4	4	CD
38536	1013	7	and	and	CC
38536	1013	8	16	16	CD
38536	1013	9	;	;	SYM
38536	1013	10	18	18	CD
38536	1013	11	and	and	CC
38536	1013	12	50	50	CD
38536	1013	13	;	;	SYM
38536	1013	14	12m^2n	12m^2n	CD
38536	1013	15	and	and	CC
38536	1013	16	3mn^2	3mn^2	CD
38536	1013	17	.	.	.
38536	1014	1	3	3	LS
38536	1014	2	.	.	.
38536	1015	1	Find	find	VB
38536	1015	2	a	a	DT
38536	1015	3	third	third	JJ
38536	1015	4	proportional	proportional	NN
38536	1015	5	to	to	IN
38536	1015	6	4	4	CD
38536	1015	7	and	and	CC
38536	1015	8	7	7	CD
38536	1015	9	;	;	SYM
38536	1015	10	5	5	CD
38536	1015	11	and	and	CC
38536	1015	12	10	10	CD
38536	1015	13	;	;	:
38536	1015	14	a^2	a^2	CD
38536	1015	15	-	-	SYM
38536	1015	16	9	9	CD
38536	1015	17	and	and	CC
38536	1015	18	a	a	DT
38536	1015	19	-	-	SYM
38536	1015	20	3	3	CD
38536	1015	21	.	.	.
38536	1016	1	4	4	LS
38536	1016	2	.	.	.
38536	1017	1	Find	find	VB
38536	1017	2	a	a	DT
38536	1017	3	fourth	fourth	JJ
38536	1017	4	proportional	proportional	NN
38536	1017	5	to	to	IN
38536	1017	6	2	2	CD
38536	1017	7	,	,	,
38536	1017	8	5	5	CD
38536	1017	9	,	,	,
38536	1017	10	and	and	CC
38536	1017	11	4	4	CD
38536	1017	12	;	;	SYM
38536	1017	13	35	35	CD
38536	1017	14	,	,	,
38536	1017	15	20	20	CD
38536	1017	16	,	,	,
38536	1017	17	and	and	CC
38536	1017	18	14	14	CD
38536	1017	19	.	.	.
38536	1018	1	5	5	CD
38536	1018	2	.	.	.
38536	1019	1	Write	write	VB
38536	1019	2	out	out	RP
38536	1019	3	the	the	DT
38536	1019	4	proofs	proof	NNS
38536	1019	5	for	for	IN
38536	1019	6	the	the	DT
38536	1019	7	following	following	NN
38536	1019	8	,	,	,
38536	1019	9	stating	state	VBG
38536	1019	10	the	the	DT
38536	1019	11	theorem	theorem	NN
38536	1019	12	in	in	IN
38536	1019	13	full	full	JJ
38536	1019	14	in	in	IN
38536	1019	15	each	each	DT
38536	1019	16	case	case	NN
38536	1019	17	:	:	:
38536	1019	18	(	(	-LRB-
38536	1019	19	_	_	NNP
38536	1019	20	a	a	DT
38536	1019	21	_	_	NNP
38536	1019	22	)	)	-RRB-
38536	1019	23	The	the	DT
38536	1019	24	product	product	NN
38536	1019	25	of	of	IN
38536	1019	26	the	the	DT
38536	1019	27	extremes	extreme	NNS
38536	1019	28	equals	equal	VBZ
38536	1019	29	etc	etc	FW
38536	1019	30	.	.	.
38536	1020	1	(	(	-LRB-
38536	1020	2	_	_	NNP
38536	1020	3	b	b	NNP
38536	1020	4	_	_	NNP
38536	1020	5	)	)	-RRB-
38536	1020	6	If	if	IN
38536	1020	7	the	the	DT
38536	1020	8	product	product	NN
38536	1020	9	of	of	IN
38536	1020	10	two	two	CD
38536	1020	11	numbers	number	NNS
38536	1020	12	equals	equal	VBZ
38536	1020	13	the	the	DT
38536	1020	14	product	product	NN
38536	1020	15	of	of	IN
38536	1020	16	two	two	CD
38536	1020	17	other	other	JJ
38536	1020	18	numbers	number	NNS
38536	1020	19	,	,	,
38536	1020	20	either	either	CC
38536	1020	21	pair	pair	NN
38536	1020	22	etc	etc	FW
38536	1020	23	.	.	.
38536	1021	1	(	(	-LRB-
38536	1021	2	_	_	NNP
38536	1021	3	c	c	NNP
38536	1021	4	_	_	NNP
38536	1021	5	)	)	-RRB-
38536	1021	6	Alternation	Alternation	NNP
38536	1021	7	.	.	.
38536	1022	1	(	(	-LRB-
38536	1022	2	_	_	NNP
38536	1022	3	d	d	NNP
38536	1022	4	_	_	NNP
38536	1022	5	)	)	-RRB-
38536	1022	6	Inversion	Inversion	NNP
38536	1022	7	.	.	.
38536	1023	1	(	(	-LRB-
38536	1023	2	_	_	NNP
38536	1023	3	e	e	NNP
38536	1023	4	_	_	NNP
38536	1023	5	)	)	-RRB-
38536	1023	6	Composition	composition	NN
38536	1023	7	.	.	.
38536	1024	1	(	(	-LRB-
38536	1024	2	_	_	NNP
38536	1024	3	f	f	NNP
38536	1024	4	_	_	NNP
38536	1024	5	)	)	-RRB-
38536	1024	6	Division	Division	NNP
38536	1024	7	.	.	.
38536	1025	1	(	(	-LRB-
38536	1025	2	_	_	NNP
38536	1025	3	g	g	NNP
38536	1025	4	_	_	NNP
38536	1025	5	)	)	-RRB-
38536	1025	6	Composition	composition	NN
38536	1025	7	and	and	CC
38536	1025	8	division	division	NN
38536	1025	9	.	.	.
38536	1026	1	(	(	-LRB-
38536	1026	2	_	_	NNP
38536	1026	3	h	h	NNP
38536	1026	4	_	_	NNP
38536	1026	5	)	)	-RRB-
38536	1026	6	In	in	IN
38536	1026	7	a	a	DT
38536	1026	8	series	series	NN
38536	1026	9	of	of	IN
38536	1026	10	equal	equal	JJ
38536	1026	11	ratios	ratio	NNS
38536	1026	12	,	,	,
38536	1026	13	the	the	DT
38536	1026	14	sum	sum	NN
38536	1026	15	of	of	IN
38536	1026	16	the	the	DT
38536	1026	17	antecedents	antecedent	NNS
38536	1026	18	is	be	VBZ
38536	1026	19	to	to	IN
38536	1026	20	the	the	DT
38536	1026	21	sum	sum	NN
38536	1026	22	of	of	IN
38536	1026	23	the	the	DT
38536	1026	24	consequents	consequent	NNS
38536	1026	25	etc	etc	FW
38536	1026	26	.	.	.
38536	1027	1	(	(	-LRB-
38536	1027	2	_	_	NNP
38536	1027	3	i	i	PRP
38536	1027	4	_	_	NNP
38536	1027	5	)	)	-RRB-
38536	1027	6	Like	like	IN
38536	1027	7	powers	power	NNS
38536	1027	8	or	or	CC
38536	1027	9	like	like	IN
38536	1027	10	roots	root	NNS
38536	1027	11	of	of	IN
38536	1027	12	the	the	DT
38536	1027	13	terms	term	NNS
38536	1027	14	of	of	IN
38536	1027	15	a	a	DT
38536	1027	16	proportion	proportion	NN
38536	1027	17	etc	etc	NN
38536	1027	18	.	.	.
38536	1028	1	6	6	CD
38536	1028	2	.	.	.
38536	1029	1	If	if	IN
38536	1029	2	x	x	NNS
38536	1029	3	:	:	:
38536	1029	4	m	m	NN
38536	1029	5	:	:	:
38536	1029	6	:	:	:
38536	1029	7	13	13	CD
38536	1029	8	:	:	SYM
38536	1029	9	7	7	CD
38536	1029	10	,	,	,
38536	1029	11	write	write	VB
38536	1029	12	all	all	PDT
38536	1029	13	the	the	DT
38536	1029	14	possible	possible	JJ
38536	1029	15	proportions	proportion	NNS
38536	1029	16	that	that	WDT
38536	1029	17	can	can	MD
38536	1029	18	be	be	VB
38536	1029	19	derived	derive	VBN
38536	1029	20	from	from	IN
38536	1029	21	it	-PRON-	PRP
38536	1029	22	.	.	.
38536	1030	1	[	[	-LRB-
38536	1030	2	See	see	VB
38536	1030	3	(	(	-LRB-
38536	1030	4	5	5	CD
38536	1030	5	)	)	-RRB-
38536	1030	6	above	above	RB
38536	1030	7	.	.	.
38536	1030	8	]	]	-RRB-
38536	1031	1	7	7	LS
38536	1031	2	.	.	.
38536	1032	1	Given	give	VBN
38536	1032	2	rs	r	NNS
38536	1032	3	=	=	SYM
38536	1032	4	161	161	CD
38536	1032	5	m	m	NN
38536	1032	6	;	;	:
38536	1032	7	write	write	VB
38536	1032	8	the	the	DT
38536	1032	9	eight	eight	CD
38536	1032	10	proportions	proportion	NNS
38536	1032	11	that	that	WDT
38536	1032	12	may	may	MD
38536	1032	13	be	be	VB
38536	1032	14	derived	derive	VBN
38536	1032	15	from	from	IN
38536	1032	16	it	-PRON-	PRP
38536	1032	17	,	,	,
38536	1032	18	and	and	CC
38536	1032	19	quote	quote	VB
38536	1032	20	your	-PRON-	PRP$
38536	1032	21	authority	authority	NN
38536	1032	22	.	.	.
38536	1033	1	8	8	LS
38536	1033	2	.	.	.
38536	1034	1	(	(	-LRB-
38536	1034	2	_	_	NNP
38536	1034	3	a	a	DT
38536	1034	4	_	_	NNP
38536	1034	5	)	)	-RRB-
38536	1034	6	What	what	WP
38536	1034	7	theorem	theorem	NN
38536	1034	8	allows	allow	VBZ
38536	1034	9	you	-PRON-	PRP
38536	1034	10	to	to	TO
38536	1034	11	change	change	VB
38536	1034	12	any	any	DT
38536	1034	13	proportion	proportion	NN
38536	1034	14	into	into	IN
38536	1034	15	an	an	DT
38536	1034	16	equation	equation	NN
38536	1034	17	?	?	.
38536	1035	1	(	(	-LRB-
38536	1035	2	_	_	NNP
38536	1035	3	b	b	NNP
38536	1035	4	_	_	NNP
38536	1035	5	)	)	-RRB-
38536	1035	6	What	what	WP
38536	1035	7	theorem	theorem	NN
38536	1035	8	allows	allow	VBZ
38536	1035	9	you	-PRON-	PRP
38536	1035	10	to	to	TO
38536	1035	11	change	change	VB
38536	1035	12	any	any	DT
38536	1035	13	equation	equation	NN
38536	1035	14	into	into	IN
38536	1035	15	a	a	DT
38536	1035	16	proportion	proportion	NN
38536	1035	17	?	?	.
38536	1036	1	9	9	CD
38536	1036	2	.	.	.
38536	1037	1	If	if	IN
38536	1037	2	xy	xy	NNP
38536	1037	3	=	=	SYM
38536	1037	4	rg	rg	NNP
38536	1037	5	,	,	,
38536	1037	6	what	what	WP
38536	1037	7	is	be	VBZ
38536	1037	8	the	the	DT
38536	1037	9	ratio	ratio	NN
38536	1037	10	of	of	IN
38536	1037	11	x	x	NNS
38536	1037	12	to	to	IN
38536	1037	13	g	g	NN
38536	1037	14	?	?	.
38536	1038	1	of	of	IN
38536	1038	2	y	y	NNP
38536	1038	3	to	to	IN
38536	1038	4	r	r	NNP
38536	1038	5	?	?	.
38536	1039	1	of	of	IN
38536	1039	2	y	y	NNP
38536	1039	3	to	to	IN
38536	1039	4	g	g	NNP
38536	1039	5	?	?	.
38536	1040	1	10	10	CD
38536	1040	2	.	.	.
38536	1041	1	Find	find	VB
38536	1041	2	two	two	CD
38536	1041	3	numbers	number	NNS
38536	1041	4	such	such	JJ
38536	1041	5	that	that	IN
38536	1041	6	their	-PRON-	PRP$
38536	1041	7	sum	sum	NN
38536	1041	8	,	,	,
38536	1041	9	difference	difference	NN
38536	1041	10	,	,	,
38536	1041	11	and	and	CC
38536	1041	12	the	the	DT
38536	1041	13	sum	sum	NN
38536	1041	14	of	of	IN
38536	1041	15	their	-PRON-	PRP$
38536	1041	16	squares	square	NNS
38536	1041	17	are	be	VBP
38536	1041	18	in	in	IN
38536	1041	19	the	the	DT
38536	1041	20	ratio	ratio	NN
38536	1041	21	5	5	CD
38536	1041	22	:	:	SYM
38536	1041	23	3	3	CD
38536	1041	24	:	:	SYM
38536	1041	25	51	51	CD
38536	1041	26	.	.	.
38536	1042	1	(	(	-LRB-
38536	1042	2	_	_	NNP
38536	1042	3	Yale	Yale	NNP
38536	1042	4	.	.	.
38536	1042	5	_	_	NNP
38536	1042	6	)	)	-RRB-
38536	1042	7	~Reference:~	~Reference:~	NNP
38536	1042	8	The	the	DT
38536	1042	9	chapter	chapter	NN
38536	1042	10	on	on	IN
38536	1042	11	Ratio	Ratio	NNP
38536	1042	12	and	and	CC
38536	1042	13	Proportion	proportion	NN
38536	1042	14	in	in	IN
38536	1042	15	any	any	DT
38536	1042	16	algebra	algebra	NN
38536	1042	17	.	.	.
38536	1043	1	An	an	DT
38536	1043	2	easy	easy	JJ
38536	1043	3	and	and	CC
38536	1043	4	powerful	powerful	JJ
38536	1043	5	method	method	NN
38536	1043	6	of	of	IN
38536	1043	7	proving	prove	VBG
38536	1043	8	four	four	CD
38536	1043	9	expressions	expression	NNS
38536	1043	10	in	in	IN
38536	1043	11	proportion	proportion	NN
38536	1043	12	is	be	VBZ
38536	1043	13	illustrated	illustrate	VBN
38536	1043	14	by	by	IN
38536	1043	15	the	the	DT
38536	1043	16	following	follow	VBG
38536	1043	17	example	example	NN
38536	1043	18	:	:	:
38536	1043	19	Given	give	VBN
38536	1043	20	a	a	NN
38536	1043	21	:	:	:
38536	1043	22	b	b	NN
38536	1043	23	=	=	SYM
38536	1043	24	c	c	NN
38536	1043	25	:	:	:
38536	1043	26	d	d	LS
38536	1043	27	;	;	:
38536	1043	28	prove	prove	VB
38536	1043	29	that	that	IN
38536	1043	30	3a^3	3a^3	CD
38536	1043	31	+	+	SYM
38536	1043	32	5ab^2	5ab^2	CD
38536	1043	33	:	:	:
38536	1043	34	3a^3	3a^3	CD
38536	1043	35	-	-	HYPH
38536	1043	36	5ab^2	5ab^2	NNS
38536	1043	37	=	=	SYM
38536	1043	38	3c^3	3c^3	CD
38536	1043	39	+	+	CC
38536	1043	40	5cd^2	5cd^2	CD
38536	1043	41	:	:	:
38536	1043	42	3c^3	3c^3	CD
38536	1043	43	-	-	SYM
38536	1043	44	5cd^2	5cd^2	CD
38536	1043	45	.	.	.
38536	1044	1	Let	let	VB
38536	1044	2	a	a	NN
38536	1044	3	/	/	SYM
38536	1044	4	b	b	NN
38536	1044	5	=	=	SYM
38536	1044	6	r.	r.	NNP
38536	1044	7	Therefore	therefore	RB
38536	1044	8	a	a	DT
38536	1044	9	=	=	NFP
38536	1044	10	br	br	XX
38536	1044	11	.	.	.
38536	1045	1	Also	also	RB
38536	1045	2	c	c	NN
38536	1045	3	/	/	SYM
38536	1045	4	d	d	NNP
38536	1045	5	=	=	SYM
38536	1045	6	r.	r.	NNP
38536	1045	7	Therefore	Therefore	NNP
38536	1045	8	c	c	NNP
38536	1045	9	=	=	SYM
38536	1045	10	dr	dr	NNP
38536	1045	11	.	.	.
38536	1046	1	Substitute	substitute	VB
38536	1046	2	the	the	DT
38536	1046	3	value	value	NN
38536	1046	4	of	of	IN
38536	1046	5	a	a	DT
38536	1046	6	in	in	IN
38536	1046	7	the	the	DT
38536	1046	8	first	first	JJ
38536	1046	9	ratio	ratio	NN
38536	1046	10	,	,	,
38536	1046	11	and	and	CC
38536	1046	12	c	c	NN
38536	1046	13	in	in	IN
38536	1046	14	the	the	DT
38536	1046	15	second	second	NN
38536	1046	16	:	:	:
38536	1046	17	Then	then	RB
38536	1046	18	(	(	-LRB-
38536	1046	19	3a^3	3a^3	CD
38536	1046	20	+	+	SYM
38536	1046	21	5ab^2)/(3a^3	5ab^2)/(3a^3	CD
38536	1046	22	-	-	HYPH
38536	1046	23	5ab^2	5ab^2	NN
38536	1046	24	)	)	-RRB-
38536	1046	25	=	=	NFP
38536	1046	26	(	(	-LRB-
38536	1046	27	3b^3r^3	3b^3r^3	CD
38536	1046	28	+	+	CD
38536	1046	29	5b^3r)/(3b^3r^3	5b^3r)/(3b^3r^3	CD
38536	1046	30	-	-	:
38536	1046	31	5b^3r	5b^3r	NN
38536	1046	32	)	)	-RRB-
38536	1046	33	=	=	NFP
38536	1046	34	[	[	-LRB-
38536	1046	35	b^3r(3r^2	b^3r(3r^2	NN
38536	1046	36	+	+	CC
38536	1046	37	5)]/[b^3r(3r^2	5)]/[b^3r(3r^2	NNP
38536	1046	38	-	-	HYPH
38536	1046	39	5	5	CD
38536	1046	40	)	)	-RRB-
38536	1046	41	]	]	-RRB-
38536	1046	42	=	=	FW
38536	1046	43	(	(	-LRB-
38536	1046	44	3r^2	3r^2	CD
38536	1046	45	+	+	CD
38536	1046	46	5)/(3r^2	5)/(3r^2	CD
38536	1046	47	-	-	SYM
38536	1046	48	5	5	CD
38536	1046	49	)	)	-RRB-
38536	1046	50	.	.	.
38536	1047	1	Also	also	RB
38536	1047	2	(	(	-LRB-
38536	1047	3	3c^3	3c^3	CD
38536	1047	4	+	+	SYM
38536	1047	5	5cd^2)/(3c^3	5cd^2)/(3c^3	CD
38536	1047	6	-	-	SYM
38536	1047	7	5cd^2	5cd^2	CD
38536	1047	8	)	)	-RRB-
38536	1047	9	=	=	NFP
38536	1047	10	(	(	-LRB-
38536	1047	11	3d^3r^3	3d^3r^3	NNP
38536	1047	12	+	+	SYM
38536	1047	13	5d^3r)/(3d^3r^3	5d^3r)/(3d^3r^3	CD
38536	1047	14	-	-	:
38536	1047	15	5d^3r	5d^3r	CD
38536	1047	16	)	)	-RRB-
38536	1047	17	=	=	NFP
38536	1047	18	[	[	-LRB-
38536	1047	19	d^3r(3r^2	d^3r(3r^2	NN
38536	1047	20	+	+	CC
38536	1047	21	5)]/[d^3r(3r^2	5)]/[d^3r(3r^2	NNP
38536	1047	22	-	-	HYPH
38536	1047	23	5	5	CD
38536	1047	24	)	)	-RRB-
38536	1047	25	]	]	-RRB-
38536	1047	26	=	=	FW
38536	1047	27	(	(	-LRB-
38536	1047	28	3r^2	3r^2	CD
38536	1047	29	+	+	CD
38536	1047	30	5)/(3r^2	5)/(3r^2	CD
38536	1047	31	-	-	SYM
38536	1047	32	5	5	CD
38536	1047	33	)	)	-RRB-
38536	1047	34	.	.	.
38536	1048	1	Therefore	therefore	RB
38536	1048	2	(	(	-LRB-
38536	1048	3	3a^3	3a^3	CD
38536	1048	4	+	+	SYM
38536	1048	5	5ab^2)/(3a^3	5ab^2)/(3a^3	CD
38536	1048	6	-	-	HYPH
38536	1048	7	5ab^2	5ab^2	NN
38536	1048	8	)	)	-RRB-
38536	1048	9	=	=	NFP
38536	1048	10	(	(	-LRB-
38536	1048	11	3c^3	3c^3	CD
38536	1048	12	+	+	SYM
38536	1048	13	5cd^2)/(3c^3	5cd^2)/(3c^3	CD
38536	1048	14	-	-	SYM
38536	1048	15	5cd^2	5cd^2	CD
38536	1048	16	)	)	-RRB-
38536	1048	17	.	.	.
38536	1049	1	Axiom	Axiom	NNP
38536	1049	2	1	1	CD
38536	1049	3	.	.	.
38536	1050	1	Or	or	CC
38536	1050	2	,	,	,
38536	1050	3	3a^3	3a^3	CD
38536	1050	4	+	+	SYM
38536	1050	5	5ab^2	5ab^2	CD
38536	1050	6	:	:	:
38536	1050	7	3a^3	3a^3	CD
38536	1050	8	-	-	HYPH
38536	1050	9	5ab^2	5ab^2	NNS
38536	1050	10	=	=	SYM
38536	1050	11	3c^3	3c^3	CD
38536	1050	12	+	+	CC
38536	1050	13	5cd^2	5cd^2	CD
38536	1050	14	:	:	:
38536	1050	15	3c^3	3c^3	CD
38536	1050	16	-	-	SYM
38536	1050	17	5cd^2	5cd^2	CD
38536	1050	18	.	.	.
38536	1051	1	If	if	IN
38536	1051	2	a	a	DT
38536	1051	3	:	:	:
38536	1051	4	b	b	NN
38536	1051	5	=	=	SYM
38536	1051	6	c	c	NN
38536	1051	7	:	:	:
38536	1051	8	d	d	NN
38536	1051	9	,	,	,
38536	1051	10	prove	prove	NN
38536	1051	11	:	:	:
38536	1051	12	1	1	LS
38536	1051	13	.	.	.
38536	1051	14	a^2	a^2	NNS
38536	1051	15	+	+	CC
38536	1051	16	b^2	b^2	NNS
38536	1051	17	:	:	:
38536	1051	18	a^2	a^2	NNP
38536	1051	19	=	=	SYM
38536	1051	20	c^2	c^2	NNS
38536	1051	21	+	+	SYM
38536	1051	22	d^2	d^2	NNP
38536	1051	23	:	:	:
38536	1051	24	c^2	c^2	NNP
38536	1051	25	.	.	.
38536	1052	1	2	2	LS
38536	1052	2	.	.	.
38536	1052	3	a^2	a^2	NNS
38536	1052	4	+	+	CC
38536	1052	5	3b^2	3b^2	CD
38536	1052	6	:	:	:
38536	1052	7	a^2	a^2	CD
38536	1052	8	-	-	HYPH
38536	1052	9	3b^2	3b^2	CD
38536	1052	10	=	=	SYM
38536	1052	11	c^2	c^2	NNP
38536	1052	12	+	+	CC
38536	1052	13	3d^2	3d^2	CD
38536	1052	14	:	:	:
38536	1052	15	c^2	c^2	NNP
38536	1052	16	-	-	SYM
38536	1052	17	3d^2	3d^2	CD
38536	1052	18	.	.	.
38536	1053	1	3	3	LS
38536	1053	2	.	.	.
38536	1053	3	a^2	a^2	NNS
38536	1053	4	+	+	CC
38536	1053	5	2b^2	2b^2	CD
38536	1053	6	:	:	:
38536	1053	7	2b^2	2b^2	CD
38536	1053	8	=	=	SYM
38536	1053	9	ac	ac	NN
38536	1053	10	+	+	CC
38536	1053	11	2bd	2bd	NN
38536	1053	12	:	:	:
38536	1053	13	2bd	2bd	NN
38536	1053	14	.	.	.
38536	1054	1	4	4	LS
38536	1054	2	.	.	.
38536	1055	1	2a	2a	CD
38536	1055	2	+	+	SYM
38536	1055	3	3c	3c	NN
38536	1055	4	:	:	:
38536	1055	5	2a	2a	CD
38536	1055	6	-	-	HYPH
38536	1055	7	3c	3c	NNP
38536	1055	8	=	=	SYM
38536	1055	9	8b	8b	CD
38536	1055	10	+	+	SYM
38536	1055	11	12d	12d	NNS
38536	1055	12	:	:	:
38536	1055	13	8b	8b	NNP
38536	1055	14	-	-	HYPH
38536	1055	15	12d	12d	NNS
38536	1055	16	.	.	.
38536	1056	1	5	5	CD
38536	1056	2	.	.	.
38536	1056	3	a^2	a^2	NNP
38536	1056	4	-	-	HYPH
38536	1056	5	ab	ab	NNP
38536	1056	6	+	+	DT
38536	1056	7	b^2	b^2	NNS
38536	1056	8	:	:	:
38536	1056	9	(	(	-LRB-
38536	1056	10	a^3	a^3	NNP
38536	1056	11	-	-	HYPH
38536	1056	12	b^3)/a	b^3)/a	NNP
38536	1056	13	=	=	SYM
38536	1056	14	c^2	c^2	NNP
38536	1056	15	-	-	HYPH
38536	1056	16	cd	cd	NN
38536	1056	17	+	+	CC
38536	1056	18	d^2	d^2	NNS
38536	1056	19	:	:	:
38536	1056	20	(	(	-LRB-
38536	1056	21	c^3	c^3	NNP
38536	1056	22	-	-	HYPH
38536	1056	23	d^3)/c	d^3)/c	ADD
38536	1056	24	.	.	.
38536	1057	1	6	6	CD
38536	1057	2	.	.	.
38536	1058	1	The	the	DT
38536	1058	2	second	second	JJ
38536	1058	3	of	of	IN
38536	1058	4	three	three	CD
38536	1058	5	numbers	number	NNS
38536	1058	6	is	be	VBZ
38536	1058	7	a	a	DT
38536	1058	8	mean	mean	JJ
38536	1058	9	proportional	proportional	JJ
38536	1058	10	between	between	IN
38536	1058	11	the	the	DT
38536	1058	12	other	other	JJ
38536	1058	13	two	two	CD
38536	1058	14	.	.	.
38536	1059	1	The	the	DT
38536	1059	2	third	third	JJ
38536	1059	3	number	number	NN
38536	1059	4	exceeds	exceed	VBZ
38536	1059	5	the	the	DT
38536	1059	6	sum	sum	NN
38536	1059	7	of	of	IN
38536	1059	8	the	the	DT
38536	1059	9	other	other	JJ
38536	1059	10	two	two	CD
38536	1059	11	by	by	IN
38536	1059	12	20	20	CD
38536	1059	13	;	;	:
38536	1059	14	and	and	CC
38536	1059	15	the	the	DT
38536	1059	16	sum	sum	NN
38536	1059	17	of	of	IN
38536	1059	18	the	the	DT
38536	1059	19	first	first	JJ
38536	1059	20	and	and	CC
38536	1059	21	third	third	JJ
38536	1059	22	exceeds	exceed	VBZ
38536	1059	23	three	three	CD
38536	1059	24	times	time	NNS
38536	1059	25	the	the	DT
38536	1059	26	second	second	JJ
38536	1059	27	by	by	IN
38536	1059	28	4	4	CD
38536	1059	29	.	.	.
38536	1060	1	Find	find	VB
38536	1060	2	the	the	DT
38536	1060	3	numbers	number	NNS
38536	1060	4	.	.	.
38536	1061	1	7	7	LS
38536	1061	2	.	.	.
38536	1062	1	Three	three	CD
38536	1062	2	numbers	number	NNS
38536	1062	3	are	be	VBP
38536	1062	4	proportional	proportional	JJ
38536	1062	5	to	to	IN
38536	1062	6	5	5	CD
38536	1062	7	,	,	,
38536	1062	8	7	7	CD
38536	1062	9	,	,	,
38536	1062	10	and	and	CC
38536	1062	11	9	9	CD
38536	1062	12	;	;	:
38536	1062	13	and	and	CC
38536	1062	14	their	-PRON-	PRP$
38536	1062	15	sum	sum	NN
38536	1062	16	is	be	VBZ
38536	1062	17	14	14	CD
38536	1062	18	.	.	.
38536	1063	1	Find	find	VB
38536	1063	2	the	the	DT
38536	1063	3	numbers	number	NNS
38536	1063	4	.	.	.
38536	1064	1	(	(	-LRB-
38536	1064	2	_	_	NNP
38536	1064	3	College	College	NNP
38536	1064	4	Entrance	Entrance	NNP
38536	1064	5	Board	Board	NNP
38536	1064	6	.	.	.
38536	1064	7	_	_	NNP
38536	1064	8	)	)	-RRB-
38536	1064	9	8	8	CD
38536	1064	10	.	.	.
38536	1065	1	A	a	DT
38536	1065	2	triangular	triangular	JJ
38536	1065	3	field	field	NN
38536	1065	4	has	have	VBZ
38536	1065	5	the	the	DT
38536	1065	6	sides	side	NNS
38536	1065	7	15	15	CD
38536	1065	8	,	,	,
38536	1065	9	18	18	CD
38536	1065	10	,	,	,
38536	1065	11	and	and	CC
38536	1065	12	27	27	CD
38536	1065	13	rods	rod	NNS
38536	1065	14	,	,	,
38536	1065	15	respectively	respectively	RB
38536	1065	16	.	.	.
38536	1066	1	Find	find	VB
38536	1066	2	the	the	DT
38536	1066	3	dimensions	dimension	NNS
38536	1066	4	of	of	IN
38536	1066	5	a	a	DT
38536	1066	6	similar	similar	JJ
38536	1066	7	field	field	NN
38536	1066	8	having	have	VBG
38536	1066	9	4	4	CD
38536	1066	10	times	time	NNS
38536	1066	11	the	the	DT
38536	1066	12	area	area	NN
38536	1066	13	.	.	.
38536	1067	1	~ARITHMETICAL	~ARITHMETICAL	NFP
38536	1067	2	PROGRESSION~	PROGRESSION~	NNP
38536	1067	3	1	1	CD
38536	1067	4	.	.	.
38536	1068	1	Define	define	VB
38536	1068	2	an	an	DT
38536	1068	3	arithmetical	arithmetical	JJ
38536	1068	4	progression	progression	NN
38536	1068	5	.	.	.
38536	1069	1	Learn	learn	VB
38536	1069	2	to	to	TO
38536	1069	3	derive	derive	VB
38536	1069	4	the	the	DT
38536	1069	5	three	three	CD
38536	1069	6	formulas	formula	NNS
38536	1069	7	in	in	IN
38536	1069	8	arithmetical	arithmetical	JJ
38536	1069	9	progression	progression	NN
38536	1069	10	:	:	:
38536	1069	11	l	l	NN
38536	1069	12	=	=	SYM
38536	1069	13	a	a	NN
38536	1069	14	+	+	SYM
38536	1069	15	(	(	-LRB-
38536	1069	16	n	n	NN
38536	1069	17	-	-	HYPH
38536	1069	18	1)d	1)d	CD
38536	1069	19	,	,	,
38536	1069	20	S	S	NNP
38536	1069	21	=	=	NFP
38536	1069	22	(	(	-LRB-
38536	1069	23	n/2)(a	n/2)(a	NNP
38536	1069	24	+	+	NNP
38536	1069	25	l	l	NN
38536	1069	26	)	)	-RRB-
38536	1069	27	,	,	,
38536	1069	28	S	S	NNP
38536	1069	29	=	=	NFP
38536	1069	30	(	(	-LRB-
38536	1069	31	n/2)[2a	n/2)[2a	NNP
38536	1069	32	+	+	CC
38536	1069	33	(	(	-LRB-
38536	1069	34	n	n	CD
38536	1069	35	-	-	HYPH
38536	1069	36	1)d	1)d	CD
38536	1069	37	]	]	-RRB-
38536	1069	38	.	.	.
38536	1070	1	2	2	LS
38536	1070	2	.	.	.
38536	1071	1	Find	find	VB
38536	1071	2	the	the	DT
38536	1071	3	sum	sum	NN
38536	1071	4	of	of	IN
38536	1071	5	the	the	DT
38536	1071	6	first	first	JJ
38536	1071	7	50	50	CD
38536	1071	8	odd	odd	JJ
38536	1071	9	numbers	number	NNS
38536	1071	10	.	.	.
38536	1072	1	3	3	LS
38536	1072	2	.	.	.
38536	1073	1	In	in	IN
38536	1073	2	the	the	DT
38536	1073	3	series	series	NN
38536	1073	4	2	2	CD
38536	1073	5	,	,	,
38536	1073	6	5	5	CD
38536	1073	7	,	,	,
38536	1073	8	8	8	CD
38536	1073	9	,	,	,
38536	1073	10	·	·	NFP
38536	1073	11	·	·	NFP
38536	1073	12	·	·	NFP
38536	1073	13	,	,	,
38536	1073	14	which	which	WDT
38536	1073	15	term	term	NN
38536	1073	16	is	be	VBZ
38536	1073	17	92	92	CD
38536	1073	18	?	?	.
38536	1074	1	4	4	LS
38536	1074	2	.	.	.
38536	1075	1	How	how	WRB
38536	1075	2	many	many	JJ
38536	1075	3	terms	term	NNS
38536	1075	4	must	must	MD
38536	1075	5	be	be	VB
38536	1075	6	taken	take	VBN
38536	1075	7	from	from	IN
38536	1075	8	the	the	DT
38536	1075	9	series	series	NN
38536	1075	10	3	3	CD
38536	1075	11	,	,	,
38536	1075	12	5	5	CD
38536	1075	13	,	,	,
38536	1075	14	7	7	CD
38536	1075	15	,	,	,
38536	1075	16	·	·	NFP
38536	1075	17	·	·	NFP
38536	1075	18	·	·	NFP
38536	1075	19	,	,	,
38536	1075	20	to	to	TO
38536	1075	21	make	make	VB
38536	1075	22	a	a	DT
38536	1075	23	total	total	NN
38536	1075	24	of	of	IN
38536	1075	25	255	255	CD
38536	1075	26	?	?	.
38536	1076	1	5	5	CD
38536	1076	2	.	.	.
38536	1077	1	Insert	Insert	NNP
38536	1077	2	5	5	CD
38536	1077	3	arithmetical	arithmetical	JJ
38536	1077	4	means	mean	NNS
38536	1077	5	between	between	IN
38536	1077	6	11	11	CD
38536	1077	7	and	and	CC
38536	1077	8	32	32	CD
38536	1077	9	.	.	.
38536	1078	1	6	6	CD
38536	1078	2	.	.	.
38536	1079	1	Insert	Insert	NNP
38536	1079	2	9	9	CD
38536	1079	3	arithmetical	arithmetical	JJ
38536	1079	4	means	mean	NNS
38536	1079	5	between	between	IN
38536	1079	6	7	7	CD
38536	1079	7	-	-	SYM
38536	1079	8	1/2	1/2	CD
38536	1079	9	and	and	CC
38536	1079	10	30	30	CD
38536	1079	11	.	.	.
38536	1080	1	7	7	LS
38536	1080	2	.	.	.
38536	1081	1	Find	find	VB
38536	1081	2	x	x	NNS
38536	1081	3	,	,	,
38536	1081	4	if	if	IN
38536	1081	5	3	3	CD
38536	1081	6	+	+	SYM
38536	1081	7	2x	2x	CD
38536	1081	8	,	,	,
38536	1081	9	5	5	CD
38536	1081	10	+	+	SYM
38536	1081	11	6x	6x	CD
38536	1081	12	,	,	,
38536	1081	13	9	9	CD
38536	1081	14	+	+	SYM
38536	1081	15	5x	5x	CD
38536	1081	16	are	be	VBP
38536	1081	17	in	in	IN
38536	1081	18	A.	a.	NN
38536	1081	19	P.	P.	NNP
38536	1081	20	8	8	CD
38536	1081	21	.	.	.
38536	1082	1	The	the	DT
38536	1082	2	7th	7th	JJ
38536	1082	3	term	term	NN
38536	1082	4	of	of	IN
38536	1082	5	an	an	DT
38536	1082	6	arithmetical	arithmetical	JJ
38536	1082	7	progression	progression	NN
38536	1082	8	is	be	VBZ
38536	1082	9	17	17	CD
38536	1082	10	,	,	,
38536	1082	11	and	and	CC
38536	1082	12	the	the	DT
38536	1082	13	13th	13th	JJ
38536	1082	14	term	term	NN
38536	1082	15	is	be	VBZ
38536	1082	16	59	59	CD
38536	1082	17	.	.	.
38536	1083	1	Find	find	VB
38536	1083	2	the	the	DT
38536	1083	3	4th	4th	JJ
38536	1083	4	term	term	NN
38536	1083	5	.	.	.
38536	1084	1	9	9	CD
38536	1084	2	.	.	.
38536	1085	1	How	how	WRB
38536	1085	2	can	can	MD
38536	1085	3	you	-PRON-	PRP
38536	1085	4	turn	turn	VB
38536	1085	5	an	an	DT
38536	1085	6	A.	a.	NN
38536	1085	7	P.	p.	NN
38536	1085	8	into	into	IN
38536	1085	9	an	an	DT
38536	1085	10	equation	equation	NN
38536	1085	11	?	?	.
38536	1086	1	10	10	CD
38536	1086	2	.	.	.
38536	1087	1	Given	give	VBN
38536	1087	2	a	a	DT
38536	1087	3	=	=	SYM
38536	1087	4	-5/3	-5/3	NN
38536	1087	5	,	,	,
38536	1087	6	n	n	NN
38536	1087	7	=	=	SYM
38536	1087	8	20	20	CD
38536	1087	9	,	,	,
38536	1087	10	S	S	NNP
38536	1087	11	=	=	SYM
38536	1087	12	-5/3	-5/3	FW
38536	1087	13	,	,	,
38536	1087	14	find	find	VB
38536	1087	15	d	d	NN
38536	1087	16	and	and	CC
38536	1087	17	l.	l.	NNP
38536	1087	18	11	11	CD
38536	1087	19	.	.	.
38536	1088	1	Find	find	VB
38536	1088	2	the	the	DT
38536	1088	3	sum	sum	NN
38536	1088	4	of	of	IN
38536	1088	5	the	the	DT
38536	1088	6	first	first	JJ
38536	1088	7	n	n	JJ
38536	1088	8	odd	odd	JJ
38536	1088	9	numbers	number	NNS
38536	1088	10	.	.	.
38536	1089	1	12	12	CD
38536	1089	2	.	.	.
38536	1090	1	An	an	DT
38536	1090	2	arithmetical	arithmetical	JJ
38536	1090	3	progression	progression	NN
38536	1090	4	consists	consist	VBZ
38536	1090	5	of	of	IN
38536	1090	6	21	21	CD
38536	1090	7	terms	term	NNS
38536	1090	8	.	.	.
38536	1091	1	The	the	DT
38536	1091	2	sum	sum	NN
38536	1091	3	of	of	IN
38536	1091	4	the	the	DT
38536	1091	5	three	three	CD
38536	1091	6	terms	term	NNS
38536	1091	7	in	in	IN
38536	1091	8	the	the	DT
38536	1091	9	middle	middle	NN
38536	1091	10	is	be	VBZ
38536	1091	11	129	129	CD
38536	1091	12	;	;	:
38536	1091	13	the	the	DT
38536	1091	14	sum	sum	NN
38536	1091	15	of	of	IN
38536	1091	16	the	the	DT
38536	1091	17	last	last	JJ
38536	1091	18	three	three	CD
38536	1091	19	terms	term	NNS
38536	1091	20	is	be	VBZ
38536	1091	21	237	237	CD
38536	1091	22	.	.	.
38536	1092	1	Find	find	VB
38536	1092	2	the	the	DT
38536	1092	3	series	series	NN
38536	1092	4	.	.	.
38536	1093	1	(	(	-LRB-
38536	1093	2	Look	look	VB
38536	1093	3	up	up	RP
38536	1093	4	the	the	DT
38536	1093	5	short	short	JJ
38536	1093	6	method	method	NN
38536	1093	7	for	for	IN
38536	1093	8	such	such	JJ
38536	1093	9	problems	problem	NNS
38536	1093	10	.	.	.
38536	1093	11	)	)	-RRB-
38536	1094	1	(	(	-LRB-
38536	1094	2	_	_	NNP
38536	1094	3	Mass	Mass	NNP
38536	1094	4	.	.	.
38536	1095	1	Inst	inst	RB
38536	1095	2	.	.	.
38536	1096	1	of	of	IN
38536	1096	2	Technology	Technology	NNP
38536	1096	3	.	.	.
38536	1096	4	_	_	NNP
38536	1096	5	)	)	-RRB-
38536	1096	6	13	13	CD
38536	1096	7	.	.	.
38536	1097	1	B	b	NN
38536	1097	2	travels	travel	NNS
38536	1097	3	3	3	CD
38536	1097	4	miles	mile	NNS
38536	1097	5	the	the	DT
38536	1097	6	first	first	JJ
38536	1097	7	day	day	NN
38536	1097	8	,	,	,
38536	1097	9	7	7	CD
38536	1097	10	miles	mile	NNS
38536	1097	11	the	the	DT
38536	1097	12	second	second	JJ
38536	1097	13	day	day	NN
38536	1097	14	,	,	,
38536	1097	15	11	11	CD
38536	1097	16	miles	mile	NNS
38536	1097	17	the	the	DT
38536	1097	18	third	third	JJ
38536	1097	19	day	day	NN
38536	1097	20	,	,	,
38536	1097	21	etc	etc	FW
38536	1097	22	.	.	.
38536	1098	1	In	in	IN
38536	1098	2	how	how	WRB
38536	1098	3	many	many	JJ
38536	1098	4	days	day	NNS
38536	1098	5	will	will	MD
38536	1098	6	B	B	VBG
38536	1098	7	overtake	overtake	VB
38536	1098	8	A	a	DT
38536	1098	9	who	who	WP
38536	1098	10	started	start	VBD
38536	1098	11	from	from	IN
38536	1098	12	the	the	DT
38536	1098	13	same	same	JJ
38536	1098	14	point	point	NN
38536	1098	15	8	8	CD
38536	1098	16	days	day	NNS
38536	1098	17	in	in	IN
38536	1098	18	advance	advance	NN
38536	1098	19	and	and	CC
38536	1098	20	who	who	WP
38536	1098	21	travels	travel	VBZ
38536	1098	22	uniformly	uniformly	RB
38536	1098	23	15	15	CD
38536	1098	24	miles	mile	NNS
38536	1098	25	a	a	DT
38536	1098	26	day	day	NN
38536	1098	27	?	?	.
38536	1099	1	~Reference:~	~reference:~	VB
38536	1099	2	The	the	DT
38536	1099	3	chapter	chapter	NN
38536	1099	4	on	on	IN
38536	1099	5	Arithmetical	Arithmetical	NNP
38536	1099	6	Progression	Progression	NNP
38536	1099	7	in	in	IN
38536	1099	8	any	any	DT
38536	1099	9	algebra	algebra	NN
38536	1099	10	.	.	.
38536	1100	1	~GEOMETRICAL	~GEOMETRICAL	NNP
38536	1100	2	PROGRESSION~	PROGRESSION~	NNP
38536	1100	3	1	1	CD
38536	1100	4	.	.	.
38536	1101	1	Define	define	VB
38536	1101	2	a	a	DT
38536	1101	3	geometrical	geometrical	JJ
38536	1101	4	progression	progression	NN
38536	1101	5	.	.	.
38536	1102	1	Learn	learn	VB
38536	1102	2	to	to	TO
38536	1102	3	derive	derive	VB
38536	1102	4	the	the	DT
38536	1102	5	four	four	CD
38536	1102	6	formulas	formula	NNS
38536	1102	7	in	in	IN
38536	1102	8	geometrical	geometrical	JJ
38536	1102	9	progression	progression	NN
38536	1102	10	:	:	:
38536	1102	11	{	{	-LRB-
38536	1102	12	I.	I.	NNP
38536	1102	13	l	l	NN
38536	1102	14	=	=	SYM
38536	1102	15	ar^(n	ar^(n	NNP
38536	1102	16	-	-	HYPH
38536	1102	17	1	1	CD
38536	1102	18	)	)	-RRB-
38536	1102	19	.	.	.
38536	1103	1	{	{	-LRB-
38536	1103	2	II	ii	CD
38536	1103	3	.	.	.
38536	1104	1	S	S	NNP
38536	1104	2	=	=	NFP
38536	1104	3	(	(	-LRB-
38536	1104	4	ar^n	ar^n	NFP
38536	1104	5	-	-	HYPH
38536	1104	6	a)/(r	a)/(r	NNP
38536	1104	7	-	-	HYPH
38536	1104	8	1	1	CD
38536	1104	9	)	)	-RRB-
38536	1104	10	.	.	.
38536	1105	1	{	{	-LRB-
38536	1105	2	III	iii	CD
38536	1105	3	.	.	.
38536	1106	1	S	S	NNP
38536	1106	2	=	=	NFP
38536	1106	3	(	(	-LRB-
38536	1106	4	rl	rl	JJ
38536	1106	5	-	-	JJ
38536	1106	6	a)/(r	a)/(r	NNP
38536	1106	7	-	-	HYPH
38536	1106	8	1	1	CD
38536	1106	9	)	)	-RRB-
38536	1106	10	.	.	.
38536	1107	1	{	{	-LRB-
38536	1107	2	IV	IV	NNP
38536	1107	3	.	.	.
38536	1108	1	S_{[infinity	s_{[infinity	LS
38536	1108	2	]	]	-RRB-
38536	1108	3	}	}	-RRB-
38536	1108	4	=	=	NFP
38536	1108	5	(	(	-LRB-
38536	1108	6	a)/(1	a)/(1	NNP
38536	1108	7	-	-	HYPH
38536	1108	8	r	r	NNP
38536	1108	9	)	)	-RRB-
38536	1108	10	.	.	.
38536	1109	1	2	2	LS
38536	1109	2	.	.	.
38536	1110	1	How	how	WRB
38536	1110	2	many	many	JJ
38536	1110	3	terms	term	NNS
38536	1110	4	must	must	MD
38536	1110	5	be	be	VB
38536	1110	6	taken	take	VBN
38536	1110	7	from	from	IN
38536	1110	8	the	the	DT
38536	1110	9	series	series	NN
38536	1110	10	9	9	CD
38536	1110	11	,	,	,
38536	1110	12	18	18	CD
38536	1110	13	,	,	,
38536	1110	14	36	36	CD
38536	1110	15	,	,	,
38536	1110	16	·	·	NFP
38536	1110	17	·	·	NFP
38536	1110	18	·	·	NFP
38536	1110	19	to	to	TO
38536	1110	20	make	make	VB
38536	1110	21	a	a	DT
38536	1110	22	total	total	NN
38536	1110	23	of	of	IN
38536	1110	24	567	567	CD
38536	1110	25	?	?	.
38536	1111	1	3	3	LS
38536	1111	2	.	.	.
38536	1112	1	In	in	IN
38536	1112	2	the	the	DT
38536	1112	3	G.	G.	NNP
38536	1112	4	P.	P.	NNP
38536	1112	5	2	2	CD
38536	1112	6	,	,	,
38536	1112	7	6	6	CD
38536	1112	8	,	,	,
38536	1112	9	18	18	CD
38536	1112	10	,	,	,
38536	1112	11	·	·	NFP
38536	1112	12	·	·	NFP
38536	1112	13	·	·	NFP
38536	1112	14	,	,	,
38536	1112	15	which	which	WDT
38536	1112	16	term	term	NN
38536	1112	17	is	be	VBZ
38536	1112	18	486	486	CD
38536	1112	19	?	?	.
38536	1113	1	4	4	LS
38536	1113	2	.	.	.
38536	1114	1	Find	find	VB
38536	1114	2	x	x	NNS
38536	1114	3	,	,	,
38536	1114	4	if	if	IN
38536	1114	5	2x	2x	NNP
38536	1114	6	-	-	SYM
38536	1114	7	4	4	CD
38536	1114	8	,	,	,
38536	1114	9	5x	5x	NNP
38536	1114	10	-	-	HYPH
38536	1114	11	7	7	CD
38536	1114	12	,	,	,
38536	1114	13	10x	10x	CD
38536	1114	14	+	+	SYM
38536	1114	15	4	4	CD
38536	1114	16	are	be	VBP
38536	1114	17	in	in	IN
38536	1114	18	geometrical	geometrical	JJ
38536	1114	19	progression	progression	NN
38536	1114	20	.	.	.
38536	1115	1	5	5	CD
38536	1115	2	.	.	.
38536	1116	1	How	how	WRB
38536	1116	2	can	can	MD
38536	1116	3	you	-PRON-	PRP
38536	1116	4	turn	turn	VB
38536	1116	5	a	a	DT
38536	1116	6	G.	G.	NNP
38536	1116	7	P.	P.	NNP
38536	1116	8	into	into	IN
38536	1116	9	an	an	DT
38536	1116	10	equation	equation	NN
38536	1116	11	?	?	.
38536	1117	1	6	6	CD
38536	1117	2	.	.	.
38536	1118	1	Insert	Insert	NNP
38536	1118	2	4	4	CD
38536	1118	3	geometrical	geometrical	JJ
38536	1118	4	means	mean	NNS
38536	1118	5	between	between	IN
38536	1118	6	4	4	CD
38536	1118	7	and	and	CC
38536	1118	8	972	972	CD
38536	1118	9	.	.	.
38536	1119	1	7	7	LS
38536	1119	2	.	.	.
38536	1120	1	Insert	Insert	NNP
38536	1120	2	6	6	CD
38536	1120	3	geometrical	geometrical	JJ
38536	1120	4	means	mean	NNS
38536	1120	5	between	between	IN
38536	1120	6	5/16	5/16	CD
38536	1120	7	and	and	CC
38536	1120	8	5120	5120	CD
38536	1120	9	.	.	.
38536	1121	1	8	8	LS
38536	1121	2	.	.	.
38536	1122	1	Given	give	VBN
38536	1122	2	a	a	DT
38536	1122	3	=	=	NFP
38536	1122	4	-2	-2	NNP
38536	1122	5	,	,	,
38536	1122	6	n	n	NN
38536	1122	7	=	=	SYM
38536	1122	8	5	5	CD
38536	1122	9	,	,	,
38536	1122	10	l	l	NN
38536	1122	11	=	=	NFP
38536	1122	12	-32	-32	NNP
38536	1122	13	;	;	:
38536	1122	14	find	find	VB
38536	1122	15	r	r	NN
38536	1122	16	and	and	CC
38536	1122	17	S.	S.	NNP
38536	1122	18	9	9	CD
38536	1122	19	.	.	.
38536	1123	1	If	if	IN
38536	1123	2	the	the	DT
38536	1123	3	first	first	JJ
38536	1123	4	term	term	NN
38536	1123	5	of	of	IN
38536	1123	6	a	a	DT
38536	1123	7	geometrical	geometrical	JJ
38536	1123	8	progression	progression	NN
38536	1123	9	is	be	VBZ
38536	1123	10	12	12	CD
38536	1123	11	and	and	CC
38536	1123	12	the	the	DT
38536	1123	13	sum	sum	NN
38536	1123	14	to	to	IN
38536	1123	15	infinity	infinity	NN
38536	1123	16	is	be	VBZ
38536	1123	17	36	36	CD
38536	1123	18	,	,	,
38536	1123	19	find	find	VB
38536	1123	20	the	the	DT
38536	1123	21	4th	4th	JJ
38536	1123	22	term	term	NN
38536	1123	23	.	.	.
38536	1124	1	10	10	CD
38536	1124	2	.	.	.
38536	1125	1	If	if	IN
38536	1125	2	the	the	DT
38536	1125	3	series	series	NN
38536	1125	4	3	3	CD
38536	1125	5	-	-	SYM
38536	1125	6	1/3	1/3	CD
38536	1125	7	,	,	,
38536	1125	8	2	2	CD
38536	1125	9	-	-	SYM
38536	1125	10	1/2	1/2	CD
38536	1125	11	,	,	,
38536	1125	12	·	·	NFP
38536	1125	13	·	·	NFP
38536	1125	14	·	·	NFP
38536	1125	15	be	be	VB
38536	1125	16	an	an	DT
38536	1125	17	A.	a.	NN
38536	1125	18	P.	p.	NN
38536	1125	19	,	,	,
38536	1125	20	find	find	VB
38536	1125	21	the	the	DT
38536	1125	22	97th	97th	JJ
38536	1125	23	term	term	NN
38536	1125	24	.	.	.
38536	1126	1	If	if	IN
38536	1126	2	a	a	DT
38536	1126	3	G.	G.	NNP
38536	1126	4	P.	P.	NNP
38536	1126	5	,	,	,
38536	1126	6	find	find	VB
38536	1126	7	the	the	DT
38536	1126	8	sum	sum	NN
38536	1126	9	to	to	IN
38536	1126	10	infinity	infinity	NN
38536	1126	11	.	.	.
38536	1127	1	11	11	CD
38536	1127	2	.	.	.
38536	1128	1	The	the	DT
38536	1128	2	third	third	JJ
38536	1128	3	term	term	NN
38536	1128	4	of	of	IN
38536	1128	5	a	a	DT
38536	1128	6	geometrical	geometrical	JJ
38536	1128	7	progression	progression	NN
38536	1128	8	is	be	VBZ
38536	1128	9	36	36	CD
38536	1128	10	;	;	:
38536	1128	11	the	the	DT
38536	1128	12	6th	6th	JJ
38536	1128	13	term	term	NN
38536	1128	14	is	be	VBZ
38536	1128	15	972	972	CD
38536	1128	16	.	.	.
38536	1129	1	Find	find	VB
38536	1129	2	the	the	DT
38536	1129	3	first	first	JJ
38536	1129	4	and	and	CC
38536	1129	5	second	second	JJ
38536	1129	6	terms	term	NNS
38536	1129	7	.	.	.
38536	1130	1	12	12	CD
38536	1130	2	.	.	.
38536	1131	1	Insert	insert	NN
38536	1131	2	between	between	IN
38536	1131	3	6	6	CD
38536	1131	4	and	and	CC
38536	1131	5	16	16	CD
38536	1131	6	two	two	CD
38536	1131	7	numbers	number	NNS
38536	1131	8	,	,	,
38536	1131	9	such	such	JJ
38536	1131	10	that	that	IN
38536	1131	11	the	the	DT
38536	1131	12	first	first	JJ
38536	1131	13	three	three	CD
38536	1131	14	of	of	IN
38536	1131	15	the	the	DT
38536	1131	16	four	four	CD
38536	1131	17	shall	shall	MD
38536	1131	18	be	be	VB
38536	1131	19	in	in	IN
38536	1131	20	arithmetical	arithmetical	JJ
38536	1131	21	progression	progression	NN
38536	1131	22	,	,	,
38536	1131	23	and	and	CC
38536	1131	24	the	the	DT
38536	1131	25	last	last	JJ
38536	1131	26	three	three	CD
38536	1131	27	in	in	IN
38536	1131	28	geometrical	geometrical	JJ
38536	1131	29	progression	progression	NN
38536	1131	30	.	.	.
38536	1132	1	13	13	CD
38536	1132	2	.	.	.
38536	1133	1	A	a	DT
38536	1133	2	rubber	rubber	NN
38536	1133	3	ball	ball	NN
38536	1133	4	falls	fall	VBZ
38536	1133	5	from	from	IN
38536	1133	6	a	a	DT
38536	1133	7	height	height	NN
38536	1133	8	of	of	IN
38536	1133	9	40	40	CD
38536	1133	10	inches	inch	NNS
38536	1133	11	and	and	CC
38536	1133	12	on	on	IN
38536	1133	13	each	each	DT
38536	1133	14	rebound	rebound	NN
38536	1133	15	rises	rise	VBZ
38536	1133	16	40	40	CD
38536	1133	17	%	%	NN
38536	1133	18	of	of	IN
38536	1133	19	the	the	DT
38536	1133	20	previous	previous	JJ
38536	1133	21	height	height	NN
38536	1133	22	.	.	.
38536	1134	1	Find	find	VB
38536	1134	2	by	by	IN
38536	1134	3	formula	formula	NN
38536	1134	4	how	how	WRB
38536	1134	5	far	far	RB
38536	1134	6	it	-PRON-	PRP
38536	1134	7	falls	fall	VBZ
38536	1134	8	on	on	IN
38536	1134	9	its	-PRON-	PRP$
38536	1134	10	eighth	eighth	JJ
38536	1134	11	descent	descent	NN
38536	1134	12	.	.	.
38536	1135	1	(	(	-LRB-
38536	1135	2	_	_	NNP
38536	1135	3	Yale	Yale	NNP
38536	1135	4	.	.	.
38536	1135	5	_	_	NNP
38536	1135	6	)	)	-RRB-
38536	1135	7	~Reference:~	~Reference:~	NNP
38536	1135	8	The	the	DT
38536	1135	9	chapter	chapter	NN
38536	1135	10	on	on	IN
38536	1135	11	Geometrical	Geometrical	NNP
38536	1135	12	Progression	Progression	NNP
38536	1135	13	in	in	IN
38536	1135	14	any	any	DT
38536	1135	15	algebra	algebra	NN
38536	1135	16	.	.	.
38536	1136	1	~THE	~the	DT
38536	1136	2	BINOMIAL	BINOMIAL	NNP
38536	1136	3	THEOREM~	THEOREM~	NNP
38536	1136	4	1	1	CD
38536	1136	5	.	.	.
38536	1136	6	Review	review	VB
38536	1136	7	the	the	DT
38536	1136	8	Binomial	Binomial	NNP
38536	1136	9	Theorem	Theorem	NNP
38536	1136	10	laws	law	NNS
38536	1136	11	.	.	.
38536	1137	1	(	(	-LRB-
38536	1137	2	See	see	VB
38536	1137	3	Involution	involution	NN
38536	1137	4	.	.	.
38536	1137	5	)	)	-RRB-
38536	1138	1	Expand	expand	NN
38536	1138	2	:	:	:
38536	1138	3	2	2	CD
38536	1138	4	.	.	.
38536	1139	1	(	(	-LRB-
38536	1139	2	b	b	NN
38536	1139	3	-	-	HYPH
38536	1139	4	n)^7	n)^7	NNP
38536	1139	5	.	.	.
38536	1140	1	3	3	LS
38536	1140	2	.	.	.
38536	1141	1	(	(	-LRB-
38536	1141	2	x	x	SYM
38536	1141	3	+	+	NNP
38536	1141	4	x^(-1))^5	x^(-1))^5	NN
38536	1141	5	.	.	.
38536	1142	1	4	4	LS
38536	1142	2	.	.	.
38536	1143	1	[	[	-LRB-
38536	1143	2	a	a	DT
38536	1143	3	/	/	SYM
38536	1143	4	x	x	NN
38536	1143	5	-	-	HYPH
38536	1143	6	x	x	NN
38536	1143	7	/	/	SYM
38536	1143	8	a]^6	a]^6	NNP
38536	1143	9	.	.	.
38536	1144	1	5	5	CD
38536	1144	2	.	.	.
38536	1145	1	[	[	-LRB-
38536	1145	2	x/2y	x/2y	NNP
38536	1145	3	-	-	:
38536	1145	4	[	[	-LRB-
38536	1145	5	xy]^(1/2)]^5	xy]^(1/2)]^5	NNP
38536	1145	6	.	.	.
38536	1146	1	6	6	CD
38536	1146	2	.	.	.
38536	1147	1	(	(	-LRB-
38536	1147	2	x^2	x^2	NNP
38536	1147	3	-	-	HYPH
38536	1147	4	x	x	NNP
38536	1147	5	+	+	CD
38536	1147	6	2)^3	2)^3	CD
38536	1147	7	.	.	.
38536	1148	1	7	7	LS
38536	1148	2	.	.	.
38536	1149	1	[	[	-LRB-
38536	1149	2	(	(	-LRB-
38536	1149	3	2[b^2]^(1/3))/(y	2[b^2]^(1/3))/(y	NN
38536	1149	4	)	)	-RRB-
38536	1149	5	+	+	NFP
38536	1149	6	(	(	-LRB-
38536	1149	7	3[y^(1/2)])/(b^3)]^4	3[y^(1/2)])/(b^3)]^4	CD
38536	1149	8	.	.	.
38536	1150	1	8	8	LS
38536	1150	2	.	.	.
38536	1151	1	(	(	-LRB-
38536	1151	2	a	a	DT
38536	1151	3	+	+	SYM
38536	1151	4	b)^n	b)^n	NNP
38536	1151	5	=	=	NFP
38536	1151	6	a^n	a^n	NN
38536	1151	7	+	+	CC
38536	1151	8	na^(n	na^(n	NNP
38536	1151	9	-	-	HYPH
38536	1151	10	1)b	1)b	CD
38536	1151	11	+	+	CC
38536	1151	12	[	[	-LRB-
38536	1151	13	n(n	n(n	NN
38536	1151	14	-	-	HYPH
38536	1151	15	1)]/(1·2	1)]/(1·2	CD
38536	1151	16	)	)	-RRB-
38536	1151	17	a^(n	a^(n	CD
38536	1151	18	-	-	HYPH
38536	1151	19	2)b^2	2)b^2	CD
38536	1151	20	+	+	CC
38536	1151	21	[	[	-LRB-
38536	1151	22	n(n	n(n	NN
38536	1151	23	-	-	HYPH
38536	1151	24	1)(n	1)(n	CD
38536	1151	25	-	-	HYPH
38536	1151	26	2)]/(1·2·3	2)]/(1·2·3	CD
38536	1151	27	)	)	-RRB-
38536	1151	28	a^(n	a^(n	CD
38536	1151	29	-	-	HYPH
38536	1151	30	3)b^3	3)b^3	CD
38536	1151	31	+	+	CC
38536	1151	32	[	[	-LRB-
38536	1151	33	n(n	n(n	NN
38536	1151	34	-	-	HYPH
38536	1151	35	1)(n	1)(n	CD
38536	1151	36	-	-	HYPH
38536	1151	37	2)(n	2)(n	CD
38536	1151	38	-	-	HYPH
38536	1151	39	3)]/(1·2·3·4	3)]/(1·2·3·4	CD
38536	1151	40	)	)	-RRB-
38536	1151	41	a^(n	a^(n	CD
38536	1151	42	-	-	SYM
38536	1151	43	4	4	CD
38536	1151	44	)	)	-RRB-
38536	1151	45	b^4	b^4	NNS
38536	1151	46	+	+	SYM
38536	1151	47	·	·	NFP
38536	1151	48	·	·	NFP
38536	1151	49	·	·	NFP
38536	1151	50	.	.	.
38536	1152	1	Show	show	VB
38536	1152	2	by	by	IN
38536	1152	3	observation	observation	NN
38536	1152	4	that	that	IN
38536	1152	5	the	the	DT
38536	1152	6	formula	formula	NN
38536	1152	7	for	for	IN
38536	1152	8	the	the	DT
38536	1152	9	(	(	-LRB-
38536	1152	10	r	r	NN
38536	1152	11	+	+	SYM
38536	1152	12	1)th	1)th	CD
38536	1152	13	term	term	NN
38536	1152	14	=	=	NFP
38536	1152	15	[	[	-LRB-
38536	1152	16	n(n	n(n	NN
38536	1152	17	-	-	HYPH
38536	1152	18	1)(n	1)(n	CD
38536	1152	19	-	-	HYPH
38536	1152	20	2)···(n	2)···(n	CD
38536	1152	21	-	-	HYPH
38536	1152	22	r	r	NN
38536	1152	23	+	+	CC
38536	1152	24	1)]/[1·2·3·4	1)]/[1·2·3·4	CD
38536	1152	25	·	·	NFP
38536	1152	26	·	·	NFP
38536	1152	27	·	·	NFP
38536	1152	28	r	r	LS
38536	1152	29	]	]	-RRB-
38536	1152	30	a^(n	a^(n	CD
38536	1152	31	-	-	HYPH
38536	1152	32	r)b^r	r)b^r	NN
38536	1152	33	.	.	.
38536	1153	1	9	9	CD
38536	1153	2	.	.	.
38536	1154	1	Indicate	indicate	VB
38536	1154	2	what	what	WP
38536	1154	3	the	the	DT
38536	1154	4	97th	97th	JJ
38536	1154	5	term	term	NN
38536	1154	6	of	of	IN
38536	1154	7	(	(	-LRB-
38536	1154	8	a	a	DT
38536	1154	9	+	+	SYM
38536	1154	10	b)^n	b)^n	NNP
38536	1154	11	would	would	MD
38536	1154	12	be	be	VB
38536	1154	13	.	.	.
38536	1155	1	10	10	CD
38536	1155	2	.	.	.
38536	1156	1	Using	use	VBG
38536	1156	2	the	the	DT
38536	1156	3	expansion	expansion	NN
38536	1156	4	of	of	IN
38536	1156	5	(	(	-LRB-
38536	1156	6	a	a	DT
38536	1156	7	+	+	SYM
38536	1156	8	b)^n	b)^n	NN
38536	1156	9	in	in	IN
38536	1156	10	(	(	-LRB-
38536	1156	11	8)	8)	CD
38536	1156	12	,	,	,
38536	1156	13	derive	derive	VB
38536	1156	14	a	a	DT
38536	1156	15	formula	formula	NN
38536	1156	16	for	for	IN
38536	1156	17	the	the	DT
38536	1156	18	rth	rth	NN
38536	1156	19	term	term	NN
38536	1156	20	by	by	IN
38536	1156	21	observing	observe	VBG
38536	1156	22	how	how	WRB
38536	1156	23	each	each	DT
38536	1156	24	term	term	NN
38536	1156	25	is	be	VBZ
38536	1156	26	made	make	VBN
38536	1156	27	up	up	RP
38536	1156	28	,	,	,
38536	1156	29	then	then	RB
38536	1156	30	generalizing	generalize	VBG
38536	1156	31	.	.	.
38536	1157	1	Using	use	VBG
38536	1157	2	either	either	CC
38536	1157	3	the	the	DT
38536	1157	4	formula	formula	NN
38536	1157	5	in	in	IN
38536	1157	6	(	(	-LRB-
38536	1157	7	8)	8)	CD
38536	1157	8	or	or	CC
38536	1157	9	(	(	-LRB-
38536	1157	10	10	10	CD
38536	1157	11	)	)	-RRB-
38536	1157	12	,	,	,
38536	1157	13	whichever	whichever	WDT
38536	1157	14	you	-PRON-	PRP
38536	1157	15	are	be	VBP
38536	1157	16	familiar	familiar	JJ
38536	1157	17	with	with	IN
38536	1157	18	,	,	,
38536	1157	19	find	find	VB
38536	1157	20	:	:	:
38536	1157	21	11	11	CD
38536	1157	22	.	.	.
38536	1158	1	The	the	DT
38536	1158	2	4th	4th	JJ
38536	1158	3	term	term	NN
38536	1158	4	of	of	IN
38536	1158	5	[	[	-LRB-
38536	1158	6	a	a	DT
38536	1158	7	+	+	SYM
38536	1158	8	1	1	CD
38536	1158	9	/	/	SYM
38536	1158	10	a]^(30	a]^(30	NN
38536	1158	11	)	)	-RRB-
38536	1158	12	.	.	.
38536	1159	1	12	12	CD
38536	1159	2	.	.	.
38536	1160	1	The	the	DT
38536	1160	2	8th	8th	JJ
38536	1160	3	term	term	NN
38536	1160	4	of	of	IN
38536	1160	5	(	(	-LRB-
38536	1160	6	1	1	CD
38536	1160	7	+	+	SYM
38536	1160	8	x[y^(1/2)])^(13	x[y^(1/2)])^(13	CD
38536	1160	9	)	)	-RRB-
38536	1160	10	.	.	.
38536	1161	1	13	13	CD
38536	1161	2	.	.	.
38536	1162	1	The	the	DT
38536	1162	2	middle	middle	JJ
38536	1162	3	term	term	NN
38536	1162	4	of	of	IN
38536	1162	5	(	(	-LRB-
38536	1162	6	2a^(3/4	2a^(3/4	CD
38536	1162	7	)	)	-RRB-
38536	1162	8	-	-	HYPH
38536	1162	9	y[a^(1/3)])^(10	y[a^(1/3)])^(10	CD
38536	1162	10	)	)	-RRB-
38536	1162	11	.	.	.
38536	1163	1	14	14	CD
38536	1163	2	.	.	.
38536	1164	1	The	the	DT
38536	1164	2	term	term	NN
38536	1164	3	not	not	RB
38536	1164	4	containing	contain	VBG
38536	1164	5	x	x	NNS
38536	1164	6	in	in	IN
38536	1164	7	[	[	-LRB-
38536	1164	8	x^3	x^3	NNP
38536	1164	9	-	-	HYPH
38536	1164	10	2	2	NNP
38536	1164	11	/	/	SYM
38536	1164	12	x]^(12	x]^(12	NN
38536	1164	13	)	)	-RRB-
38536	1164	14	.	.	.
38536	1165	1	15	15	CD
38536	1165	2	.	.	.
38536	1166	1	The	the	DT
38536	1166	2	term	term	NN
38536	1166	3	containing	contain	VBG
38536	1166	4	x^(18	x^(18	.
38536	1166	5	)	)	-RRB-
38536	1166	6	in	in	IN
38536	1166	7	[	[	-LRB-
38536	1166	8	x^2	x^2	NNP
38536	1166	9	-	-	HYPH
38536	1166	10	a	a	NNP
38536	1166	11	/	/	SYM
38536	1166	12	x]^(15	x]^(15	NN
38536	1166	13	)	)	-RRB-
38536	1166	14	.	.	.
38536	1167	1	~Reference:~	~reference:~	VB
38536	1167	2	The	the	DT
38536	1167	3	chapter	chapter	NN
38536	1167	4	on	on	IN
38536	1167	5	The	the	DT
38536	1167	6	Binomial	Binomial	NNP
38536	1167	7	Theorem	Theorem	NNP
38536	1167	8	in	in	IN
38536	1167	9	any	any	DT
38536	1167	10	algebra	algebra	NN
38536	1167	11	.	.	.
38536	1168	1	~MISCELLANEOUS	~MISCELLANEOUS	NFP
38536	1168	2	EXAMPLES	EXAMPLES	NNP
38536	1168	3	,	,	,
38536	1168	4	QUADRATICS	QUADRATICS	NNP
38536	1168	5	AND	and	CC
38536	1168	6	BEYOND~	BEYOND~	NNP
38536	1168	7	1	1	CD
38536	1168	8	.	.	.
38536	1169	1	Solve	solve	VB
38536	1169	2	the	the	DT
38536	1169	3	equation	equation	NN
38536	1169	4	x^2	x^2	JJ
38536	1169	5	-	-	HYPH
38536	1169	6	1.6x	1.6x	CD
38536	1169	7	-	-	HYPH
38536	1169	8	.23	.23	CD
38536	1169	9	=	=	SYM
38536	1169	10	0	0	CD
38536	1169	11	,	,	,
38536	1169	12	obtaining	obtain	VBG
38536	1169	13	the	the	DT
38536	1169	14	values	value	NNS
38536	1169	15	of	of	IN
38536	1169	16	the	the	DT
38536	1169	17	roots	root	NNS
38536	1169	18	correct	correct	JJ
38536	1169	19	to	to	IN
38536	1169	20	three	three	CD
38536	1169	21	significant	significant	JJ
38536	1169	22	figures	figure	NNS
38536	1169	23	.	.	.
38536	1170	1	(	(	-LRB-
38536	1170	2	_	_	NNP
38536	1170	3	Harvard	Harvard	NNP
38536	1170	4	.	.	.
38536	1170	5	_	_	NNP
38536	1170	6	)	)	-RRB-
38536	1170	7	2	2	CD
38536	1170	8	.	.	.
38536	1171	1	Write	write	VB
38536	1171	2	the	the	DT
38536	1171	3	roots	root	NNS
38536	1171	4	of	of	IN
38536	1171	5	(	(	-LRB-
38536	1171	6	x^2	x^2	NNP
38536	1171	7	+	+	SYM
38536	1171	8	2x)(x^2	2x)(x^2	CD
38536	1171	9	-	-	HYPH
38536	1171	10	2x	2x	CD
38536	1171	11	-	-	HYPH
38536	1171	12	3)(x^2	3)(x^2	CD
38536	1171	13	-	-	HYPH
38536	1171	14	x	x	NN
38536	1171	15	+	+	SYM
38536	1171	16	1	1	CD
38536	1171	17	)	)	-RRB-
38536	1171	18	=	=	SYM
38536	1171	19	0	0	NFP
38536	1171	20	.	.	.
38536	1172	1	(	(	-LRB-
38536	1172	2	_	_	NNP
38536	1172	3	Sheffield	Sheffield	NNP
38536	1172	4	Scientific	Scientific	NNP
38536	1172	5	School	School	NNP
38536	1172	6	.	.	.
38536	1172	7	_	_	NNP
38536	1172	8	)	)	-RRB-
38536	1172	9	3	3	CD
38536	1172	10	.	.	.
38536	1173	1	Solve	solve	VB
38536	1173	2	2[2x	2[2x	CD
38536	1173	3	+	+	SYM
38536	1173	4	2]^(1/2	2]^(1/2	CD
38536	1173	5	)	)	-RRB-
38536	1173	6	+	+	CC
38536	1173	7	[	[	-LRB-
38536	1173	8	2x	2x	CD
38536	1173	9	+	+	SYM
38536	1173	10	1]^(1/2	1]^(1/2	NN
38536	1173	11	)	)	-RRB-
38536	1173	12	=	=	NFP
38536	1173	13	(	(	-LRB-
38536	1173	14	12x	12x	NNS
38536	1173	15	+	+	SYM
38536	1173	16	4)/([8x	4)/([8x	CD
38536	1173	17	+	+	CC
38536	1173	18	8]^{1/2	8]^{1/2	NN
38536	1173	19	}	}	-RRB-
38536	1173	20	)	)	-RRB-
38536	1173	21	.	.	.
38536	1174	1	(	(	-LRB-
38536	1174	2	_	_	NNP
38536	1174	3	Yale	Yale	NNP
38536	1174	4	.	.	.
38536	1174	5	_	_	NNP
38536	1174	6	)	)	-RRB-
38536	1174	7	4	4	CD
38536	1174	8	.	.	.
38536	1175	1	Solve	solve	VB
38536	1175	2	the	the	DT
38536	1175	3	equation	equation	NN
38536	1175	4	V	v	NN
38536	1175	5	=	=	NFP
38536	1175	6	(	(	-LRB-
38536	1175	7	H/3)(B	H/3)(B	NNS
38536	1175	8	+	+	SYM
38536	1175	9	x	x	SYM
38536	1175	10	+	+	SYM
38536	1175	11	[	[	-LRB-
38536	1175	12	Bx]^(1/2	Bx]^(1/2	NNP
38536	1175	13	)	)	-RRB-
38536	1175	14	)	)	-RRB-
38536	1175	15	for	for	IN
38536	1175	16	x	x	NNS
38536	1175	17	,	,	,
38536	1175	18	taking	take	VBG
38536	1175	19	H	h	NN
38536	1175	20	=	=	SYM
38536	1175	21	6	6	CD
38536	1175	22	,	,	,
38536	1175	23	B	b	NN
38536	1175	24	=	=	SYM
38536	1175	25	8	8	CD
38536	1175	26	,	,	,
38536	1175	27	and	and	CC
38536	1175	28	V	v	NN
38536	1175	29	=	=	SYM
38536	1175	30	28	28	CD
38536	1175	31	;	;	:
38536	1175	32	and	and	CC
38536	1175	33	verify	verify	VB
38536	1175	34	your	-PRON-	PRP$
38536	1175	35	result	result	NN
38536	1175	36	.	.	.
38536	1176	1	(	(	-LRB-
38536	1176	2	_	_	NNP
38536	1176	3	Harvard	Harvard	NNP
38536	1176	4	.	.	.
38536	1176	5	_	_	NNP
38536	1176	6	)	)	-RRB-
38536	1176	7	5	5	CD
38536	1176	8	.	.	.
38536	1177	1	Solve	solve	VB
38536	1177	2	{	{	-LRB-
38536	1177	3	x	x	NN
38536	1177	4	:	:	:
38536	1177	5	y	y	NNP
38536	1177	6	=	=	SYM
38536	1177	7	2	2	CD
38536	1177	8	:	:	SYM
38536	1177	9	3	3	CD
38536	1177	10	,	,	,
38536	1177	11	{	{	-LRB-
38536	1177	12	x^2	x^2	XX
38536	1177	13	+	+	SYM
38536	1177	14	y^2	y^2	NNS
38536	1177	15	=	=	SYM
38536	1177	16	5(x	5(x	CD
38536	1177	17	+	+	SYM
38536	1177	18	y	y	NN
38536	1177	19	)	)	-RRB-
38536	1177	20	+	+	CC
38536	1177	21	2	2	CD
38536	1177	22	.	.	.
38536	1178	1	6	6	CD
38536	1178	2	.	.	.
38536	1179	1	Solve	solve	VB
38536	1179	2	2x^2	2x^2	CD
38536	1179	3	-	-	HYPH
38536	1179	4	4x	4x	NNS
38536	1179	5	+	+	SYM
38536	1179	6	3[x^2	3[x^2	NNP
38536	1179	7	-	-	HYPH
38536	1179	8	2x	2x	CD
38536	1179	9	+	+	SYM
38536	1179	10	6]^(1/2	6]^(1/2	NN
38536	1179	11	)	)	-RRB-
38536	1179	12	=	=	SYM
38536	1179	13	15	15	CD
38536	1179	14	.	.	.
38536	1180	1	(	(	-LRB-
38536	1180	2	_	_	NNP
38536	1180	3	Coll	Coll	NNP
38536	1180	4	.	.	.
38536	1181	1	Ent	ent	NN
38536	1181	2	.	.	.
38536	1182	1	Board	Board	NNP
38536	1182	2	.	.	.
38536	1182	3	_	_	NNP
38536	1182	4	)	)	-RRB-
38536	1182	5	7	7	CD
38536	1182	6	.	.	.
38536	1183	1	Find	find	VB
38536	1183	2	all	all	DT
38536	1183	3	values	value	NNS
38536	1183	4	of	of	IN
38536	1183	5	x	x	NNS
38536	1183	6	and	and	CC
38536	1183	7	y	y	NNP
38536	1183	8	which	which	WDT
38536	1183	9	satisfy	satisfy	VBP
38536	1183	10	the	the	DT
38536	1183	11	equations	equation	NNS
38536	1183	12	:	:	:
38536	1183	13	{	{	-LRB-
38536	1183	14	x^(1/2	x^(1/2	NNP
38536	1183	15	)	)	-RRB-
38536	1183	16	+	+	SYM
38536	1183	17	y^(1/2	y^(1/2	NN
38536	1183	18	)	)	-RRB-
38536	1183	19	=	=	SYM
38536	1183	20	4	4	CD
38536	1183	21	,	,	,
38536	1183	22	{	{	-LRB-
38536	1183	23	1/[[x	1/[[x	CD
38536	1183	24	+	+	CD
38536	1183	25	1]^(1/2	1]^(1/2	CD
38536	1183	26	)	)	-RRB-
38536	1183	27	-	-	:
38536	1183	28	x^(1/2	x^(1/2	NNP
38536	1183	29	)	)	-RRB-
38536	1183	30	]	]	-RRB-
38536	1183	31	-	-	:
38536	1183	32	1/[[x	1/[[x	CD
38536	1183	33	+	+	CD
38536	1183	34	1]^(1/2	1]^(1/2	NN
38536	1183	35	)	)	-RRB-
38536	1183	36	+	+	NFP
38536	1183	37	x^(1/2	x^(1/2	NNP
38536	1183	38	)	)	-RRB-
38536	1183	39	]	]	-RRB-
38536	1183	40	=	=	NFP
38536	1183	41	y.	y.	NNP
38536	1184	1	(	(	-LRB-
38536	1184	2	_	_	NNP
38536	1184	3	Mass	Mass	NNP
38536	1184	4	.	.	.
38536	1185	1	Inst	inst	RB
38536	1185	2	.	.	.
38536	1186	1	of	of	IN
38536	1186	2	Technology	Technology	NNP
38536	1186	3	.	.	.
38536	1186	4	_	_	NNP
38536	1186	5	)	)	-RRB-
38536	1186	6	8	8	CD
38536	1186	7	.	.	.
38536	1187	1	If	if	IN
38536	1187	2	[	[	-LRB-
38536	1187	3	alpha	alpha	NN
38536	1187	4	]	]	-RRB-
38536	1187	5	and	and	CC
38536	1187	6	[	[	-LRB-
38536	1187	7	beta	beta	NN
38536	1187	8	]	]	-RRB-
38536	1187	9	represent	represent	VB
38536	1187	10	the	the	DT
38536	1187	11	roots	root	NNS
38536	1187	12	of	of	IN
38536	1187	13	px^2	px^2	NN
38536	1187	14	+	+	CC
38536	1187	15	qx	qx	NN
38536	1187	16	+	+	NNS
38536	1187	17	r	r	NN
38536	1187	18	=	=	SYM
38536	1187	19	0	0	CD
38536	1187	20	,	,	,
38536	1187	21	find	find	VB
38536	1187	22	[	[	-LRB-
38536	1187	23	alpha	alpha	NN
38536	1187	24	]	]	-RRB-
38536	1187	25	+	+	CC
38536	1187	26	[	[	-LRB-
38536	1187	27	beta	beta	NN
38536	1187	28	]	]	-RRB-
38536	1187	29	,	,	,
38536	1187	30	[	[	-LRB-
38536	1187	31	alpha	alpha	NN
38536	1187	32	]	]	-RRB-
38536	1187	33	-	-	,
38536	1187	34	[	[	-LRB-
38536	1187	35	beta	beta	NN
38536	1187	36	]	]	-RRB-
38536	1187	37	,	,	,
38536	1187	38	and	and	CC
38536	1187	39	[	[	-LRB-
38536	1187	40	alpha][beta	alpha][beta	NNP
38536	1187	41	]	]	-RRB-
38536	1187	42	in	in	IN
38536	1187	43	terms	term	NNS
38536	1187	44	of	of	IN
38536	1187	45	p	p	NN
38536	1187	46	,	,	,
38536	1187	47	q	q	NNP
38536	1187	48	,	,	,
38536	1187	49	and	and	CC
38536	1187	50	r.	r.	NNP
38536	1187	51	(	(	-LRB-
38536	1187	52	_	_	NNP
38536	1187	53	Princeton	Princeton	NNP
38536	1187	54	.	.	.
38536	1187	55	_	_	NNP
38536	1187	56	)	)	-RRB-
38536	1187	57	9	9	CD
38536	1187	58	.	.	.
38536	1188	1	Form	form	VB
38536	1188	2	the	the	DT
38536	1188	3	equation	equation	NN
38536	1188	4	whose	whose	WP$
38536	1188	5	roots	root	NNS
38536	1188	6	are	be	VBP
38536	1188	7	2	2	CD
38536	1188	8	+	+	SYM
38536	1188	9	[	[	-LRB-
38536	1188	10	3]^(1/2	3]^(1/2	NN
38536	1188	11	)	)	-RRB-
38536	1188	12	and	and	CC
38536	1188	13	2	2	CD
38536	1188	14	-	-	HYPH
38536	1188	15	[	[	-LRB-
38536	1188	16	-3]^(1/2	-3]^(1/2	NN
38536	1188	17	)	)	-RRB-
38536	1188	18	.	.	.
38536	1189	1	10	10	CD
38536	1189	2	.	.	.
38536	1190	1	Determine	Determine	NNP
38536	1190	2	,	,	,
38536	1190	3	without	without	IN
38536	1190	4	solving	solve	VBG
38536	1190	5	,	,	,
38536	1190	6	the	the	DT
38536	1190	7	character	character	NN
38536	1190	8	of	of	IN
38536	1190	9	the	the	DT
38536	1190	10	roots	root	NNS
38536	1190	11	of	of	IN
38536	1190	12	9x^2	9x^2	CD
38536	1190	13	-	-	HYPH
38536	1190	14	24x	24x	NNS
38536	1190	15	+	+	SYM
38536	1190	16	16	16	CD
38536	1190	17	=	=	SYM
38536	1190	18	0	0	CD
38536	1190	19	.	.	.
38536	1191	1	(	(	-LRB-
38536	1191	2	_	_	NNP
38536	1191	3	College	College	NNP
38536	1191	4	Entrance	Entrance	NNP
38536	1191	5	Board	Board	NNP
38536	1191	6	.	.	.
38536	1191	7	_	_	NNP
38536	1191	8	)	)	-RRB-
38536	1191	9	11	11	CD
38536	1191	10	.	.	.
38536	1192	1	If	if	IN
38536	1192	2	a	a	DT
38536	1192	3	:	:	:
38536	1192	4	b	b	NN
38536	1192	5	=	=	SYM
38536	1192	6	c	c	NN
38536	1192	7	:	:	:
38536	1192	8	d	d	NN
38536	1192	9	,	,	,
38536	1192	10	prove	prove	VBP
38536	1192	11	that	that	IN
38536	1192	12	a	a	NN
38536	1192	13	+	+	SYM
38536	1192	14	b	b	NN
38536	1192	15	:	:	:
38536	1192	16	c	c	NN
38536	1192	17	+	+	SYM
38536	1192	18	d	d	NN
38536	1192	19	=	=	SYM
38536	1192	20	[	[	-LRB-
38536	1192	21	a^2	a^2	CD
38536	1192	22	+	+	SYM
38536	1192	23	b^2]^(1/2	b^2]^(1/2	CD
38536	1192	24	)	)	-RRB-
38536	1192	25	:	:	:
38536	1192	26	[	[	-LRB-
38536	1192	27	c^2	c^2	NNP
38536	1192	28	+	+	SYM
38536	1192	29	d^2]^(1/2	d^2]^(1/2	NNP
38536	1192	30	)	)	-RRB-
38536	1192	31	.	.	.
38536	1193	1	(	(	-LRB-
38536	1193	2	_	_	NNP
38536	1193	3	College	College	NNP
38536	1193	4	Entrance	Entrance	NNP
38536	1193	5	Board	Board	NNP
38536	1193	6	.	.	.
38536	1193	7	_	_	NNP
38536	1193	8	)	)	-RRB-
38536	1193	9	12	12	CD
38536	1193	10	.	.	.
38536	1194	1	Given	give	VBN
38536	1194	2	a	a	DT
38536	1194	3	:	:	:
38536	1194	4	b	b	NN
38536	1194	5	=	=	SYM
38536	1194	6	c	c	NN
38536	1194	7	:	:	:
38536	1194	8	d.	d.	NNP
38536	1194	9	Prove	prove	VB
38536	1194	10	that	that	IN
38536	1194	11	a^2	a^2	CD
38536	1194	12	+	+	SYM
38536	1194	13	b^2	b^2	NNS
38536	1194	14	:	:	:
38536	1194	15	(	(	-LRB-
38536	1194	16	a^3)/(a	a^3)/(a	NNP
38536	1194	17	+	+	SYM
38536	1194	18	b	b	LS
38536	1194	19	)	)	-RRB-
38536	1194	20	=	=	NFP
38536	1194	21	c^2	c^2	NNS
38536	1194	22	+	+	SYM
38536	1194	23	d^2	d^2	NNP
38536	1194	24	:	:	:
38536	1194	25	(	(	-LRB-
38536	1194	26	c^3)/(c	c^3)/(c	NNP
38536	1194	27	+	+	SYM
38536	1194	28	d	d	NN
38536	1194	29	)	)	-RRB-
38536	1194	30	.	.	.
38536	1195	1	(	(	-LRB-
38536	1195	2	_	_	NNP
38536	1195	3	Sheffield	Sheffield	NNP
38536	1195	4	.	.	.
38536	1195	5	_	_	NNP
38536	1195	6	)	)	-RRB-
38536	1195	7	13	13	CD
38536	1195	8	.	.	.
38536	1196	1	The	the	DT
38536	1196	2	9th	9th	JJ
38536	1196	3	term	term	NN
38536	1196	4	of	of	IN
38536	1196	5	an	an	DT
38536	1196	6	arithmetical	arithmetical	JJ
38536	1196	7	progression	progression	NN
38536	1196	8	is	be	VBZ
38536	1196	9	1/6	1/6	CD
38536	1196	10	;	;	:
38536	1196	11	the	the	DT
38536	1196	12	16th	16th	JJ
38536	1196	13	term	term	NN
38536	1196	14	is	be	VBZ
38536	1196	15	5/2	5/2	CD
38536	1196	16	.	.	.
38536	1197	1	Find	find	VB
38536	1197	2	the	the	DT
38536	1197	3	first	first	JJ
38536	1197	4	term	term	NN
38536	1197	5	.	.	.
38536	1198	1	(	(	-LRB-
38536	1198	2	_	_	NNP
38536	1198	3	Regents	Regents	NNP
38536	1198	4	.	.	.
38536	1198	5	_	_	NNP
38536	1198	6	)	)	-RRB-
38536	1198	7	Solve	solve	VB
38536	1198	8	graphically	graphically	RB
38536	1198	9	:	:	:
38536	1198	10	1	1	LS
38536	1198	11	.	.	.
38536	1198	12	x^2	x^2	JJ
38536	1198	13	-	-	HYPH
38536	1198	14	x	x	NNP
38536	1198	15	-	-	CD
38536	1198	16	6	6	CD
38536	1198	17	=	=	SYM
38536	1198	18	0	0	CD
38536	1198	19	.	.	.
38536	1199	1	2	2	LS
38536	1199	2	.	.	.
38536	1199	3	x^2	x^2	NNP
38536	1199	4	+	+	SYM
38536	1199	5	3x	3x	CD
38536	1199	6	-	-	SYM
38536	1199	7	10	10	CD
38536	1199	8	=	=	SYM
38536	1199	9	0	0	CD
38536	1199	10	.	.	.
38536	1200	1	3	3	LS
38536	1200	2	.	.	.
38536	1201	1	Find	find	VB
38536	1201	2	four	four	CD
38536	1201	3	numbers	number	NNS
38536	1201	4	in	in	IN
38536	1201	5	arithmetical	arithmetical	JJ
38536	1201	6	progression	progression	NN
38536	1201	7	,	,	,
38536	1201	8	such	such	JJ
38536	1201	9	that	that	IN
38536	1201	10	the	the	DT
38536	1201	11	sum	sum	NN
38536	1201	12	of	of	IN
38536	1201	13	the	the	DT
38536	1201	14	first	first	JJ
38536	1201	15	two	two	CD
38536	1201	16	is	be	VBZ
38536	1201	17	1	1	CD
38536	1201	18	,	,	,
38536	1201	19	and	and	CC
38536	1201	20	the	the	DT
38536	1201	21	sum	sum	NN
38536	1201	22	of	of	IN
38536	1201	23	the	the	DT
38536	1201	24	last	last	JJ
38536	1201	25	two	two	CD
38536	1201	26	is	be	VBZ
38536	1201	27	-19	-19	NNP
38536	1201	28	.	.	.
38536	1202	1	4	4	LS
38536	1202	2	.	.	.
38536	1203	1	What	what	WDT
38536	1203	2	number	number	NN
38536	1203	3	added	add	VBN
38536	1203	4	to	to	IN
38536	1203	5	2	2	CD
38536	1203	6	,	,	,
38536	1203	7	20	20	CD
38536	1203	8	,	,	,
38536	1203	9	9	9	CD
38536	1203	10	,	,	,
38536	1203	11	34	34	CD
38536	1203	12	,	,	,
38536	1203	13	will	will	MD
38536	1203	14	make	make	VB
38536	1203	15	the	the	DT
38536	1203	16	results	result	NNS
38536	1203	17	proportional	proportional	JJ
38536	1203	18	?	?	.
38536	1204	1	5	5	CD
38536	1204	2	.	.	.
38536	1205	1	Find	find	VB
38536	1205	2	the	the	DT
38536	1205	3	middle	middle	JJ
38536	1205	4	term	term	NN
38536	1205	5	of	of	IN
38536	1205	6	[	[	-LRB-
38536	1205	7	3a^5	3a^5	CD
38536	1205	8	+	+	CC
38536	1205	9	(	(	-LRB-
38536	1205	10	b^(3/4))/(2)]^8	b^(3/4))/(2)]^8	RB
38536	1205	11	.	.	.
38536	1206	1	6	6	CD
38536	1206	2	.	.	.
38536	1207	1	Solve	solve	VB
38536	1207	2	(	(	-LRB-
38536	1207	3	x	x	NNS
38536	1207	4	+	+	SYM
38536	1207	5	1)/(3x	1)/(3x	CD
38536	1207	6	+	+	SYM
38536	1207	7	2	2	CD
38536	1207	8	)	)	-RRB-
38536	1207	9	=	=	NFP
38536	1207	10	(	(	-LRB-
38536	1207	11	2x	2x	JJ
38536	1207	12	-	-	HYPH
38536	1207	13	3)/(3x	3)/(3x	CD
38536	1207	14	-	-	HYPH
38536	1207	15	2	2	CD
38536	1207	16	)	)	-RRB-
38536	1207	17	-	-	HYPH
38536	1207	18	1	1	CD
38536	1207	19	-	-	HYPH
38536	1207	20	36/(4	36/(4	CD
38536	1207	21	-	-	HYPH
38536	1207	22	9x^2	9x^2	CD
38536	1207	23	)	)	-RRB-
38536	1207	24	.	.	.
38536	1208	1	(	(	-LRB-
38536	1208	2	_	_	NNP
38536	1208	3	Princeton	Princeton	NNP
38536	1208	4	.	.	.
38536	1208	5	_	_	NNP
38536	1208	6	)	)	-RRB-
38536	1208	7	7	7	CD
38536	1208	8	.	.	.
38536	1209	1	A	a	DT
38536	1209	2	strip	strip	NN
38536	1209	3	of	of	IN
38536	1209	4	carpet	carpet	NN
38536	1209	5	one	one	CD
38536	1209	6	half	half	NN
38536	1209	7	inch	inch	NN
38536	1209	8	thick	thick	JJ
38536	1209	9	and	and	CC
38536	1209	10	29	29	CD
38536	1209	11	-	-	SYM
38536	1209	12	6/7	6/7	CD
38536	1209	13	feet	foot	NNS
38536	1209	14	long	long	JJ
38536	1209	15	is	be	VBZ
38536	1209	16	rolled	roll	VBN
38536	1209	17	on	on	IN
38536	1209	18	a	a	DT
38536	1209	19	roller	roller	NN
38536	1209	20	four	four	CD
38536	1209	21	inches	inch	NNS
38536	1209	22	in	in	IN
38536	1209	23	diameter	diameter	NN
38536	1209	24	.	.	.
38536	1210	1	Find	find	VB
38536	1210	2	how	how	WRB
38536	1210	3	many	many	JJ
38536	1210	4	turns	turn	NNS
38536	1210	5	there	there	EX
38536	1210	6	will	will	MD
38536	1210	7	be	be	VB
38536	1210	8	,	,	,
38536	1210	9	remembering	remember	VBG
38536	1210	10	that	that	IN
38536	1210	11	each	each	DT
38536	1210	12	turn	turn	NN
38536	1210	13	increases	increase	VBZ
38536	1210	14	the	the	DT
38536	1210	15	diameter	diameter	NN
38536	1210	16	by	by	IN
38536	1210	17	one	one	CD
38536	1210	18	inch	inch	NN
38536	1210	19	,	,	,
38536	1210	20	and	and	CC
38536	1210	21	that	that	IN
38536	1210	22	the	the	DT
38536	1210	23	circumference	circumference	NN
38536	1210	24	of	of	IN
38536	1210	25	a	a	DT
38536	1210	26	circle	circle	NN
38536	1210	27	equals	equal	VBZ
38536	1210	28	(	(	-LRB-
38536	1210	29	approximately	approximately	RB
38536	1210	30	)	)	-RRB-
38536	1210	31	22/7	22/7	CD
38536	1210	32	times	time	NNS
38536	1210	33	the	the	DT
38536	1210	34	diameter	diameter	NN
38536	1210	35	.	.	.
38536	1211	1	(	(	-LRB-
38536	1211	2	_	_	NNP
38536	1211	3	Harvard	Harvard	NNP
38536	1211	4	.	.	.
38536	1211	5	_	_	NNP
38536	1211	6	)	)	-RRB-
38536	1211	7	8	8	CD
38536	1211	8	.	.	.
38536	1212	1	The	the	DT
38536	1212	2	sum	sum	NN
38536	1212	3	of	of	IN
38536	1212	4	the	the	DT
38536	1212	5	first	first	JJ
38536	1212	6	three	three	CD
38536	1212	7	terms	term	NNS
38536	1212	8	of	of	IN
38536	1212	9	a	a	DT
38536	1212	10	geometrical	geometrical	JJ
38536	1212	11	progression	progression	NN
38536	1212	12	is	be	VBZ
38536	1212	13	21	21	CD
38536	1212	14	,	,	,
38536	1212	15	and	and	CC
38536	1212	16	the	the	DT
38536	1212	17	sum	sum	NN
38536	1212	18	of	of	IN
38536	1212	19	their	-PRON-	PRP$
38536	1212	20	squares	square	NNS
38536	1212	21	is	be	VBZ
38536	1212	22	189	189	CD
38536	1212	23	.	.	.
38536	1213	1	What	what	WP
38536	1213	2	is	be	VBZ
38536	1213	3	the	the	DT
38536	1213	4	first	first	JJ
38536	1213	5	term	term	NN
38536	1213	6	?	?	.
38536	1214	1	(	(	-LRB-
38536	1214	2	_	_	NNP
38536	1214	3	Yale	Yale	NNP
38536	1214	4	.	.	.
38536	1214	5	_	_	NNP
38536	1214	6	)	)	-RRB-
38536	1214	7	9	9	CD
38536	1214	8	.	.	.
38536	1215	1	Find	find	VB
38536	1215	2	the	the	DT
38536	1215	3	geometrical	geometrical	JJ
38536	1215	4	progression	progression	NN
38536	1215	5	whose	whose	WP$
38536	1215	6	sum	sum	NN
38536	1215	7	to	to	IN
38536	1215	8	infinity	infinity	NN
38536	1215	9	is	be	VBZ
38536	1215	10	4	4	CD
38536	1215	11	,	,	,
38536	1215	12	and	and	CC
38536	1215	13	whose	whose	WP$
38536	1215	14	second	second	JJ
38536	1215	15	term	term	NN
38536	1215	16	is	be	VBZ
38536	1215	17	3/4	3/4	CD
38536	1215	18	.	.	.
38536	1216	1	10	10	CD
38536	1216	2	.	.	.
38536	1217	1	Solve	solve	VB
38536	1217	2	4x	4x	NNS
38536	1217	3	+	+	SYM
38536	1217	4	4[3x^2	4[3x^2	CD
38536	1217	5	-	-	HYPH
38536	1217	6	7x	7x	CD
38536	1217	7	+	+	CC
38536	1217	8	3]^(1/2	3]^(1/2	NN
38536	1217	9	)	)	-RRB-
38536	1217	10	=	=	NFP
38536	1217	11	3x^2	3x^2	CD
38536	1217	12	-	-	SYM
38536	1217	13	3x	3x	CD
38536	1217	14	+	+	SYM
38536	1217	15	6	6	CD
38536	1217	16	.	.	.
38536	1218	1	11	11	CD
38536	1218	2	.	.	.
38536	1219	1	Solve	solve	VB
38536	1219	2	{	{	-LRB-
38536	1219	3	2x^2	2x^2	CD
38536	1219	4	+	+	CC
38536	1219	5	3xy	3xy	NN
38536	1219	6	-	-	HYPH
38536	1219	7	5y^2	5y^2	CD
38536	1219	8	=	=	SYM
38536	1219	9	4	4	CD
38536	1219	10	,	,	,
38536	1219	11	{	{	-LRB-
38536	1219	12	2xy	2xy	JJ
38536	1219	13	+	+	SYM
38536	1219	14	3y^2	3y^2	CD
38536	1219	15	=	=	SYM
38536	1219	16	-3	-3	:
38536	1219	17	.	.	.
38536	1220	1	12	12	CD
38536	1220	2	.	.	.
38536	1221	1	Two	two	CD
38536	1221	2	hundred	hundred	CD
38536	1221	3	stones	stone	NNS
38536	1221	4	are	be	VBP
38536	1221	5	placed	place	VBN
38536	1221	6	on	on	IN
38536	1221	7	the	the	DT
38536	1221	8	ground	ground	NN
38536	1221	9	3	3	CD
38536	1221	10	feet	foot	NNS
38536	1221	11	apart	apart	RB
38536	1221	12	,	,	,
38536	1221	13	the	the	DT
38536	1221	14	first	first	JJ
38536	1221	15	being	be	VBG
38536	1221	16	3	3	CD
38536	1221	17	feet	foot	NNS
38536	1221	18	from	from	IN
38536	1221	19	a	a	DT
38536	1221	20	basket	basket	NN
38536	1221	21	.	.	.
38536	1222	1	If	if	IN
38536	1222	2	the	the	DT
38536	1222	3	basket	basket	NN
38536	1222	4	and	and	CC
38536	1222	5	all	all	PDT
38536	1222	6	the	the	DT
38536	1222	7	stones	stone	NNS
38536	1222	8	are	be	VBP
38536	1222	9	in	in	IN
38536	1222	10	a	a	DT
38536	1222	11	straight	straight	JJ
38536	1222	12	line	line	NN
38536	1222	13	,	,	,
38536	1222	14	how	how	WRB
38536	1222	15	far	far	RB
38536	1222	16	does	do	VBZ
38536	1222	17	a	a	DT
38536	1222	18	person	person	NN
38536	1222	19	travel	travel	NN
38536	1222	20	who	who	WP
38536	1222	21	starts	start	VBZ
38536	1222	22	from	from	IN
38536	1222	23	the	the	DT
38536	1222	24	basket	basket	NN
38536	1222	25	and	and	CC
38536	1222	26	brings	bring	VBZ
38536	1222	27	the	the	DT
38536	1222	28	stones	stone	NNS
38536	1222	29	to	to	IN
38536	1222	30	it	-PRON-	PRP
38536	1222	31	one	one	CD
38536	1222	32	by	by	IN
38536	1222	33	one	one	CD
38536	1222	34	?	?	.
38536	1223	1	Solve	solve	VB
38536	1223	2	graphically	graphically	RB
38536	1223	3	;	;	:
38536	1223	4	and	and	CC
38536	1223	5	check	check	VB
38536	1223	6	by	by	IN
38536	1223	7	solving	solve	VBG
38536	1223	8	algebraically	algebraically	RB
38536	1223	9	:	:	:
38536	1223	10	1	1	CD
38536	1223	11	.	.	.
38536	1224	1	{	{	-LRB-
38536	1224	2	x^2	x^2	NNP
38536	1224	3	+	+	SYM
38536	1224	4	y^2	y^2	NNS
38536	1224	5	=	=	SYM
38536	1224	6	25	25	CD
38536	1224	7	,	,	,
38536	1224	8	{	{	-LRB-
38536	1224	9	x	x	SYM
38536	1224	10	+	+	SYM
38536	1224	11	y	y	NN
38536	1224	12	=	=	SYM
38536	1224	13	1	1	CD
38536	1224	14	.	.	.
38536	1225	1	2	2	LS
38536	1225	2	.	.	.
38536	1225	3	x^2	x^2	NNP
38536	1225	4	-	-	HYPH
38536	1225	5	3x	3x	CD
38536	1225	6	-	-	HYPH
38536	1225	7	18	18	CD
38536	1225	8	=	=	SYM
38536	1225	9	0	0	CD
38536	1225	10	.	.	.
38536	1226	1	3	3	LS
38536	1226	2	.	.	.
38536	1226	3	x^2	x^2	NNP
38536	1226	4	+	+	SYM
38536	1226	5	3x	3x	CD
38536	1226	6	-	-	SYM
38536	1226	7	10	10	CD
38536	1226	8	=	=	SYM
38536	1226	9	0	0	CD
38536	1226	10	.	.	.
38536	1227	1	Determine	determine	VB
38536	1227	2	the	the	DT
38536	1227	3	value	value	NN
38536	1227	4	of	of	IN
38536	1227	5	m	m	NN
38536	1227	6	for	for	IN
38536	1227	7	which	which	WDT
38536	1227	8	the	the	DT
38536	1227	9	roots	root	NNS
38536	1227	10	of	of	IN
38536	1227	11	the	the	DT
38536	1227	12	equation	equation	NN
38536	1227	13	will	will	MD
38536	1227	14	be	be	VB
38536	1227	15	equal	equal	JJ
38536	1227	16	:	:	:
38536	1227	17	(	(	-LRB-
38536	1227	18	HINT	HINT	NNS
38536	1227	19	:	:	:
38536	1227	20	See	see	VB
38536	1227	21	page	page	NN
38536	1227	22	40	40	CD
38536	1227	23	.	.	.
38536	1228	1	To	to	TO
38536	1228	2	have	have	VB
38536	1228	3	the	the	DT
38536	1228	4	roots	root	NNS
38536	1228	5	equal	equal	JJ
38536	1228	6	,	,	,
38536	1228	7	b^2	b^2	NNS
38536	1228	8	-	-	HYPH
38536	1228	9	4ac	4ac	NN
38536	1228	10	must	must	MD
38536	1228	11	equal	equal	VB
38536	1228	12	0	0	CD
38536	1228	13	.	.	.
38536	1228	14	)	)	-RRB-
38536	1229	1	4	4	LS
38536	1229	2	.	.	.
38536	1230	1	2x^2	2x^2	CD
38536	1230	2	-	-	HYPH
38536	1230	3	mx	mx	NNS
38536	1230	4	+	+	CC
38536	1230	5	12	12	CD
38536	1230	6	-	-	SYM
38536	1230	7	1/2	1/2	CD
38536	1230	8	=	=	SYM
38536	1230	9	0	0	CD
38536	1230	10	.	.	.
38536	1231	1	5	5	CD
38536	1231	2	.	.	.
38536	1232	1	(	(	-LRB-
38536	1232	2	m	m	NNP
38536	1232	3	-	-	HYPH
38536	1232	4	1)x^2	1)x^2	CD
38536	1232	5	+	+	CD
38536	1232	6	mx	mx	NN
38536	1232	7	+	+	CC
38536	1232	8	2	2	CD
38536	1232	9	m	m	CD
38536	1232	10	-	-	HYPH
38536	1232	11	3	3	CD
38536	1232	12	=	=	SYM
38536	1232	13	0	0	CD
38536	1232	14	.	.	.
38536	1233	1	6	6	CD
38536	1233	2	.	.	.
38536	1234	1	If	if	IN
38536	1234	2	2a	2a	CD
38536	1234	3	+	+	SYM
38536	1234	4	3b	3b	NN
38536	1234	5	is	be	VBZ
38536	1234	6	a	a	DT
38536	1234	7	root	root	NN
38536	1234	8	of	of	IN
38536	1234	9	x^2	x^2	JJ
38536	1234	10	-	-	JJ
38536	1234	11	6bx	6bx	JJ
38536	1234	12	-	-	HYPH
38536	1234	13	4a^2	4a^2	CD
38536	1234	14	+	+	SYM
38536	1234	15	9b^2	9b^2	CD
38536	1234	16	=	=	SYM
38536	1234	17	0	0	CD
38536	1234	18	,	,	,
38536	1234	19	find	find	VB
38536	1234	20	the	the	DT
38536	1234	21	other	other	JJ
38536	1234	22	root	root	NN
38536	1234	23	without	without	IN
38536	1234	24	solving	solve	VBG
38536	1234	25	the	the	DT
38536	1234	26	equation	equation	NN
38536	1234	27	.	.	.
38536	1235	1	(	(	-LRB-
38536	1235	2	_	_	NNP
38536	1235	3	Univ	Univ	NNP
38536	1235	4	.	.	.
38536	1236	1	of	of	IN
38536	1236	2	Penn	Penn	NNP
38536	1236	3	.	.	.
38536	1236	4	_	_	NNP
38536	1236	5	)	)	-RRB-
38536	1236	6	7	7	CD
38536	1236	7	.	.	.
38536	1237	1	How	how	WRB
38536	1237	2	many	many	JJ
38536	1237	3	times	time	NNS
38536	1237	4	does	do	VBZ
38536	1237	5	a	a	DT
38536	1237	6	common	common	JJ
38536	1237	7	clock	clock	NN
38536	1237	8	strike	strike	NN
38536	1237	9	in	in	IN
38536	1237	10	12	12	CD
38536	1237	11	hours	hour	NNS
38536	1237	12	?	?	.
38536	1238	1	8	8	LS
38536	1238	2	.	.	.
38536	1239	1	Find	find	VB
38536	1239	2	the	the	DT
38536	1239	3	sum	sum	NN
38536	1239	4	to	to	IN
38536	1239	5	infinity	infinity	NN
38536	1239	6	of	of	IN
38536	1239	7	2/(2^(1/2	2/(2^(1/2	CD
38536	1239	8	)	)	-RRB-
38536	1239	9	)	)	-RRB-
38536	1239	10	,	,	,
38536	1239	11	1/(2^(1/2	1/(2^(1/2	CD
38536	1239	12	)	)	-RRB-
38536	1239	13	)	)	-RRB-
38536	1239	14	,	,	,
38536	1239	15	1/(2[2]^(1/2	1/(2[2]^(1/2	CD
38536	1239	16	)	)	-RRB-
38536	1239	17	)	)	-RRB-
38536	1239	18	,	,	,
38536	1239	19	·	·	NFP
38536	1239	20	·	·	NFP
38536	1239	21	·	·	NFP
38536	1239	22	.	.	.
38536	1240	1	9	9	CD
38536	1240	2	.	.	.
38536	1241	1	Solve	solve	VB
38536	1241	2	[	[	-LRB-
38536	1241	3	x/2	x/2	NNS
38536	1241	4	+	+	SYM
38536	1241	5	6	6	CD
38536	1241	6	/	/	SYM
38536	1241	7	x]^2	x]^2	NNP
38536	1241	8	-	-	HYPH
38536	1241	9	6[x/2	6[x/2	CD
38536	1241	10	+	+	CC
38536	1241	11	6	6	CD
38536	1241	12	/	/	SYM
38536	1241	13	x	x	NN
38536	1241	14	]	]	-RRB-
38536	1241	15	+	+	CC
38536	1241	16	8	8	CD
38536	1241	17	=	=	SYM
38536	1241	18	0	0	CD
38536	1241	19	.	.	.
38536	1242	1	10	10	CD
38536	1242	2	.	.	.
38536	1243	1	Find	find	VB
38536	1243	2	the	the	DT
38536	1243	3	value	value	NN
38536	1243	4	of	of	IN
38536	1243	5	the	the	DT
38536	1243	6	recurring	recur	VBG
38536	1243	7	decimal	decimal	JJ
38536	1243	8	2.214214	2.214214	CD
38536	1243	9	·	·	NFP
38536	1243	10	·	·	NFP
38536	1243	11	·	·	NFP
38536	1243	12	.	.	.
38536	1244	1	11	11	CD
38536	1244	2	.	.	.
38536	1245	1	A	a	DT
38536	1245	2	man	man	NN
38536	1245	3	purchases	purchase	VBZ
38536	1245	4	a	a	DT
38536	1245	5	$	$	$
38536	1245	6	500	500	CD
38536	1245	7	piano	piano	NN
38536	1245	8	by	by	IN
38536	1245	9	paying	pay	VBG
38536	1245	10	monthly	monthly	JJ
38536	1245	11	installments	installment	NNS
38536	1245	12	of	of	IN
38536	1245	13	$	$	$
38536	1245	14	10	10	CD
38536	1245	15	and	and	CC
38536	1245	16	interest	interest	NN
38536	1245	17	on	on	IN
38536	1245	18	the	the	DT
38536	1245	19	debt	debt	NN
38536	1245	20	.	.	.
38536	1246	1	If	if	IN
38536	1246	2	the	the	DT
38536	1246	3	yearly	yearly	JJ
38536	1246	4	rate	rate	NN
38536	1246	5	is	be	VBZ
38536	1246	6	6	6	CD
38536	1246	7	%	%	NN
38536	1246	8	,	,	,
38536	1246	9	what	what	WP
38536	1246	10	is	be	VBZ
38536	1246	11	the	the	DT
38536	1246	12	total	total	JJ
38536	1246	13	amount	amount	NN
38536	1246	14	of	of	IN
38536	1246	15	interest	interest	NN
38536	1246	16	?	?	.
38536	1247	1	12	12	CD
38536	1247	2	.	.	.
38536	1248	1	The	the	DT
38536	1248	2	arithmetical	arithmetical	JJ
38536	1248	3	mean	mean	NN
38536	1248	4	between	between	IN
38536	1248	5	two	two	CD
38536	1248	6	numbers	number	NNS
38536	1248	7	is	be	VBZ
38536	1248	8	42	42	CD
38536	1248	9	-	-	SYM
38536	1248	10	1/2	1/2	CD
38536	1248	11	,	,	,
38536	1248	12	and	and	CC
38536	1248	13	their	-PRON-	PRP$
38536	1248	14	geometrical	geometrical	JJ
38536	1248	15	mean	mean	NN
38536	1248	16	is	be	VBZ
38536	1248	17	42	42	CD
38536	1248	18	.	.	.
38536	1249	1	Find	find	VB
38536	1249	2	the	the	DT
38536	1249	3	numbers	number	NNS
38536	1249	4	.	.	.
38536	1250	1	(	(	-LRB-
38536	1250	2	_	_	NNP
38536	1250	3	College	College	NNP
38536	1250	4	Entrance	Entrance	NNP
38536	1250	5	Exam	Exam	NNP
38536	1250	6	.	.	.
38536	1251	1	Board	Board	NNP
38536	1251	2	.	.	.
38536	1251	3	_	_	NNP
38536	1251	4	)	)	-RRB-
38536	1251	5	13	13	CD
38536	1251	6	.	.	.
38536	1252	1	If	if	IN
38536	1252	2	the	the	DT
38536	1252	3	middle	middle	JJ
38536	1252	4	term	term	NN
38536	1252	5	of	of	IN
38536	1252	6	[	[	-LRB-
38536	1252	7	3x	3x	CD
38536	1252	8	-	-	,
38536	1252	9	(	(	-LRB-
38536	1252	10	1)/(2[x^(1/2)])]^4	1)/(2[x^(1/2)])]^4	CD
38536	1252	11	is	be	VBZ
38536	1252	12	equal	equal	JJ
38536	1252	13	to	to	IN
38536	1252	14	the	the	DT
38536	1252	15	fourth	fourth	JJ
38536	1252	16	term	term	NN
38536	1252	17	of	of	IN
38536	1252	18	[	[	-LRB-
38536	1252	19	2[x^(1/2	2[x^(1/2	NNP
38536	1252	20	)	)	-RRB-
38536	1252	21	]	]	-RRB-
38536	1252	22	+	+	CC
38536	1252	23	1/2x]^7	1/2x]^7	CD
38536	1252	24	,	,	,
38536	1252	25	find	find	VB
38536	1252	26	the	the	DT
38536	1252	27	value	value	NN
38536	1252	28	of	of	IN
38536	1252	29	x.	x.	NNP
38536	1253	1	(	(	-LRB-
38536	1253	2	_	_	NNP
38536	1253	3	M.	M.	NNP
38536	1254	1	I.	I.	NNP
38536	1254	2	T.	T.	NNP
38536	1254	3	_	_	NNP
38536	1254	4	)	)	-RRB-
38536	1254	5	~PROBLEMS~	~PROBLEMS~	NNP
38536	1254	6	~Linear	~Linear	NFP
38536	1254	7	Equations	equation	NNS
38536	1254	8	,	,	,
38536	1254	9	One	one	CD
38536	1254	10	Unknown~	unknown~	NN
38536	1254	11	1	1	CD
38536	1254	12	.	.	.
38536	1255	1	A	a	DT
38536	1255	2	train	train	NN
38536	1255	3	running	run	VBG
38536	1255	4	30	30	CD
38536	1255	5	miles	mile	NNS
38536	1255	6	an	an	DT
38536	1255	7	hour	hour	NN
38536	1255	8	requires	require	VBZ
38536	1255	9	21	21	CD
38536	1255	10	minutes	minute	NNS
38536	1255	11	longer	long	RBR
38536	1255	12	to	to	TO
38536	1255	13	go	go	VB
38536	1255	14	a	a	DT
38536	1255	15	certain	certain	JJ
38536	1255	16	distance	distance	NN
38536	1255	17	than	than	IN
38536	1255	18	does	do	VBZ
38536	1255	19	a	a	DT
38536	1255	20	train	train	NN
38536	1255	21	running	run	VBG
38536	1255	22	36	36	CD
38536	1255	23	miles	mile	NNS
38536	1255	24	an	an	DT
38536	1255	25	hour	hour	NN
38536	1255	26	.	.	.
38536	1256	1	How	how	WRB
38536	1256	2	great	great	JJ
38536	1256	3	is	be	VBZ
38536	1256	4	the	the	DT
38536	1256	5	distance	distance	NN
38536	1256	6	?	?	.
38536	1257	1	(	(	-LRB-
38536	1257	2	_	_	NNP
38536	1257	3	Cornell	Cornell	NNP
38536	1257	4	.	.	.
38536	1257	5	_	_	NNP
38536	1257	6	)	)	-RRB-
38536	1257	7	2	2	CD
38536	1257	8	.	.	.
38536	1258	1	A	a	DT
38536	1258	2	man	man	NN
38536	1258	3	can	can	MD
38536	1258	4	walk	walk	VB
38536	1258	5	2	2	CD
38536	1258	6	-	-	SYM
38536	1258	7	1/2	1/2	CD
38536	1258	8	miles	mile	NNS
38536	1258	9	an	an	DT
38536	1258	10	hour	hour	NN
38536	1258	11	up	up	IN
38536	1258	12	hill	hill	NNP
38536	1258	13	and	and	CC
38536	1258	14	3	3	CD
38536	1258	15	-	-	SYM
38536	1258	16	1/2	1/2	CD
38536	1258	17	miles	mile	NNS
38536	1258	18	an	an	DT
38536	1258	19	hour	hour	NN
38536	1258	20	down	down	IN
38536	1258	21	hill	hill	NNP
38536	1258	22	.	.	.
38536	1259	1	He	-PRON-	PRP
38536	1259	2	walks	walk	VBZ
38536	1259	3	56	56	CD
38536	1259	4	miles	mile	NNS
38536	1259	5	in	in	IN
38536	1259	6	20	20	CD
38536	1259	7	hours	hour	NNS
38536	1259	8	on	on	IN
38536	1259	9	a	a	DT
38536	1259	10	road	road	NN
38536	1259	11	no	no	DT
38536	1259	12	part	part	NN
38536	1259	13	of	of	IN
38536	1259	14	which	which	WDT
38536	1259	15	is	be	VBZ
38536	1259	16	level	level	NN
38536	1259	17	.	.	.
38536	1260	1	How	how	WRB
38536	1260	2	much	much	JJ
38536	1260	3	of	of	IN
38536	1260	4	it	-PRON-	PRP
38536	1260	5	is	be	VBZ
38536	1260	6	up	up	RB
38536	1260	7	hill	hill	NN
38536	1260	8	?	?	.
38536	1261	1	(	(	-LRB-
38536	1261	2	_	_	NNP
38536	1261	3	Yale	Yale	NNP
38536	1261	4	.	.	.
38536	1261	5	_	_	NNP
38536	1261	6	)	)	-RRB-
38536	1261	7	3	3	CD
38536	1261	8	.	.	.
38536	1262	1	A	a	DT
38536	1262	2	physician	physician	NN
38536	1262	3	having	have	VBG
38536	1262	4	100	100	CD
38536	1262	5	cubic	cubic	JJ
38536	1262	6	centimeters	centimeter	NNS
38536	1262	7	of	of	IN
38536	1262	8	a	a	DT
38536	1262	9	6	6	CD
38536	1262	10	%	%	NN
38536	1262	11	solution	solution	NN
38536	1262	12	of	of	IN
38536	1262	13	a	a	DT
38536	1262	14	certain	certain	JJ
38536	1262	15	medicine	medicine	NN
38536	1262	16	wishes	wish	VBZ
38536	1262	17	to	to	TO
38536	1262	18	dilute	dilute	VB
38536	1262	19	it	-PRON-	PRP
38536	1262	20	to	to	IN
38536	1262	21	a	a	DT
38536	1262	22	3	3	CD
38536	1262	23	-	-	SYM
38536	1262	24	1/2	1/2	CD
38536	1262	25	%	%	NN
38536	1262	26	solution	solution	NN
38536	1262	27	.	.	.
38536	1263	1	How	how	WRB
38536	1263	2	much	much	JJ
38536	1263	3	water	water	NN
38536	1263	4	must	must	MD
38536	1263	5	he	-PRON-	PRP
38536	1263	6	add	add	VB
38536	1263	7	?	?	.
38536	1264	1	(	(	-LRB-
38536	1264	2	A	a	DT
38536	1264	3	6	6	CD
38536	1264	4	%	%	NN
38536	1264	5	solution	solution	NN
38536	1264	6	contains	contain	VBZ
38536	1264	7	6	6	CD
38536	1264	8	%	%	NN
38536	1264	9	of	of	IN
38536	1264	10	medicine	medicine	NN
38536	1264	11	and	and	CC
38536	1264	12	94	94	CD
38536	1264	13	%	%	NN
38536	1264	14	of	of	IN
38536	1264	15	water	water	NN
38536	1264	16	.	.	.
38536	1264	17	)	)	-RRB-
38536	1265	1	(	(	-LRB-
38536	1265	2	_	_	NNP
38536	1265	3	Case	Case	NNP
38536	1265	4	.	.	.
38536	1265	5	_	_	NNP
38536	1265	6	)	)	-RRB-
38536	1265	7	4	4	CD
38536	1265	8	.	.	.
38536	1266	1	A	a	DT
38536	1266	2	clerk	clerk	NN
38536	1266	3	earned	earn	VBD
38536	1266	4	$	$	$
38536	1266	5	504	504	CD
38536	1266	6	in	in	IN
38536	1266	7	a	a	DT
38536	1266	8	certain	certain	JJ
38536	1266	9	number	number	NN
38536	1266	10	of	of	IN
38536	1266	11	months	month	NNS
38536	1266	12	.	.	.
38536	1267	1	His	-PRON-	PRP$
38536	1267	2	salary	salary	NN
38536	1267	3	was	be	VBD
38536	1267	4	increased	increase	VBN
38536	1267	5	25	25	CD
38536	1267	6	%	%	NN
38536	1267	7	,	,	,
38536	1267	8	and	and	CC
38536	1267	9	he	-PRON-	PRP
38536	1267	10	then	then	RB
38536	1267	11	earned	earn	VBD
38536	1267	12	$	$	$
38536	1267	13	450	450	CD
38536	1267	14	in	in	IN
38536	1267	15	two	two	CD
38536	1267	16	months	month	NNS
38536	1267	17	less	less	JJR
38536	1267	18	time	time	NN
38536	1267	19	than	than	IN
38536	1267	20	it	-PRON-	PRP
38536	1267	21	had	have	VBD
38536	1267	22	previously	previously	RB
38536	1267	23	taken	take	VBN
38536	1267	24	him	-PRON-	PRP
38536	1267	25	to	to	TO
38536	1267	26	earn	earn	VB
38536	1267	27	$	$	$
38536	1267	28	504	504	CD
38536	1267	29	.	.	.
38536	1268	1	What	what	WP
38536	1268	2	was	be	VBD
38536	1268	3	his	-PRON-	PRP$
38536	1268	4	original	original	JJ
38536	1268	5	salary	salary	NN
38536	1268	6	per	per	IN
38536	1268	7	month	month	NN
38536	1268	8	?	?	.
38536	1269	1	(	(	-LRB-
38536	1269	2	_	_	NNP
38536	1269	3	College	College	NNP
38536	1269	4	Entrance	Entrance	NNP
38536	1269	5	Board	Board	NNP
38536	1269	6	.	.	.
38536	1269	7	_	_	NNP
38536	1269	8	)	)	-RRB-
38536	1269	9	5	5	CD
38536	1269	10	.	.	.
38536	1270	1	A	a	DT
38536	1270	2	person	person	NN
38536	1270	3	who	who	WP
38536	1270	4	possesses	possess	VBZ
38536	1270	5	$	$	$
38536	1270	6	15,000	15,000	CD
38536	1270	7	employs	employ	NNS
38536	1270	8	a	a	DT
38536	1270	9	part	part	NN
38536	1270	10	of	of	IN
38536	1270	11	the	the	DT
38536	1270	12	money	money	NN
38536	1270	13	in	in	IN
38536	1270	14	building	build	VBG
38536	1270	15	a	a	DT
38536	1270	16	house	house	NN
38536	1270	17	.	.	.
38536	1271	1	He	-PRON-	PRP
38536	1271	2	invests	invest	VBZ
38536	1271	3	one	one	CD
38536	1271	4	third	third	NN
38536	1271	5	of	of	IN
38536	1271	6	the	the	DT
38536	1271	7	money	money	NN
38536	1271	8	which	which	WDT
38536	1271	9	remains	remain	VBZ
38536	1271	10	at	at	IN
38536	1271	11	6	6	CD
38536	1271	12	%	%	NN
38536	1271	13	,	,	,
38536	1271	14	and	and	CC
38536	1271	15	the	the	DT
38536	1271	16	other	other	JJ
38536	1271	17	two	two	CD
38536	1271	18	thirds	third	NNS
38536	1271	19	at	at	IN
38536	1271	20	9	9	CD
38536	1271	21	%	%	NN
38536	1271	22	,	,	,
38536	1271	23	and	and	CC
38536	1271	24	from	from	IN
38536	1271	25	these	these	DT
38536	1271	26	investments	investment	NNS
38536	1271	27	he	-PRON-	PRP
38536	1271	28	obtains	obtain	VBZ
38536	1271	29	an	an	DT
38536	1271	30	annual	annual	JJ
38536	1271	31	income	income	NN
38536	1271	32	of	of	IN
38536	1271	33	$	$	$
38536	1271	34	500	500	CD
38536	1271	35	.	.	.
38536	1272	1	What	what	WP
38536	1272	2	was	be	VBD
38536	1272	3	the	the	DT
38536	1272	4	cost	cost	NN
38536	1272	5	of	of	IN
38536	1272	6	the	the	DT
38536	1272	7	house	house	NN
38536	1272	8	?	?	.
38536	1273	1	(	(	-LRB-
38536	1273	2	_	_	NNP
38536	1273	3	M.	M.	NNP
38536	1274	1	I.	I.	NNP
38536	1274	2	T.	T.	NNP
38536	1274	3	_	_	NNP
38536	1274	4	)	)	-RRB-
38536	1274	5	6	6	CD
38536	1274	6	.	.	.
38536	1275	1	Two	two	CD
38536	1275	2	travelers	traveler	NNS
38536	1275	3	have	have	VBP
38536	1275	4	together	together	RB
38536	1275	5	400	400	CD
38536	1275	6	pounds	pound	NNS
38536	1275	7	of	of	IN
38536	1275	8	baggage	baggage	NN
38536	1275	9	.	.	.
38536	1276	1	One	one	CD
38536	1276	2	pays	pay	VBZ
38536	1276	3	$	$	$
38536	1276	4	1.20	1.20	CD
38536	1276	5	and	and	CC
38536	1276	6	the	the	DT
38536	1276	7	other	other	JJ
38536	1276	8	$	$	$
38536	1276	9	1.80	1.80	CD
38536	1276	10	for	for	IN
38536	1276	11	excess	excess	NN
38536	1276	12	above	above	IN
38536	1276	13	the	the	DT
38536	1276	14	weight	weight	NN
38536	1276	15	carried	carry	VBN
38536	1276	16	free	free	JJ
38536	1276	17	.	.	.
38536	1277	1	If	if	IN
38536	1277	2	all	all	DT
38536	1277	3	had	have	VBD
38536	1277	4	belonged	belong	VBN
38536	1277	5	to	to	IN
38536	1277	6	one	one	CD
38536	1277	7	person	person	NN
38536	1277	8	,	,	,
38536	1277	9	he	-PRON-	PRP
38536	1277	10	would	would	MD
38536	1277	11	have	have	VB
38536	1277	12	had	have	VBN
38536	1277	13	to	to	TO
38536	1277	14	pay	pay	VB
38536	1277	15	$	$	$
38536	1277	16	4.50	4.50	CD
38536	1277	17	.	.	.
38536	1278	1	How	how	WRB
38536	1278	2	much	much	JJ
38536	1278	3	baggage	baggage	NN
38536	1278	4	is	be	VBZ
38536	1278	5	allowed	allow	VBN
38536	1278	6	to	to	TO
38536	1278	7	go	go	VB
38536	1278	8	free	free	JJ
38536	1278	9	?	?	.
38536	1279	1	(	(	-LRB-
38536	1279	2	_	_	NNP
38536	1279	3	Yale	Yale	NNP
38536	1279	4	.	.	.
38536	1279	5	_	_	NNP
38536	1279	6	)	)	-RRB-
38536	1279	7	7	7	CD
38536	1279	8	.	.	.
38536	1280	1	A	a	DT
38536	1280	2	man	man	NN
38536	1280	3	who	who	WP
38536	1280	4	can	can	MD
38536	1280	5	row	row	VB
38536	1280	6	4	4	CD
38536	1280	7	-	-	SYM
38536	1280	8	1/3	1/3	CD
38536	1280	9	miles	mile	NNS
38536	1280	10	an	an	DT
38536	1280	11	hour	hour	NN
38536	1280	12	in	in	IN
38536	1280	13	still	still	RB
38536	1280	14	water	water	NN
38536	1280	15	rows	row	NNS
38536	1280	16	downstream	downstream	JJ
38536	1280	17	and	and	CC
38536	1280	18	returns	return	NNS
38536	1280	19	.	.	.
38536	1281	1	The	the	DT
38536	1281	2	rate	rate	NN
38536	1281	3	of	of	IN
38536	1281	4	the	the	DT
38536	1281	5	current	current	NN
38536	1281	6	is	be	VBZ
38536	1281	7	2	2	CD
38536	1281	8	-	-	SYM
38536	1281	9	1/4	1/4	CD
38536	1281	10	miles	mile	NNS
38536	1281	11	per	per	IN
38536	1281	12	hour	hour	NN
38536	1281	13	,	,	,
38536	1281	14	and	and	CC
38536	1281	15	the	the	DT
38536	1281	16	time	time	NN
38536	1281	17	required	require	VBN
38536	1281	18	for	for	IN
38536	1281	19	the	the	DT
38536	1281	20	trip	trip	NN
38536	1281	21	is	be	VBZ
38536	1281	22	13	13	CD
38536	1281	23	hours	hour	NNS
38536	1281	24	.	.	.
38536	1282	1	How	how	WRB
38536	1282	2	many	many	JJ
38536	1282	3	hours	hour	NNS
38536	1282	4	does	do	VBZ
38536	1282	5	he	-PRON-	PRP
38536	1282	6	require	require	VB
38536	1282	7	to	to	TO
38536	1282	8	return	return	VB
38536	1282	9	?	?	.
38536	1283	1	~Simultaneous	~Simultaneous	NFP
38536	1283	2	Equations	Equations	NNP
38536	1283	3	,	,	,
38536	1283	4	Two	two	CD
38536	1283	5	and	and	CC
38536	1283	6	Three	three	CD
38536	1283	7	Unknowns~	unknowns~	NN
38536	1283	8	1	1	CD
38536	1283	9	.	.	.
38536	1284	1	A	a	DT
38536	1284	2	manual	manual	JJ
38536	1284	3	training	training	NN
38536	1284	4	student	student	NN
38536	1284	5	in	in	IN
38536	1284	6	making	make	VBG
38536	1284	7	a	a	DT
38536	1284	8	bookcase	bookcase	NN
38536	1284	9	finds	find	VBZ
38536	1284	10	that	that	IN
38536	1284	11	the	the	DT
38536	1284	12	distance	distance	NN
38536	1284	13	from	from	IN
38536	1284	14	the	the	DT
38536	1284	15	top	top	NN
38536	1284	16	of	of	IN
38536	1284	17	the	the	DT
38536	1284	18	lowest	low	JJS
38536	1284	19	shelf	shelf	NN
38536	1284	20	to	to	IN
38536	1284	21	the	the	DT
38536	1284	22	under	under	JJ
38536	1284	23	side	side	NN
38536	1284	24	of	of	IN
38536	1284	25	the	the	DT
38536	1284	26	top	top	JJ
38536	1284	27	shelf	shelf	NN
38536	1284	28	is	be	VBZ
38536	1284	29	4	4	CD
38536	1284	30	ft	ft	NN
38536	1284	31	.	.	NN
38536	1284	32	6	6	CD
38536	1284	33	in	in	IN
38536	1284	34	.	.	.
38536	1285	1	He	-PRON-	PRP
38536	1285	2	desires	desire	VBZ
38536	1285	3	to	to	TO
38536	1285	4	put	put	VB
38536	1285	5	between	between	IN
38536	1285	6	these	these	DT
38536	1285	7	four	four	CD
38536	1285	8	other	other	JJ
38536	1285	9	shelves	shelf	NNS
38536	1285	10	of	of	IN
38536	1285	11	inch	inch	NN
38536	1285	12	boards	board	NNS
38536	1285	13	in	in	IN
38536	1285	14	such	such	PDT
38536	1285	15	a	a	DT
38536	1285	16	way	way	NN
38536	1285	17	that	that	WDT
38536	1285	18	the	the	DT
38536	1285	19	book	book	NN
38536	1285	20	space	space	NN
38536	1285	21	will	will	MD
38536	1285	22	diminish	diminish	VB
38536	1285	23	one	one	CD
38536	1285	24	inch	inch	NN
38536	1285	25	for	for	IN
38536	1285	26	each	each	DT
38536	1285	27	shelf	shelf	NN
38536	1285	28	from	from	IN
38536	1285	29	the	the	DT
38536	1285	30	bottom	bottom	NN
38536	1285	31	to	to	IN
38536	1285	32	the	the	DT
38536	1285	33	top	top	NN
38536	1285	34	.	.	.
38536	1286	1	What	what	WP
38536	1286	2	will	will	MD
38536	1286	3	be	be	VB
38536	1286	4	the	the	DT
38536	1286	5	several	several	JJ
38536	1286	6	spaces	space	NNS
38536	1286	7	between	between	IN
38536	1286	8	the	the	DT
38536	1286	9	shelves	shelf	NNS
38536	1286	10	?	?	.
38536	1287	1	2	2	LS
38536	1287	2	.	.	.
38536	1288	1	A	a	DT
38536	1288	2	quantity	quantity	NN
38536	1288	3	of	of	IN
38536	1288	4	water	water	NN
38536	1288	5	,	,	,
38536	1288	6	sufficient	sufficient	JJ
38536	1288	7	to	to	TO
38536	1288	8	fill	fill	VB
38536	1288	9	three	three	CD
38536	1288	10	jars	jar	NNS
38536	1288	11	of	of	IN
38536	1288	12	different	different	JJ
38536	1288	13	sizes	size	NNS
38536	1288	14	,	,	,
38536	1288	15	will	will	MD
38536	1288	16	fill	fill	VB
38536	1288	17	the	the	DT
38536	1288	18	smallest	small	JJS
38536	1288	19	jar	jar	NN
38536	1288	20	4	4	CD
38536	1288	21	times	time	NNS
38536	1288	22	,	,	,
38536	1288	23	or	or	CC
38536	1288	24	the	the	DT
38536	1288	25	largest	large	JJS
38536	1288	26	jar	jar	NN
38536	1288	27	twice	twice	RB
38536	1288	28	with	with	IN
38536	1288	29	4	4	CD
38536	1288	30	gallons	gallon	NNS
38536	1288	31	to	to	TO
38536	1288	32	spare	spare	VB
38536	1288	33	,	,	,
38536	1288	34	or	or	CC
38536	1288	35	the	the	DT
38536	1288	36	second	second	JJ
38536	1288	37	jar	jar	NN
38536	1288	38	three	three	CD
38536	1288	39	times	time	NNS
38536	1288	40	with	with	IN
38536	1288	41	2	2	CD
38536	1288	42	gallons	gallon	NNS
38536	1288	43	to	to	TO
38536	1288	44	spare	spare	VB
38536	1288	45	.	.	.
38536	1289	1	What	what	WP
38536	1289	2	is	be	VBZ
38536	1289	3	the	the	DT
38536	1289	4	capacity	capacity	NN
38536	1289	5	of	of	IN
38536	1289	6	each	each	DT
38536	1289	7	jar	jar	NN
38536	1289	8	?	?	.
38536	1290	1	(	(	-LRB-
38536	1290	2	_	_	NNP
38536	1290	3	Case	Case	NNP
38536	1290	4	.	.	.
38536	1290	5	_	_	NNP
38536	1290	6	)	)	-RRB-
38536	1290	7	3	3	CD
38536	1290	8	.	.	.
38536	1291	1	A	a	DT
38536	1291	2	policeman	policeman	NN
38536	1291	3	is	be	VBZ
38536	1291	4	chasing	chase	VBG
38536	1291	5	a	a	DT
38536	1291	6	pickpocket	pickpocket	NN
38536	1291	7	.	.	.
38536	1292	1	When	when	WRB
38536	1292	2	the	the	DT
38536	1292	3	policeman	policeman	NN
38536	1292	4	is	be	VBZ
38536	1292	5	80	80	CD
38536	1292	6	yards	yard	NNS
38536	1292	7	behind	behind	IN
38536	1292	8	him	-PRON-	PRP
38536	1292	9	,	,	,
38536	1292	10	the	the	DT
38536	1292	11	pickpocket	pickpocket	NN
38536	1292	12	turns	turn	VBZ
38536	1292	13	up	up	RP
38536	1292	14	an	an	DT
38536	1292	15	alley	alley	NN
38536	1292	16	;	;	:
38536	1292	17	but	but	CC
38536	1292	18	coming	come	VBG
38536	1292	19	to	to	IN
38536	1292	20	the	the	DT
38536	1292	21	end	end	NN
38536	1292	22	,	,	,
38536	1292	23	he	-PRON-	PRP
38536	1292	24	finds	find	VBZ
38536	1292	25	there	there	EX
38536	1292	26	is	be	VBZ
38536	1292	27	no	no	DT
38536	1292	28	outlet	outlet	NN
38536	1292	29	,	,	,
38536	1292	30	turns	turn	VBZ
38536	1292	31	back	back	RB
38536	1292	32	,	,	,
38536	1292	33	and	and	CC
38536	1292	34	is	be	VBZ
38536	1292	35	caught	catch	VBN
38536	1292	36	just	just	RB
38536	1292	37	as	as	IN
38536	1292	38	he	-PRON-	PRP
38536	1292	39	comes	come	VBZ
38536	1292	40	out	out	IN
38536	1292	41	of	of	IN
38536	1292	42	the	the	DT
38536	1292	43	alley	alley	NN
38536	1292	44	.	.	.
38536	1293	1	If	if	IN
38536	1293	2	he	-PRON-	PRP
38536	1293	3	had	have	VBD
38536	1293	4	discovered	discover	VBN
38536	1293	5	that	that	IN
38536	1293	6	the	the	DT
38536	1293	7	alley	alley	NN
38536	1293	8	had	have	VBD
38536	1293	9	no	no	DT
38536	1293	10	outlet	outlet	NN
38536	1293	11	when	when	WRB
38536	1293	12	he	-PRON-	PRP
38536	1293	13	had	have	VBD
38536	1293	14	run	run	VBN
38536	1293	15	halfway	halfway	RB
38536	1293	16	up	up	RB
38536	1293	17	and	and	CC
38536	1293	18	had	have	VBD
38536	1293	19	then	then	RB
38536	1293	20	turned	turn	VBN
38536	1293	21	back	back	RB
38536	1293	22	,	,	,
38536	1293	23	the	the	DT
38536	1293	24	policeman	policeman	NN
38536	1293	25	would	would	MD
38536	1293	26	have	have	VB
38536	1293	27	had	have	VBN
38536	1293	28	to	to	TO
38536	1293	29	pursue	pursue	VB
38536	1293	30	the	the	DT
38536	1293	31	thief	thief	NN
38536	1293	32	120	120	CD
38536	1293	33	yards	yard	NNS
38536	1293	34	beyond	beyond	IN
38536	1293	35	the	the	DT
38536	1293	36	alley	alley	NN
38536	1293	37	before	before	IN
38536	1293	38	catching	catch	VBG
38536	1293	39	him	-PRON-	PRP
38536	1293	40	.	.	.
38536	1294	1	How	how	WRB
38536	1294	2	long	long	RB
38536	1294	3	is	be	VBZ
38536	1294	4	the	the	DT
38536	1294	5	alley	alley	NN
38536	1294	6	?	?	.
38536	1295	1	(	(	-LRB-
38536	1295	2	_	_	NNP
38536	1295	3	Harvard	Harvard	NNP
38536	1295	4	.	.	.
38536	1295	5	_	_	NNP
38536	1295	6	)	)	-RRB-
38536	1295	7	4	4	CD
38536	1295	8	.	.	.
38536	1296	1	A	a	DT
38536	1296	2	and	and	CC
38536	1296	3	B	b	NN
38536	1296	4	together	together	RB
38536	1296	5	can	can	MD
38536	1296	6	do	do	VB
38536	1296	7	a	a	DT
38536	1296	8	piece	piece	NN
38536	1296	9	of	of	IN
38536	1296	10	work	work	NN
38536	1296	11	in	in	IN
38536	1296	12	14	14	CD
38536	1296	13	days	day	NNS
38536	1296	14	.	.	.
38536	1297	1	After	after	IN
38536	1297	2	they	-PRON-	PRP
38536	1297	3	have	have	VBP
38536	1297	4	worked	work	VBN
38536	1297	5	6	6	CD
38536	1297	6	days	day	NNS
38536	1297	7	on	on	IN
38536	1297	8	it	-PRON-	PRP
38536	1297	9	,	,	,
38536	1297	10	they	-PRON-	PRP
38536	1297	11	are	be	VBP
38536	1297	12	joined	join	VBN
38536	1297	13	by	by	IN
38536	1297	14	C	C	NNP
38536	1297	15	who	who	WP
38536	1297	16	works	work	VBZ
38536	1297	17	twice	twice	RB
38536	1297	18	as	as	RB
38536	1297	19	fast	fast	RB
38536	1297	20	as	as	IN
38536	1297	21	A.	a.	NN
38536	1298	1	The	the	DT
38536	1298	2	three	three	CD
38536	1298	3	finish	finish	NN
38536	1298	4	the	the	DT
38536	1298	5	work	work	NN
38536	1298	6	in	in	IN
38536	1298	7	4	4	CD
38536	1298	8	days	day	NNS
38536	1298	9	.	.	.
38536	1299	1	How	how	WRB
38536	1299	2	long	long	RB
38536	1299	3	would	would	MD
38536	1299	4	it	-PRON-	PRP
38536	1299	5	take	take	VB
38536	1299	6	each	each	DT
38536	1299	7	man	man	NN
38536	1299	8	alone	alone	JJ
38536	1299	9	to	to	TO
38536	1299	10	do	do	VB
38536	1299	11	it	-PRON-	PRP
38536	1299	12	?	?	.
38536	1300	1	(	(	-LRB-
38536	1300	2	_	_	NNP
38536	1300	3	Columbia	Columbia	NNP
38536	1300	4	.	.	.
38536	1300	5	_	_	NNP
38536	1300	6	)	)	-RRB-
38536	1300	7	5	5	CD
38536	1300	8	.	.	.
38536	1301	1	In	in	IN
38536	1301	2	a	a	DT
38536	1301	3	certain	certain	JJ
38536	1301	4	mill	mill	NN
38536	1301	5	some	some	DT
38536	1301	6	of	of	IN
38536	1301	7	the	the	DT
38536	1301	8	workmen	workman	NNS
38536	1301	9	receive	receive	VBP
38536	1301	10	$	$	$
38536	1301	11	1.50	1.50	CD
38536	1301	12	a	a	DT
38536	1301	13	day	day	NN
38536	1301	14	,	,	,
38536	1301	15	others	other	NNS
38536	1301	16	more	more	RBR
38536	1301	17	.	.	.
38536	1302	1	The	the	DT
38536	1302	2	total	total	NN
38536	1302	3	paid	pay	VBN
38536	1302	4	in	in	IN
38536	1302	5	wages	wage	NNS
38536	1302	6	each	each	DT
38536	1302	7	day	day	NN
38536	1302	8	is	be	VBZ
38536	1302	9	$	$	$
38536	1302	10	350	350	CD
38536	1302	11	.	.	.
38536	1303	1	An	an	DT
38536	1303	2	assessment	assessment	NN
38536	1303	3	made	make	VBN
38536	1303	4	by	by	IN
38536	1303	5	a	a	DT
38536	1303	6	labor	labor	NN
38536	1303	7	union	union	NN
38536	1303	8	to	to	TO
38536	1303	9	raise	raise	VB
38536	1303	10	$	$	$
38536	1303	11	200	200	CD
38536	1303	12	requires	require	VBZ
38536	1303	13	$	$	$
38536	1303	14	1.00	1.00	CD
38536	1303	15	from	from	IN
38536	1303	16	each	each	DT
38536	1303	17	man	man	NN
38536	1303	18	receiving	receive	VBG
38536	1303	19	$	$	$
38536	1303	20	1.50	1.50	CD
38536	1303	21	a	a	DT
38536	1303	22	day	day	NN
38536	1303	23	,	,	,
38536	1303	24	and	and	CC
38536	1303	25	half	half	NN
38536	1303	26	of	of	IN
38536	1303	27	one	one	CD
38536	1303	28	day	day	NN
38536	1303	29	's	's	POS
38536	1303	30	pay	pay	NN
38536	1303	31	from	from	IN
38536	1303	32	every	every	DT
38536	1303	33	man	man	NN
38536	1303	34	receiving	receive	VBG
38536	1303	35	more	more	RBR
38536	1303	36	.	.	.
38536	1304	1	How	how	WRB
38536	1304	2	many	many	JJ
38536	1304	3	men	man	NNS
38536	1304	4	receive	receive	VBP
38536	1304	5	$	$	$
38536	1304	6	1.50	1.50	CD
38536	1304	7	a	a	DT
38536	1304	8	day	day	NN
38536	1304	9	?	?	.
38536	1305	1	(	(	-LRB-
38536	1305	2	_	_	NNP
38536	1305	3	Harvard	Harvard	NNP
38536	1305	4	.	.	.
38536	1305	5	_	_	NNP
38536	1305	6	)	)	-RRB-
38536	1305	7	6	6	CD
38536	1305	8	.	.	.
38536	1306	1	There	there	EX
38536	1306	2	are	be	VBP
38536	1306	3	two	two	CD
38536	1306	4	alloys	alloy	NNS
38536	1306	5	of	of	IN
38536	1306	6	silver	silver	NN
38536	1306	7	and	and	CC
38536	1306	8	copper	copper	NN
38536	1306	9	,	,	,
38536	1306	10	of	of	IN
38536	1306	11	which	which	WDT
38536	1306	12	one	one	NN
38536	1306	13	contains	contain	VBZ
38536	1306	14	twice	twice	PDT
38536	1306	15	as	as	RB
38536	1306	16	much	much	JJ
38536	1306	17	copper	copper	NN
38536	1306	18	as	as	IN
38536	1306	19	silver	silver	NN
38536	1306	20	,	,	,
38536	1306	21	and	and	CC
38536	1306	22	the	the	DT
38536	1306	23	other	other	JJ
38536	1306	24	three	three	CD
38536	1306	25	times	time	NNS
38536	1306	26	as	as	RB
38536	1306	27	much	much	JJ
38536	1306	28	silver	silver	NN
38536	1306	29	as	as	IN
38536	1306	30	copper	copper	NN
38536	1306	31	.	.	.
38536	1307	1	How	how	WRB
38536	1307	2	much	much	JJ
38536	1307	3	must	must	MD
38536	1307	4	be	be	VB
38536	1307	5	taken	take	VBN
38536	1307	6	from	from	IN
38536	1307	7	each	each	DT
38536	1307	8	to	to	TO
38536	1307	9	obtain	obtain	VB
38536	1307	10	a	a	DT
38536	1307	11	kilogram	kilogram	NN
38536	1307	12	of	of	IN
38536	1307	13	an	an	DT
38536	1307	14	alloy	alloy	NN
38536	1307	15	to	to	TO
38536	1307	16	contain	contain	VB
38536	1307	17	equal	equal	JJ
38536	1307	18	quantities	quantity	NNS
38536	1307	19	of	of	IN
38536	1307	20	silver	silver	NN
38536	1307	21	and	and	CC
38536	1307	22	copper	copper	NN
38536	1307	23	?	?	.
38536	1308	1	(	(	-LRB-
38536	1308	2	_	_	NNP
38536	1308	3	M.	M.	NNP
38536	1309	1	I.	I.	NNP
38536	1309	2	T.	T.	NNP
38536	1309	3	_	_	NNP
38536	1309	4	)	)	-RRB-
38536	1309	5	7	7	CD
38536	1309	6	.	.	.
38536	1310	1	Two	two	CD
38536	1310	2	automobiles	automobile	NNS
38536	1310	3	travel	travel	VBP
38536	1310	4	toward	toward	IN
38536	1310	5	each	each	DT
38536	1310	6	other	other	JJ
38536	1310	7	over	over	IN
38536	1310	8	a	a	DT
38536	1310	9	distance	distance	NN
38536	1310	10	of	of	IN
38536	1310	11	120	120	CD
38536	1310	12	miles	mile	NNS
38536	1310	13	.	.	.
38536	1311	1	A	a	DT
38536	1311	2	leaves	leave	NNS
38536	1311	3	at	at	IN
38536	1311	4	9	9	CD
38536	1311	5	A.M.	A.M.	NNP
38536	1311	6	,	,	,
38536	1311	7	1	1	CD
38536	1311	8	hour	hour	NN
38536	1311	9	before	before	IN
38536	1311	10	B	b	NN
38536	1311	11	starts	start	VBZ
38536	1311	12	to	to	TO
38536	1311	13	meet	meet	VB
38536	1311	14	him	-PRON-	PRP
38536	1311	15	,	,	,
38536	1311	16	and	and	CC
38536	1311	17	they	-PRON-	PRP
38536	1311	18	meet	meet	VBP
38536	1311	19	at	at	IN
38536	1311	20	12:00	12:00	CD
38536	1311	21	M.	M.	NNP
38536	1311	22	If	if	IN
38536	1311	23	each	each	DT
38536	1311	24	had	have	VBD
38536	1311	25	started	start	VBN
38536	1311	26	at	at	IN
38536	1311	27	9:15	9:15	CD
38536	1311	28	A.M.	A.M.	NNP
38536	1311	29	,	,	,
38536	1311	30	they	-PRON-	PRP
38536	1311	31	would	would	MD
38536	1311	32	have	have	VB
38536	1311	33	met	meet	VBN
38536	1311	34	at	at	IN
38536	1311	35	12:00	12:00	CD
38536	1311	36	M.	M.	NNP
38536	1311	37	also	also	RB
38536	1311	38	.	.	.
38536	1312	1	Find	find	VB
38536	1312	2	the	the	DT
38536	1312	3	rate	rate	NN
38536	1312	4	at	at	IN
38536	1312	5	which	which	WDT
38536	1312	6	each	each	DT
38536	1312	7	traveled	travel	VBD
38536	1312	8	.	.	.
38536	1313	1	(	(	-LRB-
38536	1313	2	_	_	NNP
38536	1313	3	M.	M.	NNP
38536	1314	1	I.	I.	NNP
38536	1314	2	T.	T.	NNP
38536	1314	3	_	_	NNP
38536	1314	4	)	)	-RRB-
38536	1314	5	~Quadratic	~Quadratic	.
38536	1314	6	Equations~	equations~	ADD
38536	1314	7	1	1	CD
38536	1314	8	.	.	.
38536	1315	1	Telegraph	telegraph	NN
38536	1315	2	poles	pole	NNS
38536	1315	3	are	be	VBP
38536	1315	4	set	set	VBN
38536	1315	5	at	at	IN
38536	1315	6	equal	equal	JJ
38536	1315	7	distances	distance	NNS
38536	1315	8	apart	apart	RB
38536	1315	9	.	.	.
38536	1316	1	In	in	IN
38536	1316	2	order	order	NN
38536	1316	3	to	to	TO
38536	1316	4	have	have	VB
38536	1316	5	two	two	CD
38536	1316	6	less	less	JJR
38536	1316	7	to	to	IN
38536	1316	8	the	the	DT
38536	1316	9	mile	mile	NN
38536	1316	10	,	,	,
38536	1316	11	it	-PRON-	PRP
38536	1316	12	will	will	MD
38536	1316	13	be	be	VB
38536	1316	14	necessary	necessary	JJ
38536	1316	15	to	to	TO
38536	1316	16	set	set	VB
38536	1316	17	them	-PRON-	PRP
38536	1316	18	20	20	CD
38536	1316	19	feet	foot	NNS
38536	1316	20	farther	far	RBR
38536	1316	21	apart	apart	RB
38536	1316	22	.	.	.
38536	1317	1	Find	find	VB
38536	1317	2	how	how	WRB
38536	1317	3	far	far	RB
38536	1317	4	apart	apart	RB
38536	1317	5	they	-PRON-	PRP
38536	1317	6	are	be	VBP
38536	1317	7	now	now	RB
38536	1317	8	.	.	.
38536	1318	1	(	(	-LRB-
38536	1318	2	_	_	NNP
38536	1318	3	Yale	Yale	NNP
38536	1318	4	.	.	.
38536	1318	5	_	_	NNP
38536	1318	6	)	)	-RRB-
38536	1318	7	2	2	CD
38536	1318	8	.	.	.
38536	1319	1	The	the	DT
38536	1319	2	distance	distance	NN
38536	1319	3	S	S	VBZ
38536	1319	4	that	that	IN
38536	1319	5	a	a	DT
38536	1319	6	body	body	NN
38536	1319	7	falls	fall	VBZ
38536	1319	8	from	from	IN
38536	1319	9	rest	rest	NN
38536	1319	10	in	in	IN
38536	1319	11	t	t	NN
38536	1319	12	seconds	second	NNS
38536	1319	13	is	be	VBZ
38536	1319	14	given	give	VBN
38536	1319	15	by	by	IN
38536	1319	16	the	the	DT
38536	1319	17	formula	formula	NN
38536	1319	18	S	S	NNP
38536	1319	19	=	=	SYM
38536	1319	20	16t^2	16t^2	CD
38536	1319	21	.	.	.
38536	1320	1	A	a	DT
38536	1320	2	man	man	NN
38536	1320	3	drops	drop	VBZ
38536	1320	4	a	a	DT
38536	1320	5	stone	stone	NN
38536	1320	6	into	into	IN
38536	1320	7	a	a	DT
38536	1320	8	well	well	NN
38536	1320	9	and	and	CC
38536	1320	10	hears	hear	VBZ
38536	1320	11	the	the	DT
38536	1320	12	splash	splash	NN
38536	1320	13	after	after	IN
38536	1320	14	3	3	CD
38536	1320	15	seconds	second	NNS
38536	1320	16	.	.	.
38536	1321	1	If	if	IN
38536	1321	2	the	the	DT
38536	1321	3	velocity	velocity	NN
38536	1321	4	of	of	IN
38536	1321	5	sound	sound	NN
38536	1321	6	in	in	IN
38536	1321	7	air	air	NN
38536	1321	8	is	be	VBZ
38536	1321	9	1086	1086	CD
38536	1321	10	feet	foot	NNS
38536	1321	11	a	a	DT
38536	1321	12	second	second	NN
38536	1321	13	,	,	,
38536	1321	14	what	what	WP
38536	1321	15	is	be	VBZ
38536	1321	16	the	the	DT
38536	1321	17	depth	depth	NN
38536	1321	18	of	of	IN
38536	1321	19	the	the	DT
38536	1321	20	well	well	NN
38536	1321	21	?	?	.
38536	1322	1	(	(	-LRB-
38536	1322	2	_	_	NNP
38536	1322	3	Yale	Yale	NNP
38536	1322	4	.	.	.
38536	1322	5	_	_	NNP
38536	1322	6	)	)	-RRB-
38536	1322	7	3	3	CD
38536	1322	8	.	.	.
38536	1323	1	It	-PRON-	PRP
38536	1323	2	requires	require	VBZ
38536	1323	3	2000	2000	CD
38536	1323	4	square	square	JJ
38536	1323	5	tiles	tile	NNS
38536	1323	6	of	of	IN
38536	1323	7	a	a	DT
38536	1323	8	certain	certain	JJ
38536	1323	9	size	size	NN
38536	1323	10	to	to	TO
38536	1323	11	pave	pave	VB
38536	1323	12	a	a	DT
38536	1323	13	hall	hall	NN
38536	1323	14	,	,	,
38536	1323	15	or	or	CC
38536	1323	16	3125	3125	CD
38536	1323	17	square	square	JJ
38536	1323	18	tiles	tile	NNS
38536	1323	19	whose	whose	WP$
38536	1323	20	dimensions	dimension	NNS
38536	1323	21	are	be	VBP
38536	1323	22	one	one	CD
38536	1323	23	inch	inch	NN
38536	1323	24	less	less	RBR
38536	1323	25	.	.	.
38536	1324	1	Find	find	VB
38536	1324	2	the	the	DT
38536	1324	3	area	area	NN
38536	1324	4	of	of	IN
38536	1324	5	the	the	DT
38536	1324	6	hall	hall	NN
38536	1324	7	.	.	.
38536	1325	1	How	how	WRB
38536	1325	2	many	many	JJ
38536	1325	3	solutions	solution	NNS
38536	1325	4	has	have	VBZ
38536	1325	5	the	the	DT
38536	1325	6	equation	equation	NN
38536	1325	7	of	of	IN
38536	1325	8	this	this	DT
38536	1325	9	problem	problem	NN
38536	1325	10	?	?	.
38536	1326	1	How	how	WRB
38536	1326	2	many	many	JJ
38536	1326	3	has	have	VBZ
38536	1326	4	the	the	DT
38536	1326	5	problem	problem	NN
38536	1326	6	itself	-PRON-	PRP
38536	1326	7	?	?	.
38536	1327	1	Explain	explain	VB
38536	1327	2	the	the	DT
38536	1327	3	apparent	apparent	JJ
38536	1327	4	discrepancy	discrepancy	NN
38536	1327	5	.	.	.
38536	1328	1	(	(	-LRB-
38536	1328	2	_	_	NNP
38536	1328	3	Cornell	Cornell	NNP
38536	1328	4	.	.	.
38536	1328	5	_	_	NNP
38536	1328	6	)	)	-RRB-
38536	1328	7	4	4	CD
38536	1328	8	.	.	.
38536	1329	1	A	a	DT
38536	1329	2	rectangular	rectangular	JJ
38536	1329	3	tract	tract	NN
38536	1329	4	of	of	IN
38536	1329	5	land	land	NN
38536	1329	6	,	,	,
38536	1329	7	800	800	CD
38536	1329	8	feet	foot	NNS
38536	1329	9	long	long	JJ
38536	1329	10	by	by	IN
38536	1329	11	600	600	CD
38536	1329	12	feet	foot	NNS
38536	1329	13	broad	broad	JJ
38536	1329	14	,	,	,
38536	1329	15	is	be	VBZ
38536	1329	16	divided	divide	VBN
38536	1329	17	into	into	IN
38536	1329	18	four	four	CD
38536	1329	19	rectangular	rectangular	JJ
38536	1329	20	blocks	block	NNS
38536	1329	21	by	by	IN
38536	1329	22	two	two	CD
38536	1329	23	streets	street	NNS
38536	1329	24	of	of	IN
38536	1329	25	equal	equal	JJ
38536	1329	26	width	width	NN
38536	1329	27	running	run	VBG
38536	1329	28	through	through	IN
38536	1329	29	it	-PRON-	PRP
38536	1329	30	at	at	IN
38536	1329	31	right	right	JJ
38536	1329	32	angles	angle	NNS
38536	1329	33	.	.	.
38536	1330	1	Find	find	VB
38536	1330	2	the	the	DT
38536	1330	3	width	width	NN
38536	1330	4	of	of	IN
38536	1330	5	the	the	DT
38536	1330	6	streets	street	NNS
38536	1330	7	,	,	,
38536	1330	8	if	if	IN
38536	1330	9	together	together	RB
38536	1330	10	they	-PRON-	PRP
38536	1330	11	cover	cover	VBP
38536	1330	12	an	an	DT
38536	1330	13	area	area	NN
38536	1330	14	of	of	IN
38536	1330	15	77,500	77,500	CD
38536	1330	16	square	square	JJ
38536	1330	17	feet	foot	NNS
38536	1330	18	.	.	.
38536	1331	1	(	(	-LRB-
38536	1331	2	_	_	NNP
38536	1331	3	M.	M.	NNP
38536	1332	1	I.	I.	NNP
38536	1332	2	T.	T.	NNP
38536	1332	3	_	_	NNP
38536	1332	4	)	)	-RRB-
38536	1332	5	5	5	CD
38536	1332	6	.	.	.
38536	1333	1	(	(	-LRB-
38536	1333	2	_	_	NNP
38536	1333	3	a	a	DT
38536	1333	4	_	_	NNP
38536	1333	5	)	)	-RRB-
38536	1333	6	The	the	DT
38536	1333	7	height	height	NN
38536	1333	8	y	y	NN
38536	1333	9	to	to	TO
38536	1333	10	which	which	WDT
38536	1333	11	a	a	DT
38536	1333	12	ball	ball	NN
38536	1333	13	thrown	throw	VBN
38536	1333	14	vertically	vertically	RB
38536	1333	15	upward	upward	RB
38536	1333	16	with	with	IN
38536	1333	17	a	a	DT
38536	1333	18	velocity	velocity	NN
38536	1333	19	of	of	IN
38536	1333	20	100	100	CD
38536	1333	21	feet	foot	NNS
38536	1333	22	per	per	IN
38536	1333	23	second	second	JJ
38536	1333	24	rises	rise	NNS
38536	1333	25	in	in	IN
38536	1333	26	x	x	DT
38536	1333	27	seconds	second	NNS
38536	1333	28	is	be	VBZ
38536	1333	29	given	give	VBN
38536	1333	30	by	by	IN
38536	1333	31	the	the	DT
38536	1333	32	formula	formula	NN
38536	1333	33	,	,	,
38536	1333	34	y	y	NNP
38536	1333	35	=	=	SYM
38536	1333	36	100x	100x	NNP
38536	1333	37	-	-	SYM
38536	1333	38	16x^2	16x^2	CD
38536	1333	39	.	.	.
38536	1334	1	In	in	IN
38536	1334	2	how	how	WRB
38536	1334	3	many	many	JJ
38536	1334	4	seconds	second	NNS
38536	1334	5	will	will	MD
38536	1334	6	the	the	DT
38536	1334	7	ball	ball	NN
38536	1334	8	rise	rise	VB
38536	1334	9	to	to	IN
38536	1334	10	a	a	DT
38536	1334	11	height	height	NN
38536	1334	12	of	of	IN
38536	1334	13	144	144	CD
38536	1334	14	feet	foot	NNS
38536	1334	15	?	?	.
38536	1335	1	(	(	-LRB-
38536	1335	2	_	_	NNP
38536	1335	3	b	b	NNP
38536	1335	4	_	_	NNP
38536	1335	5	)	)	-RRB-
38536	1335	6	Draw	draw	VB
38536	1335	7	the	the	DT
38536	1335	8	graph	graph	NN
38536	1335	9	of	of	IN
38536	1335	10	the	the	DT
38536	1335	11	equation	equation	NN
38536	1335	12	y	y	NNP
38536	1335	13	=	=	SYM
38536	1335	14	100x	100x	NNP
38536	1335	15	-	-	SYM
38536	1335	16	16x^2	16x^2	CD
38536	1335	17	.	.	.
38536	1336	1	(	(	-LRB-
38536	1336	2	_	_	NNP
38536	1336	3	College	College	NNP
38536	1336	4	Entrance	Entrance	NNP
38536	1336	5	Board	Board	NNP
38536	1336	6	.	.	.
38536	1336	7	_	_	NNP
38536	1336	8	)	)	-RRB-
38536	1336	9	6	6	CD
38536	1336	10	.	.	.
38536	1337	1	Two	two	CD
38536	1337	2	launches	launch	VBZ
38536	1337	3	race	race	NN
38536	1337	4	over	over	IN
38536	1337	5	a	a	DT
38536	1337	6	course	course	NN
38536	1337	7	of	of	IN
38536	1337	8	12	12	CD
38536	1337	9	miles	mile	NNS
38536	1337	10	.	.	.
38536	1338	1	The	the	DT
38536	1338	2	first	first	JJ
38536	1338	3	steams	steam	NNS
38536	1338	4	7	7	CD
38536	1338	5	-	-	SYM
38536	1338	6	1/2	1/2	CD
38536	1338	7	miles	mile	NNS
38536	1338	8	an	an	DT
38536	1338	9	hour	hour	NN
38536	1338	10	.	.	.
38536	1339	1	The	the	DT
38536	1339	2	other	other	JJ
38536	1339	3	has	have	VBZ
38536	1339	4	a	a	DT
38536	1339	5	start	start	NN
38536	1339	6	of	of	IN
38536	1339	7	10	10	CD
38536	1339	8	minutes	minute	NNS
38536	1339	9	,	,	,
38536	1339	10	runs	run	VBZ
38536	1339	11	over	over	IN
38536	1339	12	the	the	DT
38536	1339	13	first	first	JJ
38536	1339	14	half	half	NN
38536	1339	15	of	of	IN
38536	1339	16	the	the	DT
38536	1339	17	course	course	NN
38536	1339	18	with	with	IN
38536	1339	19	a	a	DT
38536	1339	20	certain	certain	JJ
38536	1339	21	speed	speed	NN
38536	1339	22	,	,	,
38536	1339	23	but	but	CC
38536	1339	24	increases	increase	VBZ
38536	1339	25	its	-PRON-	PRP$
38536	1339	26	speed	speed	NN
38536	1339	27	over	over	IN
38536	1339	28	the	the	DT
38536	1339	29	second	second	JJ
38536	1339	30	half	half	NN
38536	1339	31	of	of	IN
38536	1339	32	the	the	DT
38536	1339	33	course	course	NN
38536	1339	34	by	by	IN
38536	1339	35	2	2	CD
38536	1339	36	miles	mile	NNS
38536	1339	37	per	per	IN
38536	1339	38	hour	hour	NN
38536	1339	39	,	,	,
38536	1339	40	winning	win	VBG
38536	1339	41	the	the	DT
38536	1339	42	race	race	NN
38536	1339	43	by	by	IN
38536	1339	44	a	a	DT
38536	1339	45	minute	minute	NN
38536	1339	46	.	.	.
38536	1340	1	What	what	WP
38536	1340	2	is	be	VBZ
38536	1340	3	the	the	DT
38536	1340	4	speed	speed	NN
38536	1340	5	of	of	IN
38536	1340	6	the	the	DT
38536	1340	7	second	second	JJ
38536	1340	8	launch	launch	NN
38536	1340	9	?	?	.
38536	1341	1	Explain	explain	VB
38536	1341	2	the	the	DT
38536	1341	3	meaning	meaning	NN
38536	1341	4	of	of	IN
38536	1341	5	the	the	DT
38536	1341	6	negative	negative	JJ
38536	1341	7	answer	answer	NN
38536	1341	8	.	.	.
38536	1342	1	(	(	-LRB-
38536	1342	2	_	_	NNP
38536	1342	3	Sheffield	Sheffield	NNP
38536	1342	4	Scientific	Scientific	NNP
38536	1342	5	School	School	NNP
38536	1342	6	.	.	.
38536	1342	7	_	_	NNP
38536	1342	8	)	)	-RRB-
38536	1342	9	7	7	CD
38536	1342	10	.	.	.
38536	1343	1	The	the	DT
38536	1343	2	circumference	circumference	NN
38536	1343	3	of	of	IN
38536	1343	4	a	a	DT
38536	1343	5	rear	rear	JJ
38536	1343	6	wheel	wheel	NN
38536	1343	7	of	of	IN
38536	1343	8	a	a	DT
38536	1343	9	certain	certain	JJ
38536	1343	10	wagon	wagon	NN
38536	1343	11	is	be	VBZ
38536	1343	12	3	3	CD
38536	1343	13	feet	foot	NNS
38536	1343	14	more	more	JJR
38536	1343	15	than	than	IN
38536	1343	16	the	the	DT
38536	1343	17	circumference	circumference	NN
38536	1343	18	of	of	IN
38536	1343	19	a	a	DT
38536	1343	20	front	front	JJ
38536	1343	21	wheel	wheel	NN
38536	1343	22	.	.	.
38536	1344	1	The	the	DT
38536	1344	2	rear	rear	JJ
38536	1344	3	wheel	wheel	NN
38536	1344	4	performs	perform	VBZ
38536	1344	5	100	100	CD
38536	1344	6	fewer	few	JJR
38536	1344	7	revolutions	revolution	NNS
38536	1344	8	than	than	IN
38536	1344	9	the	the	DT
38536	1344	10	front	front	JJ
38536	1344	11	wheel	wheel	NN
38536	1344	12	in	in	IN
38536	1344	13	traveling	travel	VBG
38536	1344	14	a	a	DT
38536	1344	15	distance	distance	NN
38536	1344	16	of	of	IN
38536	1344	17	6000	6000	CD
38536	1344	18	feet	foot	NNS
38536	1344	19	.	.	.
38536	1345	1	How	how	WRB
38536	1345	2	large	large	JJ
38536	1345	3	are	be	VBP
38536	1345	4	the	the	DT
38536	1345	5	wheels	wheel	NNS
38536	1345	6	?	?	.
38536	1346	1	(	(	-LRB-
38536	1346	2	_	_	NNP
38536	1346	3	Harvard	Harvard	NNP
38536	1346	4	.	.	.
38536	1346	5	_	_	NNP
38536	1346	6	)	)	-RRB-
38536	1346	7	8	8	CD
38536	1346	8	.	.	.
38536	1347	1	A	a	DT
38536	1347	2	man	man	NN
38536	1347	3	starts	start	VBZ
38536	1347	4	from	from	IN
38536	1347	5	home	home	NN
38536	1347	6	to	to	TO
38536	1347	7	catch	catch	VB
38536	1347	8	a	a	DT
38536	1347	9	train	train	NN
38536	1347	10	,	,	,
38536	1347	11	walking	walk	VBG
38536	1347	12	at	at	IN
38536	1347	13	the	the	DT
38536	1347	14	rate	rate	NN
38536	1347	15	of	of	IN
38536	1347	16	1	1	CD
38536	1347	17	yard	yard	NN
38536	1347	18	in	in	IN
38536	1347	19	1	1	CD
38536	1347	20	second	second	NN
38536	1347	21	,	,	,
38536	1347	22	and	and	CC
38536	1347	23	arrives	arrive	VBZ
38536	1347	24	2	2	CD
38536	1347	25	minutes	minute	NNS
38536	1347	26	late	late	RB
38536	1347	27	.	.	.
38536	1348	1	If	if	IN
38536	1348	2	he	-PRON-	PRP
38536	1348	3	had	have	VBD
38536	1348	4	walked	walk	VBN
38536	1348	5	at	at	IN
38536	1348	6	the	the	DT
38536	1348	7	rate	rate	NN
38536	1348	8	of	of	IN
38536	1348	9	4	4	CD
38536	1348	10	yards	yard	NNS
38536	1348	11	in	in	IN
38536	1348	12	3	3	CD
38536	1348	13	seconds	second	NNS
38536	1348	14	,	,	,
38536	1348	15	he	-PRON-	PRP
38536	1348	16	would	would	MD
38536	1348	17	have	have	VB
38536	1348	18	arrived	arrive	VBN
38536	1348	19	2	2	CD
38536	1348	20	-	-	SYM
38536	1348	21	1/2	1/2	CD
38536	1348	22	minutes	minute	NNS
38536	1348	23	early	early	RB
38536	1348	24	.	.	.
38536	1349	1	Find	find	VB
38536	1349	2	the	the	DT
38536	1349	3	distance	distance	NN
38536	1349	4	from	from	IN
38536	1349	5	his	-PRON-	PRP$
38536	1349	6	home	home	NN
38536	1349	7	to	to	IN
38536	1349	8	the	the	DT
38536	1349	9	station	station	NN
38536	1349	10	.	.	.
38536	1350	1	(	(	-LRB-
38536	1350	2	_	_	NNP
38536	1350	3	College	College	NNP
38536	1350	4	Entrance	Entrance	NNP
38536	1350	5	Board	Board	NNP
38536	1350	6	.	.	.
38536	1350	7	_	_	NNP
38536	1350	8	)	)	-RRB-
38536	1350	9	~Simultaneous	~Simultaneous	NFP
38536	1350	10	Quadratics~	Quadratics~	NNP
38536	1350	11	1	1	CD
38536	1350	12	.	.	.
38536	1351	1	Two	two	CD
38536	1351	2	cubical	cubical	JJ
38536	1351	3	coal	coal	NN
38536	1351	4	bins	bin	NNS
38536	1351	5	together	together	RB
38536	1351	6	hold	hold	VBP
38536	1351	7	280	280	CD
38536	1351	8	cubic	cubic	JJ
38536	1351	9	feet	foot	NNS
38536	1351	10	of	of	IN
38536	1351	11	coal	coal	NN
38536	1351	12	,	,	,
38536	1351	13	and	and	CC
38536	1351	14	the	the	DT
38536	1351	15	sum	sum	NN
38536	1351	16	of	of	IN
38536	1351	17	their	-PRON-	PRP$
38536	1351	18	lengths	length	NNS
38536	1351	19	is	be	VBZ
38536	1351	20	10	10	CD
38536	1351	21	feet	foot	NNS
38536	1351	22	.	.	.
38536	1352	1	Find	find	VB
38536	1352	2	the	the	DT
38536	1352	3	length	length	NN
38536	1352	4	of	of	IN
38536	1352	5	each	each	DT
38536	1352	6	bin	bin	NN
38536	1352	7	.	.	.
38536	1353	1	2	2	LS
38536	1353	2	.	.	.
38536	1354	1	The	the	DT
38536	1354	2	sum	sum	NN
38536	1354	3	of	of	IN
38536	1354	4	the	the	DT
38536	1354	5	radii	radii	NN
38536	1354	6	of	of	IN
38536	1354	7	two	two	CD
38536	1354	8	circles	circle	NNS
38536	1354	9	is	be	VBZ
38536	1354	10	25	25	CD
38536	1354	11	inches	inch	NNS
38536	1354	12	,	,	,
38536	1354	13	and	and	CC
38536	1354	14	the	the	DT
38536	1354	15	difference	difference	NN
38536	1354	16	of	of	IN
38536	1354	17	their	-PRON-	PRP$
38536	1354	18	areas	area	NNS
38536	1354	19	is	be	VBZ
38536	1354	20	125[pi	125[pi	CD
38536	1354	21	]	]	-RRB-
38536	1354	22	square	square	JJ
38536	1354	23	inches	inch	NNS
38536	1354	24	.	.	.
38536	1355	1	Find	find	VB
38536	1355	2	the	the	DT
38536	1355	3	radii	radii	NN
38536	1355	4	.	.	.
38536	1356	1	3	3	LS
38536	1356	2	.	.	.
38536	1357	1	The	the	DT
38536	1357	2	area	area	NN
38536	1357	3	of	of	IN
38536	1357	4	a	a	DT
38536	1357	5	right	right	JJ
38536	1357	6	triangle	triangle	NN
38536	1357	7	is	be	VBZ
38536	1357	8	150	150	CD
38536	1357	9	square	square	JJ
38536	1357	10	feet	foot	NNS
38536	1357	11	,	,	,
38536	1357	12	and	and	CC
38536	1357	13	its	-PRON-	PRP$
38536	1357	14	hypotenuse	hypotenuse	NN
38536	1357	15	is	be	VBZ
38536	1357	16	25	25	CD
38536	1357	17	feet	foot	NNS
38536	1357	18	.	.	.
38536	1358	1	Find	find	VB
38536	1358	2	the	the	DT
38536	1358	3	arms	arm	NNS
38536	1358	4	of	of	IN
38536	1358	5	the	the	DT
38536	1358	6	triangle	triangle	NN
38536	1358	7	.	.	.
38536	1359	1	4	4	LS
38536	1359	2	.	.	.
38536	1360	1	The	the	DT
38536	1360	2	combined	combined	JJ
38536	1360	3	capacity	capacity	NN
38536	1360	4	of	of	IN
38536	1360	5	two	two	CD
38536	1360	6	cubical	cubical	JJ
38536	1360	7	tanks	tank	NNS
38536	1360	8	is	be	VBZ
38536	1360	9	637	637	CD
38536	1360	10	cubic	cubic	JJ
38536	1360	11	feet	foot	NNS
38536	1360	12	,	,	,
38536	1360	13	and	and	CC
38536	1360	14	the	the	DT
38536	1360	15	sum	sum	NN
38536	1360	16	of	of	IN
38536	1360	17	an	an	DT
38536	1360	18	edge	edge	NN
38536	1360	19	of	of	IN
38536	1360	20	one	one	CD
38536	1360	21	and	and	CC
38536	1360	22	an	an	DT
38536	1360	23	edge	edge	NN
38536	1360	24	of	of	IN
38536	1360	25	the	the	DT
38536	1360	26	other	other	JJ
38536	1360	27	is	be	VBZ
38536	1360	28	13	13	CD
38536	1360	29	feet	foot	NNS
38536	1360	30	.	.	.
38536	1361	1	(	(	-LRB-
38536	1361	2	_	_	NNP
38536	1361	3	a	a	DT
38536	1361	4	_	_	NN
38536	1361	5	)	)	-RRB-
38536	1361	6	Find	find	VB
38536	1361	7	the	the	DT
38536	1361	8	length	length	NN
38536	1361	9	of	of	IN
38536	1361	10	a	a	DT
38536	1361	11	diagonal	diagonal	JJ
38536	1361	12	of	of	IN
38536	1361	13	any	any	DT
38536	1361	14	face	face	NN
38536	1361	15	of	of	IN
38536	1361	16	each	each	DT
38536	1361	17	cube	cube	NN
38536	1361	18	.	.	.
38536	1362	1	(	(	-LRB-
38536	1362	2	_	_	NNP
38536	1362	3	b	b	NNP
38536	1362	4	_	_	NNP
38536	1362	5	)	)	-RRB-
38536	1362	6	Find	find	VB
38536	1362	7	the	the	DT
38536	1362	8	distance	distance	NN
38536	1362	9	from	from	IN
38536	1362	10	upper	upper	JJ
38536	1362	11	left	left	JJ
38536	1362	12	-	-	HYPH
38536	1362	13	hand	hand	NN
38536	1362	14	corner	corner	NN
38536	1362	15	to	to	TO
38536	1362	16	lower	low	JJR
38536	1362	17	right	right	JJ
38536	1362	18	-	-	HYPH
38536	1362	19	hand	hand	NN
38536	1362	20	corner	corner	NN
38536	1362	21	in	in	IN
38536	1362	22	either	either	DT
38536	1362	23	cube	cube	NN
38536	1362	24	.	.	.
38536	1363	1	5	5	CD
38536	1363	2	.	.	.
38536	1364	1	A	a	NN
38536	1364	2	and	and	CC
38536	1364	3	B	b	NN
38536	1364	4	run	run	NN
38536	1364	5	a	a	DT
38536	1364	6	mile	mile	NN
38536	1364	7	.	.	.
38536	1365	1	In	in	IN
38536	1365	2	the	the	DT
38536	1365	3	first	first	JJ
38536	1365	4	heat	heat	NN
38536	1365	5	A	a	NN
38536	1365	6	gives	give	VBZ
38536	1365	7	B	B	NNP
38536	1365	8	a	a	DT
38536	1365	9	start	start	NN
38536	1365	10	of	of	IN
38536	1365	11	20	20	CD
38536	1365	12	yards	yard	NNS
38536	1365	13	and	and	CC
38536	1365	14	beats	beat	VBZ
38536	1365	15	him	-PRON-	PRP
38536	1365	16	by	by	IN
38536	1365	17	30	30	CD
38536	1365	18	seconds	second	NNS
38536	1365	19	.	.	.
38536	1366	1	In	in	IN
38536	1366	2	the	the	DT
38536	1366	3	second	second	JJ
38536	1366	4	heat	heat	NN
38536	1366	5	A	A	NNP
38536	1366	6	gives	give	VBZ
38536	1366	7	B	B	NNP
38536	1366	8	a	a	DT
38536	1366	9	start	start	NN
38536	1366	10	of	of	IN
38536	1366	11	32	32	CD
38536	1366	12	seconds	second	NNS
38536	1366	13	and	and	CC
38536	1366	14	beats	beat	VBZ
38536	1366	15	him	-PRON-	PRP
38536	1366	16	by	by	IN
38536	1366	17	9	9	CD
38536	1366	18	-	-	SYM
38536	1366	19	5/11	5/11	CD
38536	1366	20	yards	yard	NNS
38536	1366	21	.	.	.
38536	1367	1	Find	find	VB
38536	1367	2	the	the	DT
38536	1367	3	rate	rate	NN
38536	1367	4	at	at	IN
38536	1367	5	which	which	WDT
38536	1367	6	each	each	DT
38536	1367	7	runs	run	VBZ
38536	1367	8	.	.	.
38536	1368	1	(	(	-LRB-
38536	1368	2	_	_	NNP
38536	1368	3	Sheffield	Sheffield	NNP
38536	1368	4	.	.	.
38536	1368	5	_	_	NNP
38536	1368	6	)	)	-RRB-
38536	1368	7	6	6	CD
38536	1368	8	.	.	.
38536	1369	1	After	after	IN
38536	1369	2	street	street	NN
38536	1369	3	improvement	improvement	NN
38536	1369	4	it	-PRON-	PRP
38536	1369	5	is	be	VBZ
38536	1369	6	found	find	VBN
38536	1369	7	that	that	IN
38536	1369	8	a	a	DT
38536	1369	9	certain	certain	JJ
38536	1369	10	corner	corner	NN
38536	1369	11	rectangular	rectangular	JJ
38536	1369	12	lot	lot	NN
38536	1369	13	has	have	VBZ
38536	1369	14	lost	lose	VBN
38536	1369	15	1/10	1/10	CD
38536	1369	16	of	of	IN
38536	1369	17	its	-PRON-	PRP$
38536	1369	18	length	length	NN
38536	1369	19	and	and	CC
38536	1369	20	1/15	1/15	CD
38536	1369	21	of	of	IN
38536	1369	22	its	-PRON-	PRP$
38536	1369	23	width	width	NN
38536	1369	24	.	.	.
38536	1370	1	Its	-PRON-	PRP$
38536	1370	2	perimeter	perimeter	NN
38536	1370	3	has	have	VBZ
38536	1370	4	been	be	VBN
38536	1370	5	decreased	decrease	VBN
38536	1370	6	by	by	IN
38536	1370	7	28	28	CD
38536	1370	8	feet	foot	NNS
38536	1370	9	,	,	,
38536	1370	10	and	and	CC
38536	1370	11	the	the	DT
38536	1370	12	new	new	JJ
38536	1370	13	area	area	NN
38536	1370	14	is	be	VBZ
38536	1370	15	3024	3024	CD
38536	1370	16	square	square	JJ
38536	1370	17	feet	foot	NNS
38536	1370	18	.	.	.
38536	1371	1	Find	find	VB
38536	1371	2	the	the	DT
38536	1371	3	reduced	reduced	JJ
38536	1371	4	dimensions	dimension	NNS
38536	1371	5	of	of	IN
38536	1371	6	the	the	DT
38536	1371	7	lot	lot	NN
38536	1371	8	.	.	.
38536	1372	1	(	(	-LRB-
38536	1372	2	_	_	NNP
38536	1372	3	College	College	NNP
38536	1372	4	Entrance	Entrance	NNP
38536	1372	5	Board	Board	NNP
38536	1372	6	.	.	.
38536	1372	7	_	_	NNP
38536	1372	8	)	)	-RRB-
38536	1372	9	7	7	CD
38536	1372	10	.	.	.
38536	1373	1	A	a	DT
38536	1373	2	man	man	NN
38536	1373	3	spends	spend	VBZ
38536	1373	4	$	$	$
38536	1373	5	539	539	CD
38536	1373	6	for	for	IN
38536	1373	7	sheep	sheep	NN
38536	1373	8	.	.	.
38536	1374	1	He	-PRON-	PRP
38536	1374	2	keeps	keep	VBZ
38536	1374	3	14	14	CD
38536	1374	4	of	of	IN
38536	1374	5	the	the	DT
38536	1374	6	flock	flock	NN
38536	1374	7	that	that	WDT
38536	1374	8	he	-PRON-	PRP
38536	1374	9	buys	buy	VBZ
38536	1374	10	,	,	,
38536	1374	11	and	and	CC
38536	1374	12	sells	sell	VBZ
38536	1374	13	the	the	DT
38536	1374	14	remainder	remainder	NN
38536	1374	15	at	at	IN
38536	1374	16	an	an	DT
38536	1374	17	advance	advance	NN
38536	1374	18	of	of	IN
38536	1374	19	$	$	$
38536	1374	20	2	2	CD
38536	1374	21	per	per	IN
38536	1374	22	head	head	NN
38536	1374	23	,	,	,
38536	1374	24	gaining	gain	VBG
38536	1374	25	$	$	$
38536	1374	26	28	28	CD
38536	1374	27	by	by	IN
38536	1374	28	the	the	DT
38536	1374	29	transaction	transaction	NN
38536	1374	30	.	.	.
38536	1375	1	How	how	WRB
38536	1375	2	many	many	JJ
38536	1375	3	sheep	sheep	NNS
38536	1375	4	did	do	VBD
38536	1375	5	he	-PRON-	PRP
38536	1375	6	buy	buy	VB
38536	1375	7	,	,	,
38536	1375	8	and	and	CC
38536	1375	9	what	what	WP
38536	1375	10	was	be	VBD
38536	1375	11	the	the	DT
38536	1375	12	cost	cost	NN
38536	1375	13	of	of	IN
38536	1375	14	each	each	DT
38536	1375	15	?	?	.
38536	1376	1	(	(	-LRB-
38536	1376	2	_	_	NNP
38536	1376	3	Yale	Yale	NNP
38536	1376	4	.	.	.
38536	1376	5	_	_	NNP
38536	1376	6	)	)	-RRB-
38536	1376	7	8	8	CD
38536	1376	8	.	.	.
38536	1377	1	A	a	DT
38536	1377	2	boat	boat	NN
38536	1377	3	's	's	POS
38536	1377	4	crew	crew	NN
38536	1377	5	,	,	,
38536	1377	6	rowing	row	VBG
38536	1377	7	at	at	IN
38536	1377	8	half	half	PDT
38536	1377	9	their	-PRON-	PRP$
38536	1377	10	usual	usual	JJ
38536	1377	11	speed	speed	NN
38536	1377	12	,	,	,
38536	1377	13	row	row	VB
38536	1377	14	3	3	CD
38536	1377	15	miles	mile	NNS
38536	1377	16	downstream	downstream	JJ
38536	1377	17	and	and	CC
38536	1377	18	back	back	RB
38536	1377	19	again	again	RB
38536	1377	20	in	in	IN
38536	1377	21	2	2	CD
38536	1377	22	hours	hour	NNS
38536	1377	23	and	and	CC
38536	1377	24	40	40	CD
38536	1377	25	minutes	minute	NNS
38536	1377	26	.	.	.
38536	1378	1	At	at	IN
38536	1378	2	full	full	JJ
38536	1378	3	speed	speed	NN
38536	1378	4	they	-PRON-	PRP
38536	1378	5	can	can	MD
38536	1378	6	go	go	VB
38536	1378	7	over	over	IN
38536	1378	8	the	the	DT
38536	1378	9	same	same	JJ
38536	1378	10	course	course	NN
38536	1378	11	in	in	IN
38536	1378	12	1	1	CD
38536	1378	13	hour	hour	NN
38536	1378	14	and	and	CC
38536	1378	15	4	4	CD
38536	1378	16	minutes	minute	NNS
38536	1378	17	.	.	.
38536	1379	1	Find	find	VB
38536	1379	2	the	the	DT
38536	1379	3	rate	rate	NN
38536	1379	4	of	of	IN
38536	1379	5	the	the	DT
38536	1379	6	crew	crew	NN
38536	1379	7	,	,	,
38536	1379	8	and	and	CC
38536	1379	9	the	the	DT
38536	1379	10	rate	rate	NN
38536	1379	11	of	of	IN
38536	1379	12	the	the	DT
38536	1379	13	current	current	NN
38536	1379	14	in	in	IN
38536	1379	15	miles	mile	NNS
38536	1379	16	per	per	IN
38536	1379	17	hour	hour	NN
38536	1379	18	.	.	.
38536	1380	1	(	(	-LRB-
38536	1380	2	_	_	NNP
38536	1380	3	College	College	NNP
38536	1380	4	Entrance	Entrance	NNP
38536	1380	5	Board	Board	NNP
38536	1380	6	.	.	.
38536	1380	7	_	_	NNP
38536	1380	8	)	)	-RRB-
38536	1380	9	9	9	CD
38536	1380	10	.	.	.
38536	1381	1	Find	find	VB
38536	1381	2	the	the	DT
38536	1381	3	sides	side	NNS
38536	1381	4	of	of	IN
38536	1381	5	a	a	DT
38536	1381	6	rectangle	rectangle	NN
38536	1381	7	whose	whose	WP$
38536	1381	8	area	area	NN
38536	1381	9	is	be	VBZ
38536	1381	10	unchanged	unchanged	JJ
38536	1381	11	if	if	IN
38536	1381	12	its	-PRON-	PRP$
38536	1381	13	length	length	NN
38536	1381	14	is	be	VBZ
38536	1381	15	increased	increase	VBN
38536	1381	16	by	by	IN
38536	1381	17	4	4	CD
38536	1381	18	feet	foot	NNS
38536	1381	19	and	and	CC
38536	1381	20	its	-PRON-	PRP$
38536	1381	21	breadth	breadth	NN
38536	1381	22	decreased	decrease	VBD
38536	1381	23	by	by	IN
38536	1381	24	3	3	CD
38536	1381	25	feet	foot	NNS
38536	1381	26	,	,	,
38536	1381	27	but	but	CC
38536	1381	28	which	which	WDT
38536	1381	29	loses	lose	VBZ
38536	1381	30	one	one	CD
38536	1381	31	third	third	NN
38536	1381	32	of	of	IN
38536	1381	33	its	-PRON-	PRP$
38536	1381	34	area	area	NN
38536	1381	35	if	if	IN
38536	1381	36	the	the	DT
38536	1381	37	length	length	NN
38536	1381	38	is	be	VBZ
38536	1381	39	increased	increase	VBN
38536	1381	40	by	by	IN
38536	1381	41	16	16	CD
38536	1381	42	feet	foot	NNS
38536	1381	43	and	and	CC
38536	1381	44	the	the	DT
38536	1381	45	breadth	breadth	NN
38536	1381	46	decreased	decrease	VBD
38536	1381	47	by	by	IN
38536	1381	48	10	10	CD
38536	1381	49	feet	foot	NNS
38536	1381	50	.	.	.
38536	1382	1	(	(	-LRB-
38536	1382	2	_	_	NNP
38536	1382	3	M.	M.	NNP
38536	1383	1	I.	I.	NNP
38536	1383	2	T.	T.	NNP
38536	1383	3	_	_	NNP
38536	1383	4	)	)	-RRB-
38536	1383	5	COLLEGE	COLLEGE	NNP
38536	1383	6	ENTRANCE	entrance	NN
38536	1383	7	EXAMINATIONS	examination	VBZ
38536	1383	8	~UNIVERSITY	~UNIVERSITY	NFP
38536	1383	9	OF	of	IN
38536	1383	10	CALIFORNIA~	CALIFORNIA~	NNP
38536	1383	11	ELEMENTARY	ELEMENTARY	NNP
38536	1383	12	ALGEBRA	ALGEBRA	NNP
38536	1383	13	1	1	CD
38536	1383	14	.	.	.
38536	1384	1	If	if	IN
38536	1384	2	a	a	DT
38536	1384	3	=	=	SYM
38536	1384	4	4	4	CD
38536	1384	5	,	,	,
38536	1384	6	b	b	NNP
38536	1384	7	=	=	SYM
38536	1384	8	-3	-3	NNP
38536	1384	9	,	,	,
38536	1384	10	c	c	NN
38536	1384	11	=	=	SYM
38536	1384	12	2	2	CD
38536	1384	13	,	,	,
38536	1384	14	and	and	CC
38536	1384	15	d	d	LS
38536	1384	16	=	=	SYM
38536	1384	17	-4	-4	:
38536	1384	18	,	,	,
38536	1384	19	find	find	VB
38536	1384	20	the	the	DT
38536	1384	21	value	value	NN
38536	1384	22	of	of	IN
38536	1384	23	:	:	:
38536	1384	24	(	(	-LRB-
38536	1384	25	_	_	NNP
38536	1384	26	a	a	DT
38536	1384	27	_	_	NNP
38536	1384	28	)	)	-RRB-
38536	1384	29	ab^3	ab^3	ADD
38536	1384	30	-	-	:
38536	1384	31	3cd^2	3cd^2	CD
38536	1384	32	+	+	SYM
38536	1384	33	2(3a	2(3a	CD
38536	1384	34	-	-	HYPH
38536	1384	35	b)(c	b)(c	NN
38536	1384	36	-	-	HYPH
38536	1384	37	2d	2d	NNP
38536	1384	38	)	)	-RRB-
38536	1384	39	.	.	.
38536	1385	1	(	(	-LRB-
38536	1385	2	_	_	NNP
38536	1385	3	b	b	NNP
38536	1385	4	_	_	NNP
38536	1385	5	)	)	-RRB-
38536	1385	6	2a^3	2a^3	CD
38536	1385	7	-	-	SYM
38536	1385	8	3b^4	3b^4	CD
38536	1385	9	+	+	CD
38536	1385	10	(	(	-LRB-
38536	1385	11	4c^3	4c^3	CD
38536	1385	12	+	+	SYM
38536	1385	13	d^3)(4c^2	d^3)(4c^2	CD
38536	1385	14	+	+	CC
38536	1385	15	d^2	d^2	NN
38536	1385	16	)	)	-RRB-
38536	1385	17	.	.	.
38536	1386	1	2	2	LS
38536	1386	2	.	.	.
38536	1387	1	Reduce	reduce	VB
38536	1387	2	to	to	IN
38536	1387	3	a	a	DT
38536	1387	4	mixed	mixed	JJ
38536	1387	5	number	number	NN
38536	1387	6	:	:	:
38536	1387	7	(	(	-LRB-
38536	1387	8	3a^4	3a^4	CD
38536	1387	9	-	-	HYPH
38536	1387	10	4a^3	4a^3	CD
38536	1387	11	-	-	HYPH
38536	1387	12	10a^2	10a^2	CD
38536	1387	13	+	+	SYM
38536	1387	14	41a	41a	CD
38536	1387	15	-	-	HYPH
38536	1387	16	28)/(a^2	28)/(a^2	CD
38536	1387	17	-	-	HYPH
38536	1387	18	3a	3a	CD
38536	1387	19	+	+	SYM
38536	1387	20	4	4	CD
38536	1387	21	)	)	-RRB-
38536	1387	22	.	.	.
38536	1388	1	Simplify	simplify	NN
38536	1388	2	:	:	:
38536	1388	3	3	3	CD
38536	1388	4	.	.	.
38536	1389	1	(	(	-LRB-
38536	1389	2	a	a	DT
38536	1389	3	+	+	$
38536	1389	4	2)/(a^2	2)/(a^2	CD
38536	1389	5	+	+	SYM
38536	1389	6	3a	3a	CD
38536	1389	7	-	-	HYPH
38536	1389	8	40	40	CD
38536	1389	9	)	)	-RRB-
38536	1389	10	-	-	:
38536	1389	11	(	(	-LRB-
38536	1389	12	b	b	NN
38536	1389	13	-	-	HYPH
38536	1389	14	2)/(ab	2)/(ab	CD
38536	1389	15	-	-	HYPH
38536	1389	16	5b	5b	NN
38536	1389	17	+	+	SYM
38536	1389	18	3a	3a	NNP
38536	1389	19	-	-	HYPH
38536	1389	20	15	15	CD
38536	1389	21	)	)	-RRB-
38536	1389	22	.	.	.
38536	1390	1	4	4	LS
38536	1390	2	.	.	.
38536	1391	1	[	[	-LRB-
38536	1391	2	1	1	CD
38536	1391	3	-	-	HYPH
38536	1391	4	(	(	-LRB-
38536	1391	5	2	2	CD
38536	1391	6	-	-	HYPH
38536	1391	7	3b	3b	JJ
38536	1391	8	-	-	HYPH
38536	1391	9	2c)/(a	2c)/(a	CD
38536	1391	10	+	+	SYM
38536	1391	11	2	2	CD
38536	1391	12	)	)	-RRB-
38536	1391	13	]	]	-RRB-
38536	1391	14	÷	÷	NNP
38536	1391	15	(	(	-LRB-
38536	1391	16	a^2	a^2	CD
38536	1391	17	-	-	HYPH
38536	1391	18	4c^2	4c^2	CD
38536	1391	19	+	+	CC
38536	1391	20	9b^2	9b^2	CD
38536	1391	21	+	+	SYM
38536	1391	22	6ab)/(2a^2	6ab)/(2a^2	NN
38536	1391	23	+	+	SYM
38536	1391	24	a	a	DT
38536	1391	25	-	-	SYM
38536	1391	26	6	6	CD
38536	1391	27	)	)	-RRB-
38536	1391	28	.	.	.
38536	1392	1	5	5	CD
38536	1392	2	.	.	.
38536	1393	1	A	a	DT
38536	1393	2	's	's	POS
38536	1393	3	age	age	NN
38536	1393	4	10	10	CD
38536	1393	5	years	year	NNS
38536	1393	6	hence	hence	RB
38536	1393	7	will	will	MD
38536	1393	8	be	be	VB
38536	1393	9	4	4	CD
38536	1393	10	times	time	NNS
38536	1393	11	what	what	WP
38536	1393	12	B	B	NNP
38536	1393	13	's	's	POS
38536	1393	14	age	age	NN
38536	1393	15	was	be	VBD
38536	1393	16	11	11	CD
38536	1393	17	years	year	NNS
38536	1393	18	ago	ago	RB
38536	1393	19	,	,	,
38536	1393	20	and	and	CC
38536	1393	21	the	the	DT
38536	1393	22	amount	amount	NN
38536	1393	23	that	that	WDT
38536	1393	24	A	A	NNP
38536	1393	25	's	's	POS
38536	1393	26	age	age	NN
38536	1393	27	exceeds	exceed	VBZ
38536	1393	28	B	b	NN
38536	1393	29	's	's	POS
38536	1393	30	age	age	NN
38536	1393	31	is	be	VBZ
38536	1393	32	one	one	CD
38536	1393	33	third	third	NN
38536	1393	34	of	of	IN
38536	1393	35	the	the	DT
38536	1393	36	sum	sum	NN
38536	1393	37	of	of	IN
38536	1393	38	their	-PRON-	PRP$
38536	1393	39	ages	age	NNS
38536	1393	40	8	8	CD
38536	1393	41	years	year	NNS
38536	1393	42	ago	ago	RB
38536	1393	43	.	.	.
38536	1394	1	Find	find	VB
38536	1394	2	their	-PRON-	PRP$
38536	1394	3	present	present	JJ
38536	1394	4	ages	age	NNS
38536	1394	5	.	.	.
38536	1395	1	6	6	CD
38536	1395	2	.	.	.
38536	1396	1	Draw	draw	VB
38536	1396	2	the	the	DT
38536	1396	3	lines	line	NNS
38536	1396	4	represented	represent	VBN
38536	1396	5	by	by	IN
38536	1396	6	the	the	DT
38536	1396	7	equations	equation	NNS
38536	1396	8	3x	3x	CD
38536	1396	9	-	-	HYPH
38536	1396	10	2y	2y	CD
38536	1396	11	=	=	SYM
38536	1396	12	13	13	CD
38536	1396	13	and	and	CC
38536	1396	14	2x	2x	CD
38536	1396	15	+	+	SYM
38536	1396	16	5y	5y	NNS
38536	1396	17	=	=	SYM
38536	1396	18	-4	-4	:
38536	1396	19	,	,	,
38536	1396	20	and	and	CC
38536	1396	21	find	find	VB
38536	1396	22	by	by	IN
38536	1396	23	algebra	algebra	NN
38536	1396	24	the	the	DT
38536	1396	25	coördinates	coördinate	NNS
38536	1396	26	of	of	IN
38536	1396	27	the	the	DT
38536	1396	28	point	point	NN
38536	1396	29	where	where	WRB
38536	1396	30	they	-PRON-	PRP
38536	1396	31	intersect	intersect	VBP
38536	1396	32	.	.	.
38536	1397	1	7	7	LS
38536	1397	2	.	.	.
38536	1398	1	Solve	solve	VB
38536	1398	2	the	the	DT
38536	1398	3	equations	equation	NNS
38536	1398	4	{	{	-LRB-
38536	1398	5	bx	bx	NN
38536	1398	6	-	-	HYPH
38536	1398	7	ay	ay	NNS
38536	1398	8	=	=	SYM
38536	1398	9	b^2	b^2	NNP
38536	1398	10	-	-	HYPH
38536	1398	11	ab	ab	NNP
38536	1398	12	,	,	,
38536	1398	13	{	{	-LRB-
38536	1398	14	y	y	NNP
38536	1398	15	-	-	HYPH
38536	1398	16	b	b	NNP
38536	1398	17	=	=	SYM
38536	1398	18	2(x	2(x	CD
38536	1398	19	-	-	HYPH
38536	1398	20	2a	2a	CD
38536	1398	21	)	)	-RRB-
38536	1398	22	.	.	.
38536	1399	1	8	8	LS
38536	1399	2	.	.	.
38536	1400	1	Solve	solve	VB
38536	1400	2	(	(	-LRB-
38536	1400	3	2x	2x	CD
38536	1400	4	+	+	SYM
38536	1400	5	1)(3x	1)(3x	CD
38536	1400	6	-	-	HYPH
38536	1400	7	2	2	CD
38536	1400	8	)	)	-RRB-
38536	1400	9	-	-	,
38536	1400	10	(	(	-LRB-
38536	1400	11	5x	5x	CD
38536	1400	12	-	-	HYPH
38536	1400	13	7)(x	7)(x	CD
38536	1400	14	-	-	HYPH
38536	1400	15	2	2	CD
38536	1400	16	)	)	-RRB-
38536	1400	17	=	=	SYM
38536	1400	18	41	41	CD
38536	1400	19	.	.	.
38536	1401	1	~COLORADO	~COLORADO	NFP
38536	1401	2	SCHOOL	SCHOOL	NNP
38536	1401	3	OF	of	IN
38536	1401	4	MINES~	MINES~	NNP
38536	1401	5	ELEMENTARY	ELEMENTARY	NNP
38536	1401	6	ALGEBRA	ALGEBRA	NNP
38536	1401	7	1	1	CD
38536	1401	8	.	.	.
38536	1402	1	Solve	solve	VB
38536	1402	2	by	by	IN
38536	1402	3	factoring	factor	VBG
38536	1402	4	:	:	:
38536	1402	5	x^3	x^3	NNS
38536	1402	6	+	+	SYM
38536	1402	7	30x	30x	NNS
38536	1402	8	=	=	SYM
38536	1402	9	11x^2	11x^2	CD
38536	1402	10	.	.	.
38536	1403	1	2	2	LS
38536	1403	2	.	.	.
38536	1404	1	Show	show	VB
38536	1404	2	that	that	IN
38536	1404	3	1	1	CD
38536	1404	4	-	-	HYPH
38536	1404	5	[	[	-LRB-
38536	1404	6	(	(	-LRB-
38536	1404	7	a^2	a^2	NNS
38536	1404	8	+	+	DT
38536	1404	9	b^2	b^2	NNS
38536	1404	10	-	-	:
38536	1404	11	c^2)/(2ab)]^2	c^2)/(2ab)]^2	NNP
38536	1404	12	=	=	NFP
38536	1404	13	(	(	-LRB-
38536	1404	14	a	a	DT
38536	1404	15	+	+	SYM
38536	1404	16	b	b	NN
38536	1404	17	+	+	CC
38536	1404	18	c)(a	c)(a	NNP
38536	1404	19	+	+	CC
38536	1404	20	b	b	NNP
38536	1404	21	-	-	HYPH
38536	1404	22	c)(a	c)(a	NNP
38536	1404	23	-	-	HYPH
38536	1404	24	b	b	NNP
38536	1404	25	+	+	CC
38536	1404	26	c)(b	c)(b	NN
38536	1404	27	+	+	SYM
38536	1404	28	c	c	NN
38536	1404	29	-	-	:
38536	1404	30	a	a	DT
38536	1404	31	)	)	-RRB-
38536	1404	32	÷	÷	NNP
38536	1404	33	4a^2b^2	4a^2b^2	CD
38536	1404	34	.	.	.
38536	1405	1	3	3	LS
38536	1405	2	.	.	.
38536	1406	1	How	how	WRB
38536	1406	2	many	many	JJ
38536	1406	3	pairs	pair	NNS
38536	1406	4	of	of	IN
38536	1406	5	numbers	number	NNS
38536	1406	6	will	will	MD
38536	1406	7	satisfy	satisfy	VB
38536	1406	8	simultaneously	simultaneously	RB
38536	1406	9	the	the	DT
38536	1406	10	two	two	CD
38536	1406	11	equations	equation	NNS
38536	1406	12	{	{	-LRB-
38536	1406	13	3x	3x	CD
38536	1406	14	+	+	SYM
38536	1406	15	2y	2y	CD
38536	1406	16	=	=	SYM
38536	1406	17	7	7	CD
38536	1406	18	,	,	,
38536	1406	19	{	{	-LRB-
38536	1406	20	x	x	SYM
38536	1406	21	+	+	SYM
38536	1406	22	y	y	NN
38536	1406	23	=	=	SYM
38536	1406	24	3	3	CD
38536	1406	25	?	?	.
38536	1407	1	Show	show	VB
38536	1407	2	by	by	IN
38536	1407	3	means	mean	NNS
38536	1407	4	of	of	IN
38536	1407	5	a	a	DT
38536	1407	6	graph	graph	NN
38536	1407	7	that	that	WDT
38536	1407	8	your	-PRON-	PRP$
38536	1407	9	answer	answer	NN
38536	1407	10	is	be	VBZ
38536	1407	11	correct	correct	JJ
38536	1407	12	.	.	.
38536	1408	1	What	what	WP
38536	1408	2	is	be	VBZ
38536	1408	3	meant	mean	VBN
38536	1408	4	by	by	IN
38536	1408	5	eliminating	eliminate	VBG
38536	1408	6	x	x	NNS
38536	1408	7	in	in	IN
38536	1408	8	the	the	DT
38536	1408	9	above	above	JJ
38536	1408	10	equations	equation	NNS
38536	1408	11	by	by	IN
38536	1408	12	substitution	substitution	NN
38536	1408	13	?	?	.
38536	1409	1	by	by	IN
38536	1409	2	comparison	comparison	NN
38536	1409	3	?	?	.
38536	1410	1	by	by	IN
38536	1410	2	subtraction	subtraction	NN
38536	1410	3	?	?	.
38536	1411	1	4	4	LS
38536	1411	2	.	.	.
38536	1412	1	Find	find	VB
38536	1412	2	the	the	DT
38536	1412	3	square	square	JJ
38536	1412	4	root	root	NN
38536	1412	5	of	of	IN
38536	1412	6	223,728	223,728	CD
38536	1412	7	.	.	.
38536	1413	1	5	5	CD
38536	1413	2	.	.	.
38536	1414	1	Simplify	simplify	NN
38536	1414	2	:	:	:
38536	1414	3	(	(	-LRB-
38536	1414	4	_	_	NNP
38536	1414	5	a	a	DT
38536	1414	6	_	_	NNP
38536	1414	7	)	)	-RRB-
38536	1414	8	[	[	-LRB-
38536	1414	9	1/3]^(1/2	1/3]^(1/2	NNP
38536	1414	10	)	)	-RRB-
38536	1414	11	+	+	CC
38536	1414	12	[	[	-LRB-
38536	1414	13	12]^(1/2	12]^(1/2	CD
38536	1414	14	)	)	-RRB-
38536	1414	15	-	-	:
38536	1414	16	[	[	-LRB-
38536	1414	17	3/4]^(1/2	3/4]^(1/2	NNP
38536	1414	18	)	)	-RRB-
38536	1414	19	.	.	.
38536	1415	1	(	(	-LRB-
38536	1415	2	_	_	NNP
38536	1415	3	b	b	NNP
38536	1415	4	_	_	NNP
38536	1415	5	)	)	-RRB-
38536	1415	6	(	(	-LRB-
38536	1415	7	-[-3[-4]^(1/2)]^(1/2))^4	-[-3[-4]^(1/2)]^(1/2))^4	:
38536	1415	8	.	.	.
38536	1416	1	6	6	CD
38536	1416	2	.	.	.
38536	1417	1	Solve	solve	VB
38536	1417	2	the	the	DT
38536	1417	3	equation	equation	NN
38536	1417	4	.03x^2	.03x^2	NNP
38536	1417	5	-	-	HYPH
38536	1417	6	2.23x	2.23x	CD
38536	1417	7	+	+	SYM
38536	1417	8	1.1075	1.1075	CD
38536	1417	9	=	=	SYM
38536	1417	10	0	0	CD
38536	1417	11	.	.	.
38536	1418	1	7	7	LS
38536	1418	2	.	.	.
38536	1419	1	How	how	WRB
38536	1419	2	far	far	RB
38536	1419	3	must	must	MD
38536	1419	4	a	a	DT
38536	1419	5	boy	boy	NN
38536	1419	6	run	run	VB
38536	1419	7	in	in	IN
38536	1419	8	a	a	DT
38536	1419	9	potato	potato	NN
38536	1419	10	race	race	NN
38536	1419	11	if	if	IN
38536	1419	12	there	there	EX
38536	1419	13	are	be	VBP
38536	1419	14	n	n	IN
38536	1419	15	potatoes	potato	NNS
38536	1419	16	in	in	IN
38536	1419	17	a	a	DT
38536	1419	18	straight	straight	JJ
38536	1419	19	line	line	NN
38536	1419	20	at	at	IN
38536	1419	21	a	a	DT
38536	1419	22	distance	distance	NN
38536	1419	23	d	d	CD
38536	1419	24	feet	foot	NNS
38536	1419	25	apart	apart	RB
38536	1419	26	,	,	,
38536	1419	27	the	the	DT
38536	1419	28	first	first	JJ
38536	1419	29	being	being	NN
38536	1419	30	at	at	IN
38536	1419	31	a	a	DT
38536	1419	32	distance	distance	NN
38536	1419	33	a	a	DT
38536	1419	34	feet	foot	NNS
38536	1419	35	from	from	IN
38536	1419	36	the	the	DT
38536	1419	37	basket	basket	NN
38536	1419	38	?	?	.
38536	1420	1	~COLUMBIA	~columbia	DT
38536	1420	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1420	3	ELEMENTARY	ELEMENTARY	NNP
38536	1420	4	ALGEBRA	ALGEBRA	NNP
38536	1420	5	COMPLETE	COMPLETE	NNP
38536	1420	6	TIME	time	NN
38536	1420	7	:	:	:
38536	1420	8	THREE	three	CD
38536	1420	9	HOURS	hours	NN
38536	1420	10	Six	six	CD
38536	1420	11	questions	question	NNS
38536	1420	12	are	be	VBP
38536	1420	13	required	require	VBN
38536	1420	14	;	;	:
38536	1420	15	two	two	CD
38536	1420	16	from	from	IN
38536	1420	17	Group	Group	NNP
38536	1420	18	_	_	NNP
38536	1420	19	A	A	NNP
38536	1420	20	_	_	NNP
38536	1420	21	,	,	,
38536	1420	22	two	two	CD
38536	1420	23	from	from	IN
38536	1420	24	Group	Group	NNP
38536	1420	25	_	_	NNP
38536	1420	26	B	B	NNP
38536	1420	27	_	_	NNP
38536	1420	28	,	,	,
38536	1420	29	and	and	CC
38536	1420	30	both	both	DT
38536	1420	31	questions	question	NNS
38536	1420	32	of	of	IN
38536	1420	33	Group	Group	NNP
38536	1420	34	_	_	NNP
38536	1420	35	C	C	NNP
38536	1420	36	_	_	NNP
38536	1420	37	.	.	.
38536	1421	1	No	no	DT
38536	1421	2	extra	extra	JJ
38536	1421	3	credit	credit	NN
38536	1421	4	will	will	MD
38536	1421	5	be	be	VB
38536	1421	6	given	give	VBN
38536	1421	7	for	for	IN
38536	1421	8	more	more	JJR
38536	1421	9	than	than	IN
38536	1421	10	six	six	CD
38536	1421	11	questions	question	NNS
38536	1421	12	.	.	.
38536	1422	1	_	_	NNP
38536	1422	2	Group	Group	NNP
38536	1422	3	A	A	NNP
38536	1422	4	_	_	NNP
38536	1422	5	1	1	CD
38536	1422	6	.	.	.
38536	1423	1	(	(	-LRB-
38536	1423	2	_	_	NNP
38536	1423	3	a	a	DT
38536	1423	4	_	_	NNP
38536	1423	5	)	)	-RRB-
38536	1423	6	Resolve	resolve	VB
38536	1423	7	the	the	DT
38536	1423	8	following	following	NN
38536	1423	9	into	into	IN
38536	1423	10	their	-PRON-	PRP$
38536	1423	11	prime	prime	JJ
38536	1423	12	factors	factor	NNS
38536	1423	13	:	:	:
38536	1423	14	(	(	-LRB-
38536	1423	15	1	1	CD
38536	1423	16	)	)	-RRB-
38536	1423	17	(	(	-LRB-
38536	1423	18	x^2	x^2	NNP
38536	1423	19	-	-	HYPH
38536	1423	20	y^2)^2	y^2)^2	NNP
38536	1423	21	-	-	HYPH
38536	1423	22	y^4	y^4	FW
38536	1423	23	.	.	.
38536	1424	1	(	(	-LRB-
38536	1424	2	2	2	LS
38536	1424	3	)	)	-RRB-
38536	1424	4	10x^2	10x^2	CD
38536	1424	5	-	-	HYPH
38536	1424	6	7x	7x	CD
38536	1424	7	-	-	HYPH
38536	1424	8	6	6	CD
38536	1424	9	.	.	.
38536	1425	1	(	(	-LRB-
38536	1425	2	_	_	NNP
38536	1425	3	b	b	NNP
38536	1425	4	_	_	NNP
38536	1425	5	)	)	-RRB-
38536	1425	6	Find	find	VB
38536	1425	7	the	the	DT
38536	1425	8	H.	H.	NNP
38536	1425	9	C.	C.	NNP
38536	1425	10	F.	F.	NNP
38536	1425	11	and	and	CC
38536	1425	12	the	the	DT
38536	1425	13	L.	L.	NNP
38536	1425	14	C.	C.	NNP
38536	1425	15	M.	M.	NNP
38536	1425	16	of	of	IN
38536	1425	17	x^3	x^3	NNP
38536	1425	18	-	-	HYPH
38536	1425	19	3x^2	3x^2	CD
38536	1425	20	+	+	SYM
38536	1425	21	x	x	SYM
38536	1425	22	-	-	SYM
38536	1425	23	3	3	CD
38536	1425	24	,	,	,
38536	1425	25	x^3	x^3	CD
38536	1425	26	-	-	HYPH
38536	1425	27	3x^2	3x^2	CD
38536	1425	28	-	-	HYPH
38536	1425	29	x	x	NN
38536	1425	30	+	+	CD
38536	1425	31	3	3	CD
38536	1425	32	.	.	.
38536	1426	1	2	2	LS
38536	1426	2	.	.	.
38536	1427	1	(	(	-LRB-
38536	1427	2	_	_	NNP
38536	1427	3	a	a	DT
38536	1427	4	_	_	NNP
38536	1427	5	)	)	-RRB-
38536	1427	6	Simplify	Simplify	NNP
38536	1427	7	[	[	-LRB-
38536	1427	8	x	x	NNP
38536	1427	9	/	/	SYM
38536	1427	10	y	y	NN
38536	1427	11	+	+	CC
38536	1427	12	y	y	NN
38536	1427	13	/	/	SYM
38536	1427	14	x	x	NNP
38536	1427	15	-	-	NNP
38536	1427	16	2]/[1	2]/[1	NNP
38536	1427	17	/	/	SYM
38536	1427	18	x	x	NNS
38536	1427	19	+	+	SYM
38536	1427	20	1	1	CD
38536	1427	21	/	/	SYM
38536	1427	22	y	y	NN
38536	1427	23	]	]	-RRB-
38536	1427	24	+	+	CC
38536	1427	25	[	[	-LRB-
38536	1427	26	x	x	NN
38536	1427	27	/	/	SYM
38536	1427	28	y	y	NN
38536	1427	29	+	+	CC
38536	1427	30	y	y	NN
38536	1427	31	/	/	SYM
38536	1427	32	x	x	SYM
38536	1427	33	+	+	CD
38536	1427	34	2]/[1	2]/[1	CD
38536	1427	35	/	/	SYM
38536	1427	36	x	x	NNP
38536	1427	37	-	-	SYM
38536	1427	38	1	1	CD
38536	1427	39	/	/	SYM
38536	1427	40	y	y	NNP
38536	1427	41	]	]	-RRB-
38536	1427	42	.	.	.
38536	1428	1	(	(	-LRB-
38536	1428	2	_	_	NNP
38536	1428	3	b	b	NNP
38536	1428	4	_	_	NNP
38536	1428	5	)	)	-RRB-
38536	1428	6	If	if	IN
38536	1428	7	x	x	LS
38536	1428	8	:	:	:
38536	1428	9	y	y	NNP
38536	1428	10	=	=	NFP
38536	1428	11	(	(	-LRB-
38536	1428	12	x	x	SYM
38536	1428	13	-	-	NNS
38536	1428	14	z)^2	z)^2	NNP
38536	1428	15	:	:	:
38536	1428	16	(	(	-LRB-
38536	1428	17	y	y	NNP
38536	1428	18	-	-	HYPH
38536	1428	19	z)^2	z)^2	NNP
38536	1428	20	,	,	,
38536	1428	21	prove	prove	VBP
38536	1428	22	that	that	IN
38536	1428	23	z	z	NN
38536	1428	24	is	be	VBZ
38536	1428	25	a	a	DT
38536	1428	26	mean	mean	JJ
38536	1428	27	proportional	proportional	NN
38536	1428	28	between	between	IN
38536	1428	29	x	x	NNP
38536	1428	30	and	and	CC
38536	1428	31	y.	y.	NNP
38536	1429	1	3	3	LS
38536	1429	2	.	.	.
38536	1430	1	A	a	DT
38536	1430	2	crew	crew	NN
38536	1430	3	can	can	MD
38536	1430	4	row	row	VB
38536	1430	5	10	10	CD
38536	1430	6	miles	mile	NNS
38536	1430	7	in	in	IN
38536	1430	8	50	50	CD
38536	1430	9	minutes	minute	NNS
38536	1430	10	downstream	downstream	JJ
38536	1430	11	,	,	,
38536	1430	12	and	and	CC
38536	1430	13	12	12	CD
38536	1430	14	miles	mile	NNS
38536	1430	15	in	in	IN
38536	1430	16	an	an	DT
38536	1430	17	hour	hour	NN
38536	1430	18	and	and	CC
38536	1430	19	a	a	DT
38536	1430	20	half	half	NN
38536	1430	21	upstream	upstream	NN
38536	1430	22	.	.	.
38536	1431	1	Find	find	VB
38536	1431	2	the	the	DT
38536	1431	3	rate	rate	NN
38536	1431	4	of	of	IN
38536	1431	5	the	the	DT
38536	1431	6	current	current	NN
38536	1431	7	and	and	CC
38536	1431	8	of	of	IN
38536	1431	9	the	the	DT
38536	1431	10	crew	crew	NN
38536	1431	11	in	in	IN
38536	1431	12	still	still	RB
38536	1431	13	water	water	NN
38536	1431	14	.	.	.
38536	1432	1	_	_	NNP
38536	1432	2	Group	Group	NNP
38536	1432	3	B	B	NNP
38536	1432	4	_	_	NNP
38536	1432	5	4	4	CD
38536	1432	6	.	.	.
38536	1433	1	(	(	-LRB-
38536	1433	2	_	_	NNP
38536	1433	3	a	a	DT
38536	1433	4	_	_	NNP
38536	1433	5	)	)	-RRB-
38536	1433	6	Determine	determine	VB
38536	1433	7	the	the	DT
38536	1433	8	values	value	NNS
38536	1433	9	of	of	IN
38536	1433	10	k	k	NN
38536	1433	11	so	so	IN
38536	1433	12	that	that	IN
38536	1433	13	the	the	DT
38536	1433	14	equation	equation	NN
38536	1433	15	(	(	-LRB-
38536	1433	16	2	2	CD
38536	1433	17	+	+	SYM
38536	1433	18	k)x^2	k)x^2	NN
38536	1433	19	+	+	CC
38536	1433	20	2kx	2kx	NN
38536	1433	21	+	+	SYM
38536	1433	22	1	1	CD
38536	1433	23	=	=	SYM
38536	1433	24	0	0	CD
38536	1433	25	shall	shall	MD
38536	1433	26	have	have	VB
38536	1433	27	equal	equal	JJ
38536	1433	28	roots	root	NNS
38536	1433	29	.	.	.
38536	1434	1	(	(	-LRB-
38536	1434	2	_	_	NNP
38536	1434	3	b	b	NNP
38536	1434	4	_	_	NNP
38536	1434	5	)	)	-RRB-
38536	1434	6	Solve	solve	VB
38536	1434	7	the	the	DT
38536	1434	8	equations	equation	NNS
38536	1434	9	x^2	x^2	NNP
38536	1434	10	-	-	HYPH
38536	1434	11	xy	xy	NN
38536	1434	12	+	+	CC
38536	1434	13	y^2	y^2	NN
38536	1434	14	=	=	SYM
38536	1434	15	7	7	CD
38536	1434	16	,	,	,
38536	1434	17	2x	2x	CD
38536	1434	18	-	-	HYPH
38536	1434	19	3y	3y	NNS
38536	1434	20	=	=	SYM
38536	1434	21	0	0	NFP
38536	1434	22	.	.	.
38536	1435	1	(	(	-LRB-
38536	1435	2	_	_	NNP
38536	1435	3	c	c	NNP
38536	1435	4	_	_	NNP
38536	1435	5	)	)	-RRB-
38536	1435	6	Plot	Plot	NNP
38536	1435	7	the	the	DT
38536	1435	8	following	follow	VBG
38536	1435	9	two	two	CD
38536	1435	10	equations	equation	NNS
38536	1435	11	,	,	,
38536	1435	12	and	and	CC
38536	1435	13	find	find	VB
38536	1435	14	from	from	IN
38536	1435	15	the	the	DT
38536	1435	16	graphs	graphs	NN
38536	1435	17	the	the	DT
38536	1435	18	approximate	approximate	JJ
38536	1435	19	values	value	NNS
38536	1435	20	of	of	IN
38536	1435	21	their	-PRON-	PRP$
38536	1435	22	common	common	JJ
38536	1435	23	solutions	solution	NNS
38536	1435	24	:	:	:
38536	1435	25	x^2	x^2	NNS
38536	1435	26	+	+	SYM
38536	1435	27	y^2	y^2	NNS
38536	1435	28	=	=	SYM
38536	1435	29	25	25	CD
38536	1435	30	,	,	,
38536	1435	31	4x^2	4x^2	CD
38536	1435	32	+	+	SYM
38536	1435	33	9y^2	9y^2	CD
38536	1435	34	=	=	SYM
38536	1435	35	144	144	CD
38536	1435	36	.	.	.
38536	1436	1	5	5	CD
38536	1436	2	.	.	.
38536	1437	1	Two	two	CD
38536	1437	2	integers	integer	NNS
38536	1437	3	are	be	VBP
38536	1437	4	in	in	IN
38536	1437	5	the	the	DT
38536	1437	6	ratio	ratio	NN
38536	1437	7	4	4	CD
38536	1437	8	:	:	SYM
38536	1437	9	5	5	CD
38536	1437	10	.	.	.
38536	1438	1	Increase	increase	VB
38536	1438	2	each	each	DT
38536	1438	3	by	by	IN
38536	1438	4	15	15	CD
38536	1438	5	,	,	,
38536	1438	6	and	and	CC
38536	1438	7	the	the	DT
38536	1438	8	difference	difference	NN
38536	1438	9	of	of	IN
38536	1438	10	their	-PRON-	PRP$
38536	1438	11	squares	square	NNS
38536	1438	12	is	be	VBZ
38536	1438	13	999	999	CD
38536	1438	14	.	.	.
38536	1439	1	What	what	WP
38536	1439	2	are	be	VBP
38536	1439	3	the	the	DT
38536	1439	4	integers	integer	NNS
38536	1439	5	?	?	.
38536	1440	1	6	6	CD
38536	1440	2	.	.	.
38536	1441	1	A	a	DT
38536	1441	2	man	man	NN
38536	1441	3	has	have	VBZ
38536	1441	4	$	$	$
38536	1441	5	539	539	CD
38536	1441	6	to	to	TO
38536	1441	7	spend	spend	VB
38536	1441	8	for	for	IN
38536	1441	9	sheep	sheep	NNS
38536	1441	10	.	.	.
38536	1442	1	He	-PRON-	PRP
38536	1442	2	wishes	wish	VBZ
38536	1442	3	to	to	TO
38536	1442	4	keep	keep	VB
38536	1442	5	14	14	CD
38536	1442	6	of	of	IN
38536	1442	7	the	the	DT
38536	1442	8	flock	flock	NN
38536	1442	9	that	that	WDT
38536	1442	10	he	-PRON-	PRP
38536	1442	11	buys	buy	VBZ
38536	1442	12	,	,	,
38536	1442	13	but	but	CC
38536	1442	14	to	to	TO
38536	1442	15	sell	sell	VB
38536	1442	16	the	the	DT
38536	1442	17	remainder	remainder	NN
38536	1442	18	at	at	IN
38536	1442	19	a	a	DT
38536	1442	20	gain	gain	NN
38536	1442	21	of	of	IN
38536	1442	22	$	$	$
38536	1442	23	2	2	CD
38536	1442	24	per	per	IN
38536	1442	25	head	head	NN
38536	1442	26	.	.	.
38536	1443	1	This	this	DT
38536	1443	2	he	-PRON-	PRP
38536	1443	3	does	do	VBZ
38536	1443	4	and	and	CC
38536	1443	5	gains	gain	VBZ
38536	1443	6	$	$	$
38536	1443	7	28	28	CD
38536	1443	8	.	.	.
38536	1444	1	How	how	WRB
38536	1444	2	many	many	JJ
38536	1444	3	sheep	sheep	NNS
38536	1444	4	did	do	VBD
38536	1444	5	he	-PRON-	PRP
38536	1444	6	buy	buy	VB
38536	1444	7	,	,	,
38536	1444	8	and	and	CC
38536	1444	9	at	at	IN
38536	1444	10	what	what	WDT
38536	1444	11	price	price	NN
38536	1444	12	each	each	DT
38536	1444	13	?	?	.
38536	1445	1	_	_	NNP
38536	1445	2	Group	Group	NNP
38536	1445	3	C	C	NNP
38536	1445	4	_	_	NNP
38536	1445	5	7	7	CD
38536	1445	6	.	.	.
38536	1446	1	(	(	-LRB-
38536	1446	2	_	_	NNP
38536	1446	3	a	a	DT
38536	1446	4	_	_	NN
38536	1446	5	)	)	-RRB-
38536	1446	6	Find	find	VB
38536	1446	7	the	the	DT
38536	1446	8	seventh	seventh	JJ
38536	1446	9	term	term	NN
38536	1446	10	of	of	IN
38536	1446	11	[	[	-LRB-
38536	1446	12	a	a	DT
38536	1446	13	+	+	SYM
38536	1446	14	1	1	CD
38536	1446	15	/	/	SYM
38536	1446	16	a]^(13	a]^(13	NN
38536	1446	17	)	)	-RRB-
38536	1446	18	.	.	.
38536	1447	1	(	(	-LRB-
38536	1447	2	_	_	NNP
38536	1447	3	b	b	NNP
38536	1447	4	_	_	NNP
38536	1447	5	)	)	-RRB-
38536	1447	6	Derive	derive	VB
38536	1447	7	the	the	DT
38536	1447	8	formula	formula	NN
38536	1447	9	for	for	IN
38536	1447	10	the	the	DT
38536	1447	11	sum	sum	NN
38536	1447	12	of	of	IN
38536	1447	13	n	n	CD
38536	1447	14	terms	term	NNS
38536	1447	15	of	of	IN
38536	1447	16	an	an	DT
38536	1447	17	arithmetic	arithmetic	JJ
38536	1447	18	progression	progression	NN
38536	1447	19	.	.	.
38536	1448	1	8	8	LS
38536	1448	2	.	.	.
38536	1449	1	A	a	DT
38536	1449	2	ball	ball	NN
38536	1449	3	falling	fall	VBG
38536	1449	4	from	from	IN
38536	1449	5	a	a	DT
38536	1449	6	height	height	NN
38536	1449	7	of	of	IN
38536	1449	8	60	60	CD
38536	1449	9	feet	foot	NNS
38536	1449	10	rebounds	rebound	NNS
38536	1449	11	after	after	IN
38536	1449	12	each	each	DT
38536	1449	13	fall	fall	NN
38536	1449	14	one	one	CD
38536	1449	15	third	third	NN
38536	1449	16	of	of	IN
38536	1449	17	its	-PRON-	PRP$
38536	1449	18	last	last	JJ
38536	1449	19	descent	descent	NN
38536	1449	20	.	.	.
38536	1450	1	What	what	WDT
38536	1450	2	distance	distance	NN
38536	1450	3	has	have	VBZ
38536	1450	4	it	-PRON-	PRP
38536	1450	5	passed	pass	VBN
38536	1450	6	over	over	RP
38536	1450	7	when	when	WRB
38536	1450	8	it	-PRON-	PRP
38536	1450	9	strikes	strike	VBZ
38536	1450	10	the	the	DT
38536	1450	11	ground	ground	NN
38536	1450	12	for	for	IN
38536	1450	13	the	the	DT
38536	1450	14	eighth	eighth	JJ
38536	1450	15	time	time	NN
38536	1450	16	?	?	.
38536	1451	1	~CORNELL	~CORNELL	NFP
38536	1451	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1451	3	ELEMENTARY	ELEMENTARY	NNP
38536	1451	4	ALGEBRA	ALGEBRA	NNP
38536	1451	5	1	1	CD
38536	1451	6	.	.	.
38536	1452	1	Find	find	VB
38536	1452	2	the	the	DT
38536	1452	3	H.	H.	NNP
38536	1452	4	C.	C.	NNP
38536	1452	5	F.	F.	NNP
38536	1452	6	:	:	:
38536	1452	7	x^4	x^4	NNP
38536	1452	8	-	-	HYPH
38536	1452	9	y^4	y^4	NNP
38536	1452	10	,	,	,
38536	1452	11	x^3	x^3	NNP
38536	1452	12	-	-	HYPH
38536	1452	13	xy^2	xy^2	NNP
38536	1452	14	+	+	SYM
38536	1452	15	x^2y	x^2y	NNP
38536	1452	16	-	-	HYPH
38536	1452	17	y^3	y^3	NNP
38536	1452	18	,	,	,
38536	1452	19	x^4	x^4	NNP
38536	1452	20	+	+	SYM
38536	1452	21	2x^2y^2	2x^2y^2	CD
38536	1452	22	-	-	HYPH
38536	1452	23	3y^4	3y^4	CD
38536	1452	24	.	.	.
38536	1453	1	2	2	LS
38536	1453	2	.	.	.
38536	1454	1	Solve	solve	VB
38536	1454	2	the	the	DT
38536	1454	3	following	follow	VBG
38536	1454	4	set	set	NN
38536	1454	5	of	of	IN
38536	1454	6	equations	equation	NNS
38536	1454	7	:	:	:
38536	1454	8	x	x	NNP
38536	1454	9	+	+	SYM
38536	1454	10	y	y	NN
38536	1454	11	=	=	SYM
38536	1454	12	-1	-1	NNP
38536	1454	13	,	,	,
38536	1454	14	x	x	NNS
38536	1454	15	+	+	SYM
38536	1454	16	3y	3y	CD
38536	1454	17	+	+	SYM
38536	1454	18	2z	2z	CD
38536	1454	19	=	=	SYM
38536	1454	20	-4	-4	:
38536	1454	21	,	,	,
38536	1454	22	x	x	NNP
38536	1454	23	-	-	NN
38536	1454	24	y	y	NNP
38536	1454	25	+	+	CC
38536	1454	26	4z	4z	CD
38536	1454	27	=	=	SYM
38536	1454	28	5	5	CD
38536	1454	29	.	.	.
38536	1455	1	3	3	LS
38536	1455	2	.	.	.
38536	1456	1	Expand	expand	VB
38536	1456	2	and	and	CC
38536	1456	3	simplify	simplify	VB
38536	1456	4	:	:	:
38536	1456	5	[	[	-LRB-
38536	1456	6	2x^3	2x^3	CD
38536	1456	7	-	-	SYM
38536	1456	8	1	1	CD
38536	1456	9	/	/	SYM
38536	1456	10	x]^7	x]^7	NNP
38536	1456	11	.	.	.
38536	1457	1	4	4	LS
38536	1457	2	.	.	.
38536	1458	1	An	an	DT
38536	1458	2	automobile	automobile	NN
38536	1458	3	goes	go	VBZ
38536	1458	4	80	80	CD
38536	1458	5	miles	mile	NNS
38536	1458	6	and	and	CC
38536	1458	7	back	back	RB
38536	1458	8	in	in	IN
38536	1458	9	9	9	CD
38536	1458	10	hours	hour	NNS
38536	1458	11	.	.	.
38536	1459	1	The	the	DT
38536	1459	2	rate	rate	NN
38536	1459	3	of	of	IN
38536	1459	4	speed	speed	NN
38536	1459	5	returning	return	VBG
38536	1459	6	was	be	VBD
38536	1459	7	4	4	CD
38536	1459	8	miles	mile	NNS
38536	1459	9	per	per	IN
38536	1459	10	hour	hour	NN
38536	1459	11	faster	fast	RBR
38536	1459	12	than	than	IN
38536	1459	13	the	the	DT
38536	1459	14	rate	rate	NN
38536	1459	15	going	go	VBG
38536	1459	16	.	.	.
38536	1460	1	Find	find	VB
38536	1460	2	the	the	DT
38536	1460	3	rate	rate	NN
38536	1460	4	each	each	DT
38536	1460	5	way	way	NN
38536	1460	6	.	.	.
38536	1461	1	5	5	CD
38536	1461	2	.	.	.
38536	1462	1	Simplify	simplify	NN
38536	1462	2	:	:	:
38536	1462	3	{	{	-LRB-
38536	1462	4	[	[	-LRB-
38536	1462	5	(	(	-LRB-
38536	1462	6	x	x	SYM
38536	1462	7	+	+	NNP
38536	1462	8	1)/(x	1)/(x	CD
38536	1462	9	-	-	HYPH
38536	1462	10	1)]^2	1)]^2	CD
38536	1462	11	-	-	HYPH
38536	1462	12	2	2	CD
38536	1462	13	+	+	CC
38536	1462	14	[	[	-LRB-
38536	1462	15	(	(	-LRB-
38536	1462	16	x	x	LS
38536	1462	17	-	-	NNP
38536	1462	18	1)/(x	1)/(x	CD
38536	1462	19	+	+	CC
38536	1462	20	1)]^2	1)]^2	CD
38536	1462	21	}	}	-RRB-
38536	1462	22	/{[(x	/{[(x	:
38536	1462	23	+	+	CC
38536	1462	24	1)/(x	1)/(x	CD
38536	1462	25	-	-	HYPH
38536	1462	26	1)]^2	1)]^2	CD
38536	1462	27	-	-	HYPH
38536	1462	28	[	[	-LRB-
38536	1462	29	(	(	-LRB-
38536	1462	30	x	x	SYM
38536	1462	31	-	-	NNP
38536	1462	32	1)/(x	1)/(x	CD
38536	1462	33	+	+	CC
38536	1462	34	1)]^2	1)]^2	CD
38536	1462	35	}	}	-RRB-
38536	1462	36	.	.	.
38536	1463	1	6	6	CD
38536	1463	2	.	.	.
38536	1464	1	Solve	solve	VB
38536	1464	2	for	for	IN
38536	1464	3	x	x	NNS
38536	1464	4	:	:	:
38536	1464	5	(	(	-LRB-
38536	1464	6	2x	2x	NN
38536	1464	7	+	+	SYM
38536	1464	8	3)/(x	3)/(x	CD
38536	1464	9	-	-	HYPH
38536	1464	10	1	1	CD
38536	1464	11	)	)	-RRB-
38536	1464	12	-	-	SYM
38536	1464	13	6	6	CD
38536	1464	14	=	=	SYM
38536	1464	15	5/(x^2	5/(x^2	CD
38536	1464	16	+	+	SYM
38536	1464	17	2x	2x	CD
38536	1464	18	-	-	SYM
38536	1464	19	3	3	CD
38536	1464	20	)	)	-RRB-
38536	1464	21	.	.	.
38536	1465	1	7	7	LS
38536	1465	2	.	.	.
38536	1466	1	A	a	DT
38536	1466	2	,	,	,
38536	1466	3	B	b	NN
38536	1466	4	,	,	,
38536	1466	5	and	and	CC
38536	1466	6	C	C	NNP
38536	1466	7	,	,	,
38536	1466	8	all	all	DT
38536	1466	9	working	work	VBG
38536	1466	10	together	together	RB
38536	1466	11	,	,	,
38536	1466	12	can	can	MD
38536	1466	13	do	do	VB
38536	1466	14	a	a	DT
38536	1466	15	piece	piece	NN
38536	1466	16	of	of	IN
38536	1466	17	work	work	NN
38536	1466	18	in	in	IN
38536	1466	19	2	2	CD
38536	1466	20	-	-	SYM
38536	1466	21	2/3	2/3	CD
38536	1466	22	days	day	NNS
38536	1466	23	.	.	.
38536	1467	1	A	a	DT
38536	1467	2	works	work	NNS
38536	1467	3	twice	twice	RB
38536	1467	4	as	as	RB
38536	1467	5	fast	fast	RB
38536	1467	6	as	as	IN
38536	1467	7	C	C	NNP
38536	1467	8	,	,	,
38536	1467	9	and	and	CC
38536	1467	10	A	A	NNP
38536	1467	11	and	and	CC
38536	1467	12	C	C	NNP
38536	1467	13	together	together	RB
38536	1467	14	could	could	MD
38536	1467	15	do	do	VB
38536	1467	16	the	the	DT
38536	1467	17	work	work	NN
38536	1467	18	in	in	IN
38536	1467	19	4	4	CD
38536	1467	20	days	day	NNS
38536	1467	21	.	.	.
38536	1468	1	How	how	WRB
38536	1468	2	long	long	RB
38536	1468	3	would	would	MD
38536	1468	4	it	-PRON-	PRP
38536	1468	5	take	take	VB
38536	1468	6	each	each	DT
38536	1468	7	one	one	CD
38536	1468	8	of	of	IN
38536	1468	9	the	the	DT
38536	1468	10	three	three	CD
38536	1468	11	to	to	TO
38536	1468	12	do	do	VB
38536	1468	13	the	the	DT
38536	1468	14	work	work	NN
38536	1468	15	alone	alone	RB
38536	1468	16	?	?	.
38536	1469	1	~CORNELL	~CORNELL	NFP
38536	1469	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1469	3	INTERMEDIATE	INTERMEDIATE	NNP
38536	1469	4	ALGEBRA	ALGEBRA	NNP
38536	1469	5	1	1	CD
38536	1469	6	.	.	.
38536	1470	1	Solve	solve	VB
38536	1470	2	the	the	DT
38536	1470	3	following	follow	VBG
38536	1470	4	set	set	NN
38536	1470	5	of	of	IN
38536	1470	6	equations	equation	NNS
38536	1470	7	:	:	:
38536	1470	8	x	x	NNP
38536	1470	9	+	+	SYM
38536	1470	10	y	y	NN
38536	1470	11	=	=	SYM
38536	1470	12	-1	-1	NNP
38536	1470	13	,	,	,
38536	1470	14	2z	2z	CD
38536	1470	15	+	+	SYM
38536	1470	16	5w	5w	NNP
38536	1470	17	=	=	SYM
38536	1470	18	1	1	CD
38536	1470	19	,	,	,
38536	1470	20	x	x	NNS
38536	1470	21	+	+	SYM
38536	1470	22	3y	3y	CD
38536	1470	23	+	+	SYM
38536	1470	24	2z	2z	CD
38536	1470	25	=	=	SYM
38536	1470	26	-4	-4	:
38536	1470	27	,	,	,
38536	1470	28	x	x	NNP
38536	1470	29	-	-	NN
38536	1470	30	y	y	NNP
38536	1470	31	+	+	CC
38536	1470	32	4z	4z	CD
38536	1470	33	+	+	SYM
38536	1470	34	4w	4w	NNS
38536	1470	35	=	=	SYM
38536	1470	36	5	5	CD
38536	1470	37	.	.	.
38536	1471	1	2	2	LS
38536	1471	2	.	.	.
38536	1472	1	Simplify	simplify	NN
38536	1472	2	:	:	:
38536	1472	3	(	(	-LRB-
38536	1472	4	_	_	NNP
38536	1472	5	a	a	DT
38536	1472	6	_	_	NNP
38536	1472	7	)	)	-RRB-
38536	1472	8	[	[	-LRB-
38536	1472	9	6	6	CD
38536	1472	10	-	-	SYM
38536	1472	11	20^(1/2)]^(1/2	20^(1/2)]^(1/2	CD
38536	1472	12	)	)	-RRB-
38536	1472	13	.	.	.
38536	1473	1	(	(	-LRB-
38536	1473	2	_	_	NNP
38536	1473	3	b	b	NNP
38536	1473	4	_	_	NNP
38536	1473	5	)	)	-RRB-
38536	1473	6	[	[	-LRB-
38536	1473	7	1	1	CD
38536	1473	8	+	+	CC
38536	1473	9	[	[	-LRB-
38536	1473	10	x^2	x^2	NNS
38536	1473	11	+	+	SYM
38536	1473	12	1]^(1/2)]/[1	1]^(1/2)]/[1	CD
38536	1473	13	+	+	SYM
38536	1473	14	[	[	-LRB-
38536	1473	15	x^2	x^2	NNS
38536	1473	16	+	+	SYM
38536	1473	17	1]^(1/2	1]^(1/2	NN
38536	1473	18	)	)	-RRB-
38536	1473	19	+	+	NFP
38536	1473	20	x^2	x^2	NNP
38536	1473	21	]	]	-RRB-
38536	1473	22	.	.	.
38536	1474	1	3	3	LS
38536	1474	2	.	.	.
38536	1475	1	Find	find	VB
38536	1475	2	,	,	,
38536	1475	3	and	and	CC
38536	1475	4	simplify	simplify	VB
38536	1475	5	,	,	,
38536	1475	6	the	the	DT
38536	1475	7	23d	23d	CD
38536	1475	8	term	term	NN
38536	1475	9	in	in	IN
38536	1475	10	the	the	DT
38536	1475	11	expansion	expansion	NN
38536	1475	12	of	of	IN
38536	1475	13	[	[	-LRB-
38536	1475	14	(	(	-LRB-
38536	1475	15	2x^2)/(3	2x^2)/(3	CD
38536	1475	16	)	)	-RRB-
38536	1475	17	-	-	:
38536	1475	18	3/4]^(28	3/4]^(28	CD
38536	1475	19	)	)	-RRB-
38536	1475	20	.	.	.
38536	1476	1	4	4	LS
38536	1476	2	.	.	.
38536	1477	1	The	the	DT
38536	1477	2	weight	weight	NN
38536	1477	3	of	of	IN
38536	1477	4	an	an	DT
38536	1477	5	object	object	NN
38536	1477	6	varies	vary	VBZ
38536	1477	7	directly	directly	RB
38536	1477	8	as	as	IN
38536	1477	9	its	-PRON-	PRP$
38536	1477	10	distance	distance	NN
38536	1477	11	from	from	IN
38536	1477	12	the	the	DT
38536	1477	13	center	center	NN
38536	1477	14	of	of	IN
38536	1477	15	the	the	DT
38536	1477	16	earth	earth	NN
38536	1477	17	when	when	WRB
38536	1477	18	it	-PRON-	PRP
38536	1477	19	is	be	VBZ
38536	1477	20	below	below	IN
38536	1477	21	the	the	DT
38536	1477	22	earth	earth	NN
38536	1477	23	's	's	POS
38536	1477	24	surface	surface	NN
38536	1477	25	,	,	,
38536	1477	26	and	and	CC
38536	1477	27	inversely	inversely	RB
38536	1477	28	as	as	IN
38536	1477	29	the	the	DT
38536	1477	30	square	square	NN
38536	1477	31	of	of	IN
38536	1477	32	its	-PRON-	PRP$
38536	1477	33	distance	distance	NN
38536	1477	34	from	from	IN
38536	1477	35	the	the	DT
38536	1477	36	center	center	NN
38536	1477	37	when	when	WRB
38536	1477	38	it	-PRON-	PRP
38536	1477	39	is	be	VBZ
38536	1477	40	above	above	IN
38536	1477	41	the	the	DT
38536	1477	42	surface	surface	NN
38536	1477	43	.	.	.
38536	1478	1	If	if	IN
38536	1478	2	an	an	DT
38536	1478	3	object	object	NN
38536	1478	4	weighs	weigh	VBZ
38536	1478	5	10	10	CD
38536	1478	6	pounds	pound	NNS
38536	1478	7	at	at	IN
38536	1478	8	the	the	DT
38536	1478	9	surface	surface	NN
38536	1478	10	,	,	,
38536	1478	11	how	how	WRB
38536	1478	12	far	far	RB
38536	1478	13	above	above	RB
38536	1478	14	,	,	,
38536	1478	15	and	and	CC
38536	1478	16	how	how	WRB
38536	1478	17	far	far	RB
38536	1478	18	below	below	IN
38536	1478	19	the	the	DT
38536	1478	20	surface	surface	NN
38536	1478	21	will	will	MD
38536	1478	22	it	-PRON-	PRP
38536	1478	23	weigh	weigh	VB
38536	1478	24	9	9	CD
38536	1478	25	pounds	pound	NNS
38536	1478	26	?	?	.
38536	1479	1	(	(	-LRB-
38536	1479	2	The	the	DT
38536	1479	3	radius	radius	NN
38536	1479	4	of	of	IN
38536	1479	5	the	the	DT
38536	1479	6	earth	earth	NN
38536	1479	7	may	may	MD
38536	1479	8	be	be	VB
38536	1479	9	taken	take	VBN
38536	1479	10	as	as	IN
38536	1479	11	4000	4000	CD
38536	1479	12	miles	mile	NNS
38536	1479	13	.	.	.
38536	1479	14	)	)	-RRB-
38536	1480	1	5	5	CD
38536	1480	2	.	.	.
38536	1481	1	Solve	solve	VB
38536	1481	2	the	the	DT
38536	1481	3	following	follow	VBG
38536	1481	4	pair	pair	NN
38536	1481	5	of	of	IN
38536	1481	6	equations	equation	NNS
38536	1481	7	for	for	IN
38536	1481	8	x	x	NNP
38536	1481	9	and	and	CC
38536	1481	10	y	y	NNP
38536	1481	11	:	:	:
38536	1481	12	x^2	x^2	NNP
38536	1481	13	+	+	SYM
38536	1481	14	y^2	y^2	NNS
38536	1481	15	=	=	SYM
38536	1481	16	4	4	CD
38536	1481	17	,	,	,
38536	1481	18	x	x	SYM
38536	1481	19	=	=	NNS
38536	1481	20	(	(	-LRB-
38536	1481	21	1	1	CD
38536	1481	22	+	+	CD
38536	1481	23	2^(1/2))y	2^(1/2))y	CD
38536	1481	24	-	-	SYM
38536	1481	25	2	2	CD
38536	1481	26	.	.	.
38536	1482	1	6	6	CD
38536	1482	2	.	.	.
38536	1483	1	Find	find	VB
38536	1483	2	the	the	DT
38536	1483	3	value	value	NN
38536	1483	4	of	of	IN
38536	1483	5	[	[	-LRB-
38536	1483	6	1	1	CD
38536	1483	7	+	+	SYM
38536	1483	8	8^(-x/3)]/[(8x)^(1/2	8^(-x/3)]/[(8x)^(1/2	CD
38536	1483	9	)	)	-RRB-
38536	1483	10	+	+	CC
38536	1483	11	10^(x	10^(x	CD
38536	1483	12	-	-	HYPH
38536	1483	13	2	2	CD
38536	1483	14	)	)	-RRB-
38536	1483	15	]	]	-RRB-
38536	1483	16	,	,	,
38536	1483	17	when	when	WRB
38536	1483	18	x	x	NNP
38536	1483	19	=	=	SYM
38536	1483	20	2	2	CD
38536	1483	21	.	.	.
38536	1484	1	7	7	LS
38536	1484	2	.	.	.
38536	1485	1	From	from	IN
38536	1485	2	a	a	DT
38536	1485	3	square	square	NN
38536	1485	4	of	of	IN
38536	1485	5	pasteboard	pasteboard	NN
38536	1485	6	,	,	,
38536	1485	7	12	12	CD
38536	1485	8	inches	inch	NNS
38536	1485	9	on	on	IN
38536	1485	10	a	a	DT
38536	1485	11	side	side	NN
38536	1485	12	,	,	,
38536	1485	13	square	square	JJ
38536	1485	14	corners	corner	NNS
38536	1485	15	are	be	VBP
38536	1485	16	cut	cut	VBN
38536	1485	17	,	,	,
38536	1485	18	and	and	CC
38536	1485	19	the	the	DT
38536	1485	20	sides	side	NNS
38536	1485	21	are	be	VBP
38536	1485	22	turned	turn	VBN
38536	1485	23	up	up	RP
38536	1485	24	to	to	TO
38536	1485	25	form	form	VB
38536	1485	26	a	a	DT
38536	1485	27	rectangular	rectangular	JJ
38536	1485	28	box	box	NN
38536	1485	29	.	.	.
38536	1486	1	If	if	IN
38536	1486	2	the	the	DT
38536	1486	3	squares	square	NNS
38536	1486	4	cut	cut	VBD
38536	1486	5	out	out	RP
38536	1486	6	from	from	IN
38536	1486	7	the	the	DT
38536	1486	8	corners	corner	NNS
38536	1486	9	had	have	VBD
38536	1486	10	been	be	VBN
38536	1486	11	1	1	CD
38536	1486	12	inch	inch	NN
38536	1486	13	larger	large	JJR
38536	1486	14	on	on	IN
38536	1486	15	a	a	DT
38536	1486	16	side	side	NN
38536	1486	17	,	,	,
38536	1486	18	the	the	DT
38536	1486	19	volume	volume	NN
38536	1486	20	of	of	IN
38536	1486	21	the	the	DT
38536	1486	22	box	box	NN
38536	1486	23	would	would	MD
38536	1486	24	have	have	VB
38536	1486	25	been	be	VBN
38536	1486	26	increased	increase	VBN
38536	1486	27	28	28	CD
38536	1486	28	cubic	cubic	JJ
38536	1486	29	inches	inch	NNS
38536	1486	30	.	.	.
38536	1487	1	What	what	WP
38536	1487	2	is	be	VBZ
38536	1487	3	the	the	DT
38536	1487	4	size	size	NN
38536	1487	5	of	of	IN
38536	1487	6	the	the	DT
38536	1487	7	square	square	JJ
38536	1487	8	corners	corner	NNS
38536	1487	9	cut	cut	VBD
38536	1487	10	out	out	RP
38536	1487	11	?	?	.
38536	1488	1	(	(	-LRB-
38536	1488	2	See	see	VB
38536	1488	3	the	the	DT
38536	1488	4	figure	figure	NN
38536	1488	5	on	on	IN
38536	1488	6	the	the	DT
38536	1488	7	blackboard	blackboard	NN
38536	1488	8	.	.	.
38536	1488	9	)	)	-RRB-
38536	1489	1	~HARVARD	~harvard	VB
38536	1489	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1489	3	ELEMENTARY	ELEMENTARY	NNP
38536	1489	4	ALGEBRA	ALGEBRA	NNP
38536	1489	5	TIME	time	NN
38536	1489	6	:	:	:
38536	1489	7	ONE	one	CD
38536	1489	8	HOUR	hour	NN
38536	1489	9	AND	and	CC
38536	1489	10	A	a	DT
38536	1489	11	HALF	half	NN
38536	1489	12	Arrange	arrange	NN
38536	1489	13	your	-PRON-	PRP$
38536	1489	14	work	work	NN
38536	1489	15	neatly	neatly	RB
38536	1489	16	and	and	CC
38536	1489	17	clearly	clearly	RB
38536	1489	18	,	,	,
38536	1489	19	beginning	begin	VBG
38536	1489	20	each	each	DT
38536	1489	21	question	question	NN
38536	1489	22	on	on	IN
38536	1489	23	a	a	DT
38536	1489	24	separate	separate	JJ
38536	1489	25	page	page	NN
38536	1489	26	.	.	.
38536	1490	1	1	1	LS
38536	1490	2	.	.	.
38536	1491	1	Simplify	simplify	VB
38536	1491	2	the	the	DT
38536	1491	3	following	following	JJ
38536	1491	4	expression	expression	NN
38536	1491	5	:	:	:
38536	1491	6	[	[	-LRB-
38536	1491	7	[	[	-LRB-
38536	1491	8	1	1	CD
38536	1491	9	/	/	SYM
38536	1491	10	a	a	NN
38536	1491	11	+	+	SYM
38536	1491	12	1/(b	1/(b	CD
38536	1491	13	+	+	SYM
38536	1491	14	c)]/[1	c)]/[1	NNP
38536	1491	15	/	/	SYM
38536	1491	16	a	a	NN
38536	1491	17	-	-	HYPH
38536	1491	18	1/(b	1/(b	CD
38536	1491	19	+	+	SYM
38536	1491	20	c	c	NN
38536	1491	21	)	)	-RRB-
38536	1491	22	]	]	-RRB-
38536	1491	23	[	[	-LRB-
38536	1491	24	1	1	CD
38536	1491	25	+	+	SYM
38536	1491	26	(	(	-LRB-
38536	1491	27	b^2	b^2	NNS
38536	1491	28	+	+	SYM
38536	1491	29	c^2	c^2	NNP
38536	1491	30	-	-	HYPH
38536	1491	31	a^2)/(2bc	a^2)/(2bc	NNP
38536	1491	32	)	)	-RRB-
38536	1491	33	]	]	-RRB-
38536	1491	34	.	.	.
38536	1492	1	2	2	LS
38536	1492	2	.	.	.
38536	1493	1	(	(	-LRB-
38536	1493	2	_	_	NNP
38536	1493	3	a	a	DT
38536	1493	4	_	_	NN
38536	1493	5	)	)	-RRB-
38536	1493	6	Write	write	VB
38536	1493	7	the	the	DT
38536	1493	8	middle	middle	JJ
38536	1493	9	term	term	NN
38536	1493	10	of	of	IN
38536	1493	11	the	the	DT
38536	1493	12	expansion	expansion	NN
38536	1493	13	of	of	IN
38536	1493	14	(	(	-LRB-
38536	1493	15	a	a	DT
38536	1493	16	-	-	HYPH
38536	1493	17	b)^14	b)^14	NN
38536	1493	18	by	by	IN
38536	1493	19	the	the	DT
38536	1493	20	binomial	binomial	JJ
38536	1493	21	theorem	theorem	NN
38536	1493	22	.	.	.
38536	1494	1	(	(	-LRB-
38536	1494	2	_	_	NNP
38536	1494	3	b	b	NNP
38536	1494	4	_	_	NNP
38536	1494	5	)	)	-RRB-
38536	1494	6	Find	find	VB
38536	1494	7	the	the	DT
38536	1494	8	value	value	NN
38536	1494	9	of	of	IN
38536	1494	10	a^7b^7	a^7b^7	NNS
38536	1494	11	,	,	,
38536	1494	12	if	if	IN
38536	1494	13	a	a	DT
38536	1494	14	=	=	SYM
38536	1494	15	x^(2/7)y^(-3/2	x^(2/7)y^(-3/2	NN
38536	1494	16	)	)	-RRB-
38536	1494	17	and	and	CC
38536	1494	18	b	b	LS
38536	1494	19	=	=	SYM
38536	1494	20	(	(	-LRB-
38536	1494	21	1/2	1/2	CD
38536	1494	22	)	)	-RRB-
38536	1494	23	x^(-1/7)y^(1/2	x^(-1/7)y^(1/2	NNP
38536	1494	24	)	)	-RRB-
38536	1494	25	,	,	,
38536	1494	26	and	and	CC
38536	1494	27	reduce	reduce	VB
38536	1494	28	the	the	DT
38536	1494	29	result	result	NN
38536	1494	30	to	to	IN
38536	1494	31	a	a	DT
38536	1494	32	form	form	NN
38536	1494	33	having	have	VBG
38536	1494	34	only	only	RB
38536	1494	35	positive	positive	JJ
38536	1494	36	exponents	exponent	NNS
38536	1494	37	.	.	.
38536	1495	1	3	3	LS
38536	1495	2	.	.	.
38536	1496	1	Find	find	VB
38536	1496	2	correct	correct	JJ
38536	1496	3	to	to	IN
38536	1496	4	three	three	CD
38536	1496	5	significant	significant	JJ
38536	1496	6	figures	figure	NNS
38536	1496	7	the	the	DT
38536	1496	8	negative	negative	JJ
38536	1496	9	root	root	NN
38536	1496	10	of	of	IN
38536	1496	11	the	the	DT
38536	1496	12	equation	equation	NN
38536	1496	13	1	1	CD
38536	1496	14	-	-	HYPH
38536	1496	15	2/(x	2/(x	CD
38536	1496	16	+	+	SYM
38536	1496	17	1	1	CD
38536	1496	18	)	)	-RRB-
38536	1496	19	+	+	CC
38536	1496	20	4x/{(x	4x/{(x	NN
38536	1496	21	+	+	CD
38536	1496	22	1)^2	1)^2	NN
38536	1496	23	}	}	-RRB-
38536	1496	24	=	=	SYM
38536	1496	25	0	0	NFP
38536	1496	26	.	.	.
38536	1497	1	4	4	LS
38536	1497	2	.	.	.
38536	1498	1	Prove	prove	VB
38536	1498	2	the	the	DT
38536	1498	3	rule	rule	NN
38536	1498	4	for	for	IN
38536	1498	5	finding	find	VBG
38536	1498	6	the	the	DT
38536	1498	7	sum	sum	NN
38536	1498	8	of	of	IN
38536	1498	9	n	n	CD
38536	1498	10	terms	term	NNS
38536	1498	11	of	of	IN
38536	1498	12	a	a	DT
38536	1498	13	geometrical	geometrical	JJ
38536	1498	14	progression	progression	NN
38536	1498	15	of	of	IN
38536	1498	16	which	which	WDT
38536	1498	17	the	the	DT
38536	1498	18	first	first	JJ
38536	1498	19	term	term	NN
38536	1498	20	is	be	VBZ
38536	1498	21	a	a	DT
38536	1498	22	and	and	CC
38536	1498	23	the	the	DT
38536	1498	24	constant	constant	JJ
38536	1498	25	ratio	ratio	NN
38536	1498	26	is	be	VBZ
38536	1498	27	r.	r.	NNP
38536	1498	28	Find	find	VB
38536	1498	29	the	the	DT
38536	1498	30	sum	sum	NN
38536	1498	31	of	of	IN
38536	1498	32	8	8	CD
38536	1498	33	terms	term	NNS
38536	1498	34	of	of	IN
38536	1498	35	the	the	DT
38536	1498	36	progression	progression	NN
38536	1498	37	5	5	CD
38536	1498	38	+	+	SYM
38536	1498	39	3	3	CD
38536	1498	40	-	-	SYM
38536	1498	41	1/3	1/3	CD
38536	1498	42	+	+	SYM
38536	1498	43	2	2	CD
38536	1498	44	-	-	SYM
38536	1498	45	2/9	2/9	CD
38536	1498	46	+	+	SYM
38536	1498	47	·	·	NFP
38536	1498	48	·	·	NFP
38536	1498	49	·	·	NFP
38536	1498	50	.	.	.
38536	1499	1	5	5	CD
38536	1499	2	.	.	.
38536	1500	1	A	a	DT
38536	1500	2	goldsmith	goldsmith	NN
38536	1500	3	has	have	VBZ
38536	1500	4	two	two	CD
38536	1500	5	alloys	alloy	NNS
38536	1500	6	of	of	IN
38536	1500	7	gold	gold	NN
38536	1500	8	,	,	,
38536	1500	9	the	the	DT
38536	1500	10	first	first	JJ
38536	1500	11	being	be	VBG
38536	1500	12	3/4	3/4	CD
38536	1500	13	pure	pure	JJ
38536	1500	14	gold	gold	NN
38536	1500	15	,	,	,
38536	1500	16	the	the	DT
38536	1500	17	second	second	JJ
38536	1500	18	5/12	5/12	CD
38536	1500	19	pure	pure	JJ
38536	1500	20	gold	gold	NN
38536	1500	21	.	.	.
38536	1501	1	How	how	WRB
38536	1501	2	much	much	JJ
38536	1501	3	of	of	IN
38536	1501	4	each	each	DT
38536	1501	5	must	must	MD
38536	1501	6	he	-PRON-	PRP
38536	1501	7	take	take	VB
38536	1501	8	to	to	TO
38536	1501	9	produce	produce	VB
38536	1501	10	100	100	CD
38536	1501	11	ounces	ounce	NNS
38536	1501	12	of	of	IN
38536	1501	13	an	an	DT
38536	1501	14	alloy	alloy	NN
38536	1501	15	which	which	WDT
38536	1501	16	shall	shall	MD
38536	1501	17	be	be	VB
38536	1501	18	2/3	2/3	CD
38536	1501	19	pure	pure	JJ
38536	1501	20	gold	gold	NN
38536	1501	21	?	?	.
38536	1502	1	~HARVARD	~harvard	VB
38536	1502	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1502	3	ELEMENTARY	ELEMENTARY	NNP
38536	1502	4	ALGEBRA	ALGEBRA	NNP
38536	1502	5	TIME	time	NN
38536	1502	6	:	:	:
38536	1502	7	ONE	one	CD
38536	1502	8	HOUR	hour	NN
38536	1502	9	AND	and	CC
38536	1502	10	A	a	DT
38536	1502	11	HALF	half	NN
38536	1502	12	1	1	CD
38536	1502	13	.	.	.
38536	1503	1	Solve	solve	VB
38536	1503	2	the	the	DT
38536	1503	3	simultaneous	simultaneous	JJ
38536	1503	4	equations	equation	NNS
38536	1503	5	x	x	SYM
38536	1503	6	+	+	SYM
38536	1503	7	y	y	NN
38536	1503	8	=	=	SYM
38536	1503	9	a	a	NN
38536	1503	10	+	+	SYM
38536	1503	11	b	b	NN
38536	1503	12	,	,	,
38536	1503	13	(	(	-LRB-
38536	1503	14	y	y	NNP
38536	1503	15	+	+	CC
38536	1503	16	b)/(x	b)/(x	CD
38536	1503	17	+	+	CC
38536	1503	18	a	a	NN
38536	1503	19	)	)	-RRB-
38536	1503	20	=	=	NFP
38536	1503	21	a	a	NN
38536	1503	22	/	/	SYM
38536	1503	23	b	b	NN
38536	1503	24	,	,	,
38536	1503	25	and	and	CC
38536	1503	26	verify	verify	VB
38536	1503	27	your	-PRON-	PRP$
38536	1503	28	results	result	NNS
38536	1503	29	.	.	.
38536	1504	1	2	2	LS
38536	1504	2	.	.	.
38536	1505	1	Solve	solve	VB
38536	1505	2	the	the	DT
38536	1505	3	equation	equation	NN
38536	1505	4	x^2	x^2	JJ
38536	1505	5	-	-	HYPH
38536	1505	6	1.6x	1.6x	CD
38536	1505	7	-	-	HYPH
38536	1505	8	0.23	0.23	CD
38536	1505	9	=	=	SYM
38536	1505	10	0	0	CD
38536	1505	11	,	,	,
38536	1505	12	obtaining	obtain	VBG
38536	1505	13	the	the	DT
38536	1505	14	values	value	NNS
38536	1505	15	of	of	IN
38536	1505	16	the	the	DT
38536	1505	17	roots	root	NNS
38536	1505	18	correct	correct	JJ
38536	1505	19	to	to	IN
38536	1505	20	three	three	CD
38536	1505	21	significant	significant	JJ
38536	1505	22	figures	figure	NNS
38536	1505	23	.	.	.
38536	1506	1	3	3	LS
38536	1506	2	.	.	.
38536	1507	1	Write	write	VB
38536	1507	2	out	out	RP
38536	1507	3	the	the	DT
38536	1507	4	first	first	JJ
38536	1507	5	four	four	CD
38536	1507	6	terms	term	NNS
38536	1507	7	of	of	IN
38536	1507	8	(	(	-LRB-
38536	1507	9	a	a	NN
38536	1507	10	-	-	HYPH
38536	1507	11	b)^7	b)^7	NNP
38536	1507	12	.	.	.
38536	1508	1	Find	find	VB
38536	1508	2	the	the	DT
38536	1508	3	fourth	fourth	JJ
38536	1508	4	term	term	NN
38536	1508	5	of	of	IN
38536	1508	6	this	this	DT
38536	1508	7	expansion	expansion	NN
38536	1508	8	when	when	WRB
38536	1508	9	a	a	DT
38536	1508	10	=	=	-RRB-
38536	1508	11	[	[	-LRB-
38536	1508	12	x^(-1	x^(-1	NNP
38536	1508	13	)	)	-RRB-
38536	1508	14	y^(1/2)]^(1/3	y^(1/2)]^(1/3	NNP
38536	1508	15	)	)	-RRB-
38536	1508	16	,	,	,
38536	1508	17	b	b	NN
38536	1508	18	=	=	SYM
38536	1508	19	[	[	-LRB-
38536	1508	20	9xy^(-4)]^(1/6	9xy^(-4)]^(1/6	CD
38536	1508	21	)	)	-RRB-
38536	1508	22	,	,	,
38536	1508	23	expressing	express	VBG
38536	1508	24	the	the	DT
38536	1508	25	result	result	NN
38536	1508	26	in	in	IN
38536	1508	27	terms	term	NNS
38536	1508	28	of	of	IN
38536	1508	29	a	a	DT
38536	1508	30	single	single	JJ
38536	1508	31	radical	radical	JJ
38536	1508	32	,	,	,
38536	1508	33	and	and	CC
38536	1508	34	without	without	IN
38536	1508	35	fractional	fractional	JJ
38536	1508	36	or	or	CC
38536	1508	37	negative	negative	JJ
38536	1508	38	exponents	exponent	NNS
38536	1508	39	.	.	.
38536	1509	1	4	4	LS
38536	1509	2	.	.	.
38536	1510	1	Reduce	reduce	VB
38536	1510	2	the	the	DT
38536	1510	3	following	follow	VBG
38536	1510	4	expression	expression	NN
38536	1510	5	to	to	IN
38536	1510	6	a	a	DT
38536	1510	7	polynomial	polynomial	NN
38536	1510	8	in	in	IN
38536	1510	9	a	a	NN
38536	1510	10	and	and	CC
38536	1510	11	b	b	NN
38536	1510	12	:	:	:
38536	1510	13	(	(	-LRB-
38536	1510	14	6a^3	6a^3	CD
38536	1510	15	+	+	SYM
38536	1510	16	7ab^2	7ab^2	CD
38536	1510	17	+	+	SYM
38536	1510	18	12b^3)/(3a^2	12b^3)/(3a^2	CD
38536	1510	19	-	-	HYPH
38536	1510	20	5ab	5ab	JJ
38536	1510	21	-	-	HYPH
38536	1510	22	4b^2	4b^2	CD
38536	1510	23	)	)	-RRB-
38536	1510	24	-	-	:
38536	1510	25	1/[3/19b	1/[3/19b	CD
38536	1510	26	-	-	HYPH
38536	1510	27	(	(	-LRB-
38536	1510	28	5a	5a	CD
38536	1510	29	+	+	NNP
38536	1510	30	4b)/(19a^2	4b)/(19a^2	NNP
38536	1510	31	)	)	-RRB-
38536	1510	32	]	]	-RRB-
38536	1510	33	.	.	.
38536	1511	1	5	5	CD
38536	1511	2	.	.	.
38536	1512	1	The	the	DT
38536	1512	2	cost	cost	NN
38536	1512	3	of	of	IN
38536	1512	4	publishing	publish	VBG
38536	1512	5	a	a	DT
38536	1512	6	book	book	NN
38536	1512	7	consists	consist	VBZ
38536	1512	8	of	of	IN
38536	1512	9	two	two	CD
38536	1512	10	main	main	JJ
38536	1512	11	items	item	NNS
38536	1512	12	:	:	:
38536	1512	13	first	first	RB
38536	1512	14	,	,	,
38536	1512	15	the	the	DT
38536	1512	16	fixed	fix	VBN
38536	1512	17	expense	expense	NN
38536	1512	18	of	of	IN
38536	1512	19	setting	set	VBG
38536	1512	20	up	up	RP
38536	1512	21	the	the	DT
38536	1512	22	type	type	NN
38536	1512	23	;	;	:
38536	1512	24	and	and	CC
38536	1512	25	,	,	,
38536	1512	26	second	second	JJ
38536	1512	27	,	,	,
38536	1512	28	the	the	DT
38536	1512	29	running	running	NN
38536	1512	30	expenses	expense	NNS
38536	1512	31	of	of	IN
38536	1512	32	presswork	presswork	NN
38536	1512	33	,	,	,
38536	1512	34	binding	binding	NN
38536	1512	35	,	,	,
38536	1512	36	etc	etc	FW
38536	1512	37	.	.	FW
38536	1512	38	,	,	,
38536	1512	39	which	which	WDT
38536	1512	40	may	may	MD
38536	1512	41	be	be	VB
38536	1512	42	assumed	assume	VBN
38536	1512	43	to	to	TO
38536	1512	44	be	be	VB
38536	1512	45	proportional	proportional	JJ
38536	1512	46	to	to	IN
38536	1512	47	the	the	DT
38536	1512	48	number	number	NN
38536	1512	49	of	of	IN
38536	1512	50	copies	copy	NNS
38536	1512	51	.	.	.
38536	1513	1	A	a	DT
38536	1513	2	certain	certain	JJ
38536	1513	3	book	book	NN
38536	1513	4	costs	cost	VBZ
38536	1513	5	35	35	CD
38536	1513	6	cents	cent	NNS
38536	1513	7	a	a	DT
38536	1513	8	copy	copy	NN
38536	1513	9	if	if	IN
38536	1513	10	1000	1000	CD
38536	1513	11	copies	copy	NNS
38536	1513	12	are	be	VBP
38536	1513	13	published	publish	VBN
38536	1513	14	at	at	IN
38536	1513	15	one	one	CD
38536	1513	16	time	time	NN
38536	1513	17	,	,	,
38536	1513	18	but	but	CC
38536	1513	19	only	only	RB
38536	1513	20	19	19	CD
38536	1513	21	cents	cent	NNS
38536	1513	22	a	a	DT
38536	1513	23	copy	copy	NN
38536	1513	24	if	if	IN
38536	1513	25	5000	5000	CD
38536	1513	26	copies	copy	NNS
38536	1513	27	are	be	VBP
38536	1513	28	published	publish	VBN
38536	1513	29	at	at	IN
38536	1513	30	one	one	CD
38536	1513	31	time	time	NN
38536	1513	32	.	.	.
38536	1514	1	Find	find	VB
38536	1514	2	(	(	-LRB-
38536	1514	3	_	_	NNP
38536	1514	4	a	a	DT
38536	1514	5	_	_	NNP
38536	1514	6	)	)	-RRB-
38536	1514	7	the	the	DT
38536	1514	8	cost	cost	NN
38536	1514	9	of	of	IN
38536	1514	10	setting	set	VBG
38536	1514	11	up	up	RP
38536	1514	12	the	the	DT
38536	1514	13	type	type	NN
38536	1514	14	for	for	IN
38536	1514	15	the	the	DT
38536	1514	16	book	book	NN
38536	1514	17	,	,	,
38536	1514	18	and	and	CC
38536	1514	19	(	(	-LRB-
38536	1514	20	_	_	NNP
38536	1514	21	b	b	NNP
38536	1514	22	_	_	NNP
38536	1514	23	)	)	-RRB-
38536	1514	24	the	the	DT
38536	1514	25	cost	cost	NN
38536	1514	26	of	of	IN
38536	1514	27	presswork	presswork	NN
38536	1514	28	,	,	,
38536	1514	29	binding	binding	NN
38536	1514	30	,	,	,
38536	1514	31	etc	etc	FW
38536	1514	32	.	.	FW
38536	1514	33	,	,	,
38536	1514	34	per	per	IN
38536	1514	35	thousand	thousand	CD
38536	1514	36	copies	copy	NNS
38536	1514	37	.	.	.
38536	1515	1	~HARVARD	~harvard	VB
38536	1515	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1515	3	ELEMENTARY	ELEMENTARY	NNP
38536	1515	4	ALGEBRA	ALGEBRA	NNP
38536	1515	5	TIME	time	NN
38536	1515	6	:	:	:
38536	1515	7	ONE	one	CD
38536	1515	8	HOUR	hour	NN
38536	1515	9	AND	and	CC
38536	1515	10	A	a	DT
38536	1515	11	HALF	half	NN
38536	1515	12	1	1	CD
38536	1515	13	.	.	.
38536	1516	1	Find	find	VB
38536	1516	2	the	the	DT
38536	1516	3	highest	high	JJS
38536	1516	4	common	common	JJ
38536	1516	5	factor	factor	NN
38536	1516	6	and	and	CC
38536	1516	7	the	the	DT
38536	1516	8	lowest	low	JJS
38536	1516	9	common	common	JJ
38536	1516	10	multiple	multiple	NN
38536	1516	11	of	of	IN
38536	1516	12	the	the	DT
38536	1516	13	three	three	CD
38536	1516	14	expressions	expression	NNS
38536	1516	15	a^4	a^4	NNP
38536	1516	16	-	-	HYPH
38536	1516	17	b^4	b^4	NNS
38536	1516	18	;	;	:
38536	1516	19	a^3	a^3	NNP
38536	1516	20	+	+	SYM
38536	1516	21	b^3	b^3	NNS
38536	1516	22	;	;	:
38536	1516	23	a^3	a^3	NNP
38536	1516	24	+	+	SYM
38536	1516	25	2a^2	2a^2	CD
38536	1516	26	b	b	NN
38536	1516	27	+	+	CC
38536	1516	28	2ab^2	2ab^2	CD
38536	1516	29	+	+	SYM
38536	1516	30	b^3	b^3	NNS
38536	1516	31	.	.	.
38536	1517	1	2	2	LS
38536	1517	2	.	.	.
38536	1518	1	Solve	solve	VB
38536	1518	2	the	the	DT
38536	1518	3	quadratic	quadratic	JJ
38536	1518	4	equation	equation	NN
38536	1518	5	x^2	x^2	NNP
38536	1518	6	-	-	HYPH
38536	1518	7	1.6x	1.6x	CD
38536	1518	8	+	+	SYM
38536	1518	9	0.3	0.3	CD
38536	1518	10	=	=	SYM
38536	1518	11	0	0	CD
38536	1518	12	,	,	,
38536	1518	13	computing	compute	VBG
38536	1518	14	the	the	DT
38536	1518	15	value	value	NN
38536	1518	16	of	of	IN
38536	1518	17	the	the	DT
38536	1518	18	larger	large	JJR
38536	1518	19	root	root	NN
38536	1518	20	correct	correct	JJ
38536	1518	21	to	to	IN
38536	1518	22	three	three	CD
38536	1518	23	significant	significant	JJ
38536	1518	24	figures	figure	NNS
38536	1518	25	.	.	.
38536	1519	1	3	3	LS
38536	1519	2	.	.	.
38536	1520	1	In	in	IN
38536	1520	2	the	the	DT
38536	1520	3	expression	expression	NN
38536	1520	4	x^2	x^2	JJ
38536	1520	5	-	-	HYPH
38536	1520	6	2xy	2xy	JJ
38536	1520	7	+	+	SYM
38536	1520	8	y^2	y^2	NN
38536	1520	9	-	-	HYPH
38536	1520	10	4[2^(1/2)](x	4[2^(1/2)](x	NN
38536	1520	11	+	+	SYM
38536	1520	12	y	y	NN
38536	1520	13	)	)	-RRB-
38536	1520	14	+	+	CC
38536	1520	15	8	8	CD
38536	1520	16	,	,	,
38536	1520	17	substitute	substitute	VBP
38536	1520	18	for	for	IN
38536	1520	19	x	x	NNP
38536	1520	20	and	and	CC
38536	1520	21	y	y	NNP
38536	1520	22	the	the	DT
38536	1520	23	values	value	NNS
38536	1520	24	x	x	SYM
38536	1520	25	=	=	NFP
38536	1520	26	(	(	-LRB-
38536	1520	27	u	u	NN
38536	1520	28	+	+	SYM
38536	1520	29	v	v	NN
38536	1520	30	+	+	CD
38536	1520	31	1)/[2^(1/2	1)/[2^(1/2	CD
38536	1520	32	)	)	-RRB-
38536	1520	33	]	]	-RRB-
38536	1520	34	,	,	,
38536	1520	35	y	y	NNP
38536	1520	36	=	=	NFP
38536	1520	37	(	(	-LRB-
38536	1520	38	u	u	NNP
38536	1520	39	-	-	HYPH
38536	1520	40	v	v	NNP
38536	1520	41	+	+	CD
38536	1520	42	1)/[2^(1/2	1)/[2^(1/2	CD
38536	1520	43	)	)	-RRB-
38536	1520	44	]	]	-RRB-
38536	1520	45	,	,	,
38536	1520	46	and	and	CC
38536	1520	47	reduce	reduce	VB
38536	1520	48	the	the	DT
38536	1520	49	resulting	result	VBG
38536	1520	50	expression	expression	NN
38536	1520	51	to	to	IN
38536	1520	52	its	-PRON-	PRP$
38536	1520	53	simplest	simple	JJS
38536	1520	54	form	form	NN
38536	1520	55	.	.	.
38536	1521	1	4	4	LS
38536	1521	2	.	.	.
38536	1522	1	State	state	NN
38536	1522	2	and	and	CC
38536	1522	3	prove	prove	VB
38536	1522	4	the	the	DT
38536	1522	5	formula	formula	NN
38536	1522	6	for	for	IN
38536	1522	7	the	the	DT
38536	1522	8	sum	sum	NN
38536	1522	9	of	of	IN
38536	1522	10	the	the	DT
38536	1522	11	first	first	JJ
38536	1522	12	n	n	JJ
38536	1522	13	terms	term	NNS
38536	1522	14	of	of	IN
38536	1522	15	a	a	DT
38536	1522	16	geometric	geometric	JJ
38536	1522	17	progression	progression	NN
38536	1522	18	in	in	IN
38536	1522	19	which	which	WDT
38536	1522	20	a	a	DT
38536	1522	21	is	be	VBZ
38536	1522	22	the	the	DT
38536	1522	23	first	first	JJ
38536	1522	24	term	term	NN
38536	1522	25	and	and	CC
38536	1522	26	r	r	NN
38536	1522	27	the	the	DT
38536	1522	28	constant	constant	JJ
38536	1522	29	ratio	ratio	NN
38536	1522	30	.	.	.
38536	1523	1	5	5	CD
38536	1523	2	.	.	.
38536	1524	1	A	a	DT
38536	1524	2	state	state	NN
38536	1524	3	legislature	legislature	NN
38536	1524	4	is	be	VBZ
38536	1524	5	to	to	TO
38536	1524	6	elect	elect	VB
38536	1524	7	a	a	DT
38536	1524	8	United	United	NNP
38536	1524	9	States	States	NNP
38536	1524	10	senator	senator	NN
38536	1524	11	,	,	,
38536	1524	12	a	a	DT
38536	1524	13	majority	majority	NN
38536	1524	14	of	of	IN
38536	1524	15	all	all	PDT
38536	1524	16	the	the	DT
38536	1524	17	votes	vote	NNS
38536	1524	18	cast	cast	VBD
38536	1524	19	being	be	VBG
38536	1524	20	necessary	necessary	JJ
38536	1524	21	for	for	IN
38536	1524	22	a	a	DT
38536	1524	23	choice	choice	NN
38536	1524	24	.	.	.
38536	1525	1	There	there	EX
38536	1525	2	are	be	VBP
38536	1525	3	three	three	CD
38536	1525	4	candidates	candidate	NNS
38536	1525	5	,	,	,
38536	1525	6	A	a	NN
38536	1525	7	,	,	,
38536	1525	8	B	b	NN
38536	1525	9	,	,	,
38536	1525	10	and	and	CC
38536	1525	11	C	C	NNP
38536	1525	12	,	,	,
38536	1525	13	and	and	CC
38536	1525	14	100	100	CD
38536	1525	15	members	member	NNS
38536	1525	16	vote	vote	VBP
38536	1525	17	.	.	.
38536	1526	1	On	on	IN
38536	1526	2	the	the	DT
38536	1526	3	first	first	JJ
38536	1526	4	ballot	ballot	NN
38536	1526	5	A	a	NN
38536	1526	6	has	have	VBZ
38536	1526	7	the	the	DT
38536	1526	8	largest	large	JJS
38536	1526	9	number	number	NN
38536	1526	10	of	of	IN
38536	1526	11	votes	vote	NNS
38536	1526	12	,	,	,
38536	1526	13	receiving	receive	VBG
38536	1526	14	9	9	CD
38536	1526	15	more	more	JJR
38536	1526	16	votes	vote	NNS
38536	1526	17	than	than	IN
38536	1526	18	his	-PRON-	PRP$
38536	1526	19	nearest	near	JJS
38536	1526	20	competitor	competitor	NN
38536	1526	21	,	,	,
38536	1526	22	B	b	NN
38536	1526	23	;	;	:
38536	1526	24	but	but	CC
38536	1526	25	he	-PRON-	PRP
38536	1526	26	fails	fail	VBZ
38536	1526	27	of	of	IN
38536	1526	28	the	the	DT
38536	1526	29	necessary	necessary	JJ
38536	1526	30	majority	majority	NN
38536	1526	31	.	.	.
38536	1527	1	On	on	IN
38536	1527	2	the	the	DT
38536	1527	3	second	second	JJ
38536	1527	4	ballot	ballot	NN
38536	1527	5	C	C	NNP
38536	1527	6	's	's	POS
38536	1527	7	name	name	NN
38536	1527	8	is	be	VBZ
38536	1527	9	withdrawn	withdraw	VBN
38536	1527	10	,	,	,
38536	1527	11	and	and	CC
38536	1527	12	all	all	PDT
38536	1527	13	the	the	DT
38536	1527	14	members	member	NNS
38536	1527	15	who	who	WP
38536	1527	16	voted	vote	VBD
38536	1527	17	for	for	IN
38536	1527	18	C	C	NNP
38536	1527	19	now	now	RB
38536	1527	20	vote	vote	NN
38536	1527	21	for	for	IN
38536	1527	22	B	b	NN
38536	1527	23	,	,	,
38536	1527	24	whereupon	whereupon	NNP
38536	1527	25	B	B	NNP
38536	1527	26	is	be	VBZ
38536	1527	27	elected	elect	VBN
38536	1527	28	by	by	IN
38536	1527	29	a	a	DT
38536	1527	30	majority	majority	NN
38536	1527	31	of	of	IN
38536	1527	32	2	2	CD
38536	1527	33	.	.	.
38536	1528	1	How	how	WRB
38536	1528	2	many	many	JJ
38536	1528	3	votes	vote	NNS
38536	1528	4	were	be	VBD
38536	1528	5	cast	cast	VBN
38536	1528	6	for	for	IN
38536	1528	7	each	each	DT
38536	1528	8	candidate	candidate	NN
38536	1528	9	on	on	IN
38536	1528	10	the	the	DT
38536	1528	11	first	first	JJ
38536	1528	12	ballot	ballot	NN
38536	1528	13	?	?	.
38536	1529	1	~MASSACHUSETTS	~MASSACHUSETTS	NFP
38536	1529	2	INSTITUTE	INSTITUTE	NNP
38536	1529	3	OF	of	IN
38536	1529	4	TECHNOLOGY~	TECHNOLOGY~	NNP
38536	1529	5	ALGEBRA	ALGEBRA	NNP
38536	1529	6	A	a	DT
38536	1529	7	TIME	time	NN
38536	1529	8	:	:	:
38536	1529	9	ONE	one	CD
38536	1529	10	HOUR	hour	NN
38536	1529	11	AND	and	CC
38536	1529	12	THREE	three	CD
38536	1529	13	QUARTERS	quarter	NNS
38536	1529	14	1	1	CD
38536	1529	15	.	.	.
38536	1530	1	Factor	factor	VB
38536	1530	2	the	the	DT
38536	1530	3	expressions	expression	NNS
38536	1530	4	:	:	:
38536	1530	5	x^3	x^3	NNS
38536	1530	6	+	+	SYM
38536	1530	7	x^2	x^2	NNS
38536	1530	8	=	=	SYM
38536	1530	9	2x	2x	CD
38536	1530	10	.	.	.
38536	1531	1	x^3	x^3	NNS
38536	1531	2	+	+	SYM
38536	1531	3	x^2	x^2	NNP
38536	1531	4	-	-	HYPH
38536	1531	5	4x	4x	NNP
38536	1531	6	-	-	SYM
38536	1531	7	4	4	CD
38536	1531	8	.	.	.
38536	1532	1	2	2	LS
38536	1532	2	.	.	.
38536	1533	1	Simplify	simplify	VB
38536	1533	2	the	the	DT
38536	1533	3	expression	expression	NN
38536	1533	4	:	:	:
38536	1533	5	[	[	-LRB-
38536	1533	6	1	1	CD
38536	1533	7	-	-	HYPH
38536	1533	8	(	(	-LRB-
38536	1533	9	b^2)/(a^2)][1	b^2)/(a^2)][1	HYPH
38536	1533	10	-	-	HYPH
38536	1533	11	(	(	-LRB-
38536	1533	12	ab	ab	NNP
38536	1533	13	-	-	HYPH
38536	1533	14	b^2)/(a^2)](a^4)/(a^3	b^2)/(a^2)](a^4)/(a^3	NNP
38536	1533	15	+	+	CC
38536	1533	16	b^3	b^3	NNP
38536	1533	17	)	)	-RRB-
38536	1533	18	·	·	NFP
38536	1533	19	(	(	-LRB-
38536	1533	20	a	a	DT
38536	1533	21	-	-	HYPH
38536	1533	22	b)/(a^2	b)/(a^2	NN
38536	1533	23	+	+	SYM
38536	1533	24	b^2	b^2	NNS
38536	1533	25	)	)	-RRB-
38536	1533	26	.	.	.
38536	1534	1	3	3	LS
38536	1534	2	.	.	.
38536	1535	1	Find	find	VB
38536	1535	2	the	the	DT
38536	1535	3	value	value	NN
38536	1535	4	of	of	IN
38536	1535	5	x	x	NN
38536	1535	6	+	+	SYM
38536	1535	7	[	[	-LRB-
38536	1535	8	1	1	CD
38536	1535	9	+	+	SYM
38536	1535	10	x^2]^(1/2	x^2]^(1/2	CD
38536	1535	11	)	)	-RRB-
38536	1535	12	,	,	,
38536	1535	13	when	when	WRB
38536	1535	14	x	x	LS
38536	1535	15	=	=	SYM
38536	1535	16	(	(	-LRB-
38536	1535	17	1/2)[[a	1/2)[[a	CD
38536	1535	18	/	/	SYM
38536	1535	19	b]^(1/2	b]^(1/2	NN
38536	1535	20	)	)	-RRB-
38536	1535	21	-	-	:
38536	1535	22	[	[	-LRB-
38536	1535	23	b	b	NN
38536	1535	24	/	/	SYM
38536	1535	25	a]^(1/2	a]^(1/2	NN
38536	1535	26	)	)	-RRB-
38536	1535	27	]	]	-RRB-
38536	1535	28	.	.	.
38536	1536	1	4	4	LS
38536	1536	2	.	.	.
38536	1537	1	Solve	solve	VB
38536	1537	2	the	the	DT
38536	1537	3	equations	equation	NNS
38536	1537	4	:	:	:
38536	1537	5	(	(	-LRB-
38536	1537	6	7x	7x	NN
38536	1537	7	+	+	SYM
38536	1537	8	6)/11	6)/11	NNS
38536	1537	9	+	+	SYM
38536	1537	10	y	y	NN
38536	1537	11	-	-	HYPH
38536	1537	12	16	16	NNP
38536	1537	13	=	=	SYM
38536	1537	14	(	(	-LRB-
38536	1537	15	5x	5x	CD
38536	1537	16	-	-	HYPH
38536	1537	17	13)/2	13)/2	CD
38536	1537	18	-	-	HYPH
38536	1537	19	(	(	-LRB-
38536	1537	20	8y	8y	NNP
38536	1537	21	-	-	HYPH
38536	1537	22	x)/5	x)/5	NNP
38536	1537	23	,	,	,
38536	1537	24	3(3x	3(3x	CD
38536	1537	25	+	+	SYM
38536	1537	26	4	4	CD
38536	1537	27	)	)	-RRB-
38536	1537	28	=	=	SYM
38536	1537	29	10y	10y	NNS
38536	1537	30	-	-	SYM
38536	1537	31	15	15	CD
38536	1537	32	.	.	.
38536	1538	1	5	5	CD
38536	1538	2	.	.	.
38536	1539	1	Solve	solve	VB
38536	1539	2	the	the	DT
38536	1539	3	equations	equation	NNS
38536	1539	4	:	:	:
38536	1539	5	A	a	NN
38536	1539	6	+	+	SYM
38536	1539	7	C	c	NN
38536	1539	8	=	=	SYM
38536	1539	9	2	2	CD
38536	1539	10	,	,	,
38536	1539	11	-A	-A	.
38536	1539	12	+	+	CC
38536	1539	13	B	b	NN
38536	1539	14	+	+	CC
38536	1539	15	C	c	NN
38536	1539	16	+	+	CC
38536	1539	17	D	d	NN
38536	1539	18	=	=	SYM
38536	1539	19	1	1	CD
38536	1539	20	,	,	,
38536	1539	21	2A	2a	NN
38536	1539	22	-	-	HYPH
38536	1539	23	B	b	NN
38536	1539	24	+	+	CC
38536	1539	25	2C	2c	NN
38536	1539	26	+	+	SYM
38536	1539	27	D	d	NN
38536	1539	28	=	=	SYM
38536	1539	29	5	5	CD
38536	1539	30	,	,	,
38536	1539	31	B	b	NN
38536	1539	32	+	+	CC
38536	1539	33	D	d	NN
38536	1539	34	=	=	SYM
38536	1539	35	1	1	CD
38536	1539	36	.	.	.
38536	1540	1	6	6	CD
38536	1540	2	.	.	.
38536	1541	1	Two	two	CD
38536	1541	2	squares	square	NNS
38536	1541	3	are	be	VBP
38536	1541	4	formed	form	VBN
38536	1541	5	with	with	IN
38536	1541	6	a	a	DT
38536	1541	7	combined	combined	JJ
38536	1541	8	perimeter	perimeter	NN
38536	1541	9	of	of	IN
38536	1541	10	16	16	CD
38536	1541	11	inches	inch	NNS
38536	1541	12	.	.	.
38536	1542	1	One	one	CD
38536	1542	2	square	square	NN
38536	1542	3	contains	contain	VBZ
38536	1542	4	4	4	CD
38536	1542	5	square	square	JJ
38536	1542	6	inches	inch	NNS
38536	1542	7	more	more	JJR
38536	1542	8	than	than	IN
38536	1542	9	the	the	DT
38536	1542	10	other	other	JJ
38536	1542	11	.	.	.
38536	1543	1	Find	find	VB
38536	1543	2	the	the	DT
38536	1543	3	area	area	NN
38536	1543	4	of	of	IN
38536	1543	5	each	each	DT
38536	1543	6	.	.	.
38536	1544	1	7	7	LS
38536	1544	2	.	.	.
38536	1545	1	A	a	DT
38536	1545	2	man	man	NN
38536	1545	3	walked	walk	VBD
38536	1545	4	to	to	IN
38536	1545	5	a	a	DT
38536	1545	6	railway	railway	NN
38536	1545	7	station	station	NN
38536	1545	8	at	at	IN
38536	1545	9	the	the	DT
38536	1545	10	rate	rate	NN
38536	1545	11	of	of	IN
38536	1545	12	4	4	CD
38536	1545	13	miles	mile	NNS
38536	1545	14	an	an	DT
38536	1545	15	hour	hour	NN
38536	1545	16	and	and	CC
38536	1545	17	traveled	travel	VBN
38536	1545	18	by	by	IN
38536	1545	19	train	train	NN
38536	1545	20	at	at	IN
38536	1545	21	the	the	DT
38536	1545	22	rate	rate	NN
38536	1545	23	of	of	IN
38536	1545	24	30	30	CD
38536	1545	25	miles	mile	NNS
38536	1545	26	an	an	DT
38536	1545	27	hour	hour	NN
38536	1545	28	,	,	,
38536	1545	29	reaching	reach	VBG
38536	1545	30	his	-PRON-	PRP$
38536	1545	31	destination	destination	NN
38536	1545	32	in	in	IN
38536	1545	33	20	20	CD
38536	1545	34	hours	hour	NNS
38536	1545	35	.	.	.
38536	1546	1	If	if	IN
38536	1546	2	he	-PRON-	PRP
38536	1546	3	had	have	VBD
38536	1546	4	walked	walk	VBN
38536	1546	5	3	3	CD
38536	1546	6	miles	mile	NNS
38536	1546	7	an	an	DT
38536	1546	8	hour	hour	NN
38536	1546	9	and	and	CC
38536	1546	10	ridden	ride	VBD
38536	1546	11	35	35	CD
38536	1546	12	miles	mile	NNS
38536	1546	13	an	an	DT
38536	1546	14	hour	hour	NN
38536	1546	15	,	,	,
38536	1546	16	he	-PRON-	PRP
38536	1546	17	would	would	MD
38536	1546	18	have	have	VB
38536	1546	19	made	make	VBN
38536	1546	20	the	the	DT
38536	1546	21	journey	journey	NN
38536	1546	22	in	in	IN
38536	1546	23	18	18	CD
38536	1546	24	hours	hour	NNS
38536	1546	25	.	.	.
38536	1547	1	Required	require	VBN
38536	1547	2	the	the	DT
38536	1547	3	total	total	JJ
38536	1547	4	distance	distance	NN
38536	1547	5	traveled	travel	VBD
38536	1547	6	.	.	.
38536	1548	1	~MASSACHUSETTS	~MASSACHUSETTS	NFP
38536	1548	2	INSTITUTE	INSTITUTE	NNP
38536	1548	3	OF	of	IN
38536	1548	4	TECHNOLOGY~	TECHNOLOGY~	NNP
38536	1548	5	ALGEBRA	ALGEBRA	NNP
38536	1548	6	B	B	NNP
38536	1548	7	TIME	time	NN
38536	1548	8	:	:	:
38536	1548	9	ONE	one	CD
38536	1548	10	HOUR	hour	NN
38536	1548	11	AND	and	CC
38536	1548	12	THREE	three	CD
38536	1548	13	QUARTERS	quarter	NNS
38536	1548	14	1	1	CD
38536	1548	15	.	.	.
38536	1549	1	How	how	WRB
38536	1549	2	many	many	JJ
38536	1549	3	terms	term	NNS
38536	1549	4	must	must	MD
38536	1549	5	be	be	VB
38536	1549	6	taken	take	VBN
38536	1549	7	in	in	IN
38536	1549	8	the	the	DT
38536	1549	9	series	series	NN
38536	1549	10	2	2	CD
38536	1549	11	,	,	,
38536	1549	12	5	5	CD
38536	1549	13	,	,	,
38536	1549	14	8	8	CD
38536	1549	15	,	,	,
38536	1549	16	11	11	CD
38536	1549	17	,	,	,
38536	1549	18	·	·	NFP
38536	1549	19	·	·	NFP
38536	1549	20	·	·	NFP
38536	1549	21	so	so	IN
38536	1549	22	that	that	IN
38536	1549	23	the	the	DT
38536	1549	24	sum	sum	NN
38536	1549	25	shall	shall	MD
38536	1549	26	be	be	VB
38536	1549	27	345	345	CD
38536	1549	28	?	?	.
38536	1550	1	2	2	LS
38536	1550	2	.	.	.
38536	1551	1	Prove	prove	VB
38536	1551	2	the	the	DT
38536	1551	3	formula	formula	NN
38536	1551	4	x	x	SYM
38536	1551	5	=	=	SYM
38536	1551	6	[	[	-LRB-
38536	1551	7	-b	-b	:
38536	1551	8	±	±	CD
38536	1551	9	[	[	-LRB-
38536	1551	10	b^2	b^2	NNS
38536	1551	11	-	-	SYM
38536	1551	12	4ac]^(1/2)]/(2a	4ac]^(1/2)]/(2a	CD
38536	1551	13	)	)	-RRB-
38536	1551	14	for	for	IN
38536	1551	15	solving	solve	VBG
38536	1551	16	the	the	DT
38536	1551	17	quadratic	quadratic	JJ
38536	1551	18	equation	equation	NN
38536	1551	19	ax^2	ax^2	CD
38536	1551	20	+	+	SYM
38536	1551	21	bx	bx	NN
38536	1551	22	+	+	SYM
38536	1551	23	c	c	NN
38536	1551	24	=	=	SYM
38536	1551	25	0	0	NFP
38536	1551	26	.	.	.
38536	1552	1	3	3	LS
38536	1552	2	.	.	.
38536	1553	1	Find	find	VB
38536	1553	2	all	all	DT
38536	1553	3	values	value	NNS
38536	1553	4	of	of	IN
38536	1553	5	a	a	DT
38536	1553	6	for	for	IN
38536	1553	7	which	which	WDT
38536	1553	8	[	[	-LRB-
38536	1553	9	\sq]a	\sq]a	CD
38536	1553	10	is	be	VBZ
38536	1553	11	a	a	DT
38536	1553	12	root	root	NN
38536	1553	13	of	of	IN
38536	1553	14	x^2	x^2	NNS
38536	1553	15	+	+	SYM
38536	1553	16	x	x	SYM
38536	1553	17	+	+	SYM
38536	1553	18	20	20	CD
38536	1553	19	=	=	SYM
38536	1553	20	2a	2a	CD
38536	1553	21	,	,	,
38536	1553	22	and	and	CC
38536	1553	23	check	check	VB
38536	1553	24	your	-PRON-	PRP$
38536	1553	25	results	result	NNS
38536	1553	26	.	.	.
38536	1554	1	4	4	LS
38536	1554	2	.	.	.
38536	1555	1	Solve	solve	VB
38536	1555	2	{	{	-LRB-
38536	1555	3	x^2	x^2	NNS
38536	1555	4	+	+	SYM
38536	1555	5	3y^2	3y^2	CD
38536	1555	6	=	=	SYM
38536	1555	7	10	10	CD
38536	1555	8	,	,	,
38536	1555	9	x	x	NNP
38536	1555	10	-	-	NN
38536	1555	11	y	y	NNP
38536	1555	12	=	=	SYM
38536	1555	13	2	2	CD
38536	1555	14	,	,	,
38536	1555	15	}	}	-RRB-
38536	1555	16	and	and	CC
38536	1555	17	sketch	sketch	VB
38536	1555	18	the	the	DT
38536	1555	19	graphs	graphs	NN
38536	1555	20	.	.	.
38536	1556	1	5	5	CD
38536	1556	2	.	.	.
38536	1557	1	The	the	DT
38536	1557	2	sum	sum	NN
38536	1557	3	of	of	IN
38536	1557	4	two	two	CD
38536	1557	5	numbers	number	NNS
38536	1557	6	x	x	NNPS
38536	1557	7	and	and	CC
38536	1557	8	y	y	NNP
38536	1557	9	is	be	VBZ
38536	1557	10	5	5	CD
38536	1557	11	,	,	,
38536	1557	12	and	and	CC
38536	1557	13	the	the	DT
38536	1557	14	sum	sum	NN
38536	1557	15	of	of	IN
38536	1557	16	the	the	DT
38536	1557	17	two	two	CD
38536	1557	18	middle	middle	JJ
38536	1557	19	terms	term	NNS
38536	1557	20	in	in	IN
38536	1557	21	the	the	DT
38536	1557	22	expansion	expansion	NN
38536	1557	23	of	of	IN
38536	1557	24	(	(	-LRB-
38536	1557	25	x	x	SYM
38536	1557	26	+	+	SYM
38536	1557	27	y)^3	y)^3	NNP
38536	1557	28	is	be	VBZ
38536	1557	29	equal	equal	JJ
38536	1557	30	to	to	IN
38536	1557	31	the	the	DT
38536	1557	32	sum	sum	NN
38536	1557	33	of	of	IN
38536	1557	34	the	the	DT
38536	1557	35	first	first	JJ
38536	1557	36	and	and	CC
38536	1557	37	last	last	JJ
38536	1557	38	terms	term	NNS
38536	1557	39	.	.	.
38536	1558	1	Find	find	VB
38536	1558	2	the	the	DT
38536	1558	3	numbers	number	NNS
38536	1558	4	.	.	.
38536	1559	1	6	6	CD
38536	1559	2	.	.	.
38536	1560	1	Solve	solve	VB
38536	1560	2	x^4	x^4	NNP
38536	1560	3	-	-	HYPH
38536	1560	4	2x^3	2x^3	CD
38536	1560	5	+	+	SYM
38536	1560	6	3x^2	3x^2	CD
38536	1560	7	-	-	HYPH
38536	1560	8	2x	2x	CD
38536	1560	9	+	+	SYM
38536	1560	10	1	1	CD
38536	1560	11	=	=	SYM
38536	1560	12	0	0	CD
38536	1560	13	.	.	.
38536	1561	1	(	(	-LRB-
38536	1561	2	HINT	HINT	NNS
38536	1561	3	:	:	:
38536	1561	4	Divide	divide	NN
38536	1561	5	by	by	IN
38536	1561	6	x^2	x^2	NNS
38536	1561	7	and	and	CC
38536	1561	8	substitute	substitute	NN
38536	1561	9	x	x	NNS
38536	1561	10	+	+	SYM
38536	1561	11	1	1	CD
38536	1561	12	/	/	SYM
38536	1561	13	x	x	NNS
38536	1561	14	=	=	SYM
38536	1561	15	z.	z.	NNP
38536	1561	16	)	)	-RRB-
38536	1562	1	7	7	LS
38536	1562	2	.	.	.
38536	1563	1	In	in	IN
38536	1563	2	anticipation	anticipation	NN
38536	1563	3	of	of	IN
38536	1563	4	a	a	DT
38536	1563	5	holiday	holiday	NN
38536	1563	6	a	a	DT
38536	1563	7	merchant	merchant	NN
38536	1563	8	makes	make	VBZ
38536	1563	9	an	an	DT
38536	1563	10	outlay	outlay	NN
38536	1563	11	of	of	IN
38536	1563	12	$	$	$
38536	1563	13	50	50	CD
38536	1563	14	,	,	,
38536	1563	15	which	which	WDT
38536	1563	16	will	will	MD
38536	1563	17	be	be	VB
38536	1563	18	a	a	DT
38536	1563	19	total	total	JJ
38536	1563	20	loss	loss	NN
38536	1563	21	in	in	IN
38536	1563	22	case	case	NN
38536	1563	23	of	of	IN
38536	1563	24	rain	rain	NN
38536	1563	25	,	,	,
38536	1563	26	but	but	CC
38536	1563	27	which	which	WDT
38536	1563	28	will	will	MD
38536	1563	29	bring	bring	VB
38536	1563	30	him	-PRON-	PRP
38536	1563	31	a	a	DT
38536	1563	32	clear	clear	JJ
38536	1563	33	profit	profit	NN
38536	1563	34	of	of	IN
38536	1563	35	$	$	$
38536	1563	36	150	150	CD
38536	1563	37	above	above	IN
38536	1563	38	the	the	DT
38536	1563	39	outlay	outlay	NN
38536	1563	40	if	if	IN
38536	1563	41	the	the	DT
38536	1563	42	day	day	NN
38536	1563	43	is	be	VBZ
38536	1563	44	pleasant	pleasant	JJ
38536	1563	45	.	.	.
38536	1564	1	To	to	TO
38536	1564	2	insure	insure	VB
38536	1564	3	against	against	IN
38536	1564	4	loss	loss	NN
38536	1564	5	he	-PRON-	PRP
38536	1564	6	takes	take	VBZ
38536	1564	7	out	out	RP
38536	1564	8	an	an	DT
38536	1564	9	insurance	insurance	NN
38536	1564	10	policy	policy	NN
38536	1564	11	against	against	IN
38536	1564	12	rain	rain	NN
38536	1564	13	for	for	IN
38536	1564	14	a	a	DT
38536	1564	15	certain	certain	JJ
38536	1564	16	sum	sum	NN
38536	1564	17	of	of	IN
38536	1564	18	money	money	NN
38536	1564	19	for	for	IN
38536	1564	20	which	which	WDT
38536	1564	21	he	-PRON-	PRP
38536	1564	22	has	have	VBZ
38536	1564	23	to	to	TO
38536	1564	24	pay	pay	VB
38536	1564	25	a	a	DT
38536	1564	26	certain	certain	JJ
38536	1564	27	percentage	percentage	NN
38536	1564	28	.	.	.
38536	1565	1	He	-PRON-	PRP
38536	1565	2	then	then	RB
38536	1565	3	finds	find	VBZ
38536	1565	4	that	that	IN
38536	1565	5	whether	whether	IN
38536	1565	6	the	the	DT
38536	1565	7	day	day	NN
38536	1565	8	be	be	VB
38536	1565	9	rainy	rainy	JJ
38536	1565	10	or	or	CC
38536	1565	11	pleasant	pleasant	JJ
38536	1565	12	he	-PRON-	PRP
38536	1565	13	will	will	MD
38536	1565	14	make	make	VB
38536	1565	15	$	$	$
38536	1565	16	80	80	CD
38536	1565	17	clear	clear	JJ
38536	1565	18	.	.	.
38536	1566	1	What	what	WP
38536	1566	2	is	be	VBZ
38536	1566	3	the	the	DT
38536	1566	4	amount	amount	NN
38536	1566	5	of	of	IN
38536	1566	6	the	the	DT
38536	1566	7	policy	policy	NN
38536	1566	8	,	,	,
38536	1566	9	and	and	CC
38536	1566	10	what	what	WDT
38536	1566	11	rate	rate	NN
38536	1566	12	did	do	VBD
38536	1566	13	the	the	DT
38536	1566	14	company	company	NN
38536	1566	15	charge	charge	VB
38536	1566	16	him	-PRON-	PRP
38536	1566	17	?	?	.
38536	1567	1	~MASSACHUSETTS	~MASSACHUSETTS	NFP
38536	1567	2	INSTITUTE	INSTITUTE	NNP
38536	1567	3	OF	of	IN
38536	1567	4	TECHNOLOGY~	TECHNOLOGY~	NNP
38536	1567	5	ALGEBRA	ALGEBRA	NNP
38536	1567	6	A	a	DT
38536	1567	7	TIME	time	NN
38536	1567	8	:	:	:
38536	1567	9	TWO	two	CD
38536	1567	10	HOURS	hours	NN
38536	1567	11	1	1	CD
38536	1567	12	.	.	.
38536	1568	1	Simplify	Simplify	NNP
38536	1568	2	[	[	-LRB-
38536	1568	3	m	m	NN
38536	1568	4	+	+	SYM
38536	1568	5	1	1	CD
38536	1568	6	/	/	SYM
38536	1568	7	m]^2	m]^2	NNP
38536	1568	8	+	+	CC
38536	1568	9	[	[	-LRB-
38536	1568	10	n	n	CD
38536	1568	11	+	+	SYM
38536	1568	12	1	1	CD
38536	1568	13	/	/	SYM
38536	1568	14	n]^2	n]^2	NNP
38536	1568	15	+	+	NFP
38536	1568	16	[	[	-LRB-
38536	1568	17	mn	mn	NNP
38536	1568	18	+	+	CC
38536	1568	19	1	1	CD
38536	1568	20	/	/	SYM
38536	1568	21	mn]^2	mn]^2	NN
38536	1568	22	-	-	HYPH
38536	1568	23	[	[	-LRB-
38536	1568	24	m	m	NN
38536	1568	25	+	+	SYM
38536	1568	26	1	1	CD
38536	1568	27	/	/	SYM
38536	1568	28	m][n	m][n	CD
38536	1568	29	+	+	SYM
38536	1568	30	1	1	CD
38536	1568	31	/	/	SYM
38536	1568	32	n][mn	n][mn	NNP
38536	1568	33	+	+	CC
38536	1568	34	1	1	CD
38536	1568	35	/	/	SYM
38536	1568	36	mn	mn	NNP
38536	1568	37	]	]	-RRB-
38536	1568	38	.	.	.
38536	1569	1	2	2	LS
38536	1569	2	.	.	.
38536	1570	1	Find	find	VB
38536	1570	2	the	the	DT
38536	1570	3	prime	prime	JJ
38536	1570	4	factors	factor	NNS
38536	1570	5	of	of	IN
38536	1570	6	(	(	-LRB-
38536	1570	7	_	_	NNP
38536	1570	8	a	a	DT
38536	1570	9	_	_	NNP
38536	1570	10	)	)	-RRB-
38536	1570	11	(	(	-LRB-
38536	1570	12	x	x	NN
38536	1570	13	-	-	NN
38536	1570	14	x^2)^3	x^2)^3	NN
38536	1570	15	+	+	NFP
38536	1570	16	(	(	-LRB-
38536	1570	17	x^2	x^2	NNP
38536	1570	18	-	-	HYPH
38536	1570	19	1)^3	1)^3	JJ
38536	1570	20	+	+	SYM
38536	1570	21	(	(	-LRB-
38536	1570	22	1	1	CD
38536	1570	23	-	-	HYPH
38536	1570	24	x)^3	x)^3	NNP
38536	1570	25	.	.	.
38536	1571	1	(	(	-LRB-
38536	1571	2	_	_	NNP
38536	1571	3	b	b	NNP
38536	1571	4	_	_	NNP
38536	1571	5	)	)	-RRB-
38536	1571	6	(	(	-LRB-
38536	1571	7	2x	2x	CD
38536	1571	8	+	+	CC
38536	1571	9	a	a	DT
38536	1571	10	-	-	HYPH
38536	1571	11	b)^4	b)^4	NN
38536	1571	12	-	-	HYPH
38536	1571	13	(	(	-LRB-
38536	1571	14	x	x	LS
38536	1571	15	-	-	:
38536	1571	16	a	a	NN
38536	1571	17	+	+	NN
38536	1571	18	b)^4	b)^4	NN
38536	1571	19	.	.	.
38536	1572	1	3	3	LS
38536	1572	2	.	.	.
38536	1573	1	(	(	-LRB-
38536	1573	2	_	_	NNP
38536	1573	3	a	a	DT
38536	1573	4	_	_	NNP
38536	1573	5	)	)	-RRB-
38536	1573	6	Simplify	Simplify	NNP
38536	1573	7	[	[	-LRB-
38536	1573	8	(	(	-LRB-
38536	1573	9	x^q)/(x^r)]^(q	x^q)/(x^r)]^(q	NNP
38536	1573	10	+	+	SYM
38536	1573	11	r	r	NN
38536	1573	12	)	)	-RRB-
38536	1573	13	[	[	-LRB-
38536	1573	14	(	(	-LRB-
38536	1573	15	x^r)/(x^p)]^(r	x^r)/(x^p)]^(r	NNP
38536	1573	16	+	+	NNP
38536	1573	17	p)[{x^p/{x^q}]^(p	p)[{x^p/{x^q}]^(p	NNP
38536	1573	18	+	+	SYM
38536	1573	19	q	q	NN
38536	1573	20	)	)	-RRB-
38536	1573	21	.	.	.
38536	1574	1	(	(	-LRB-
38536	1574	2	_	_	NNP
38536	1574	3	b	b	NNP
38536	1574	4	_	_	NNP
38536	1574	5	)	)	-RRB-
38536	1574	6	Show	show	VB
38536	1574	7	that	that	DT
38536	1574	8	(	(	-LRB-
38536	1574	9	[	[	-LRB-
38536	1574	10	[	[	-LRB-
38536	1574	11	x]^[1/(n+1)]]^(1	x]^[1/(n+1)]]^(1	NNP
38536	1574	12	/	/	SYM
38536	1574	13	n))/([[x]^[1/(n+2)]]^[1/(n+1	n))/([[x]^[1/(n+2)]]^[1/(n+1	NNP
38536	1574	14	)	)	-RRB-
38536	1574	15	]	]	-RRB-
38536	1574	16	)	)	-RRB-
38536	1574	17	=	=	NFP
38536	1574	18	{	{	-LRB-
38536	1574	19	x^(1	x^(1	ADD
38536	1574	20	/	/	SYM
38536	1574	21	n	n	NN
38536	1574	22	)	)	-RRB-
38536	1574	23	·	·	NFP
38536	1574	24	[	[	-LRB-
38536	1574	25	x]^[1/(n+2)]}/{[x^2]^[1/(n+1	x]^[1/(n+2)]}/{[x^2]^[1/(n+1	NNP
38536	1574	26	)	)	-RRB-
38536	1574	27	]	]	-RRB-
38536	1574	28	}	}	-RRB-
38536	1574	29	.	.	.
38536	1575	1	4	4	LS
38536	1575	2	.	.	.
38536	1576	1	Define	Define	NNP
38536	1576	2	_	_	NNP
38536	1576	3	homogeneous	homogeneous	JJ
38536	1576	4	terms	term	NNS
38536	1576	5	_	_	NNP
38536	1576	6	.	.	.
38536	1577	1	For	for	IN
38536	1577	2	what	what	WDT
38536	1577	3	value	value	NN
38536	1577	4	of	of	IN
38536	1577	5	n	n	NN
38536	1577	6	is	be	VBZ
38536	1577	7	x^n	x^n	NNP
38536	1577	8	y^(5	y^(5	,
38536	1577	9	-	-	HYPH
38536	1577	10	n/2	n/2	NNS
38536	1577	11	)	)	-RRB-
38536	1577	12	+	+	CC
38536	1577	13	x^(n	x^(n	NN
38536	1577	14	+	+	SYM
38536	1577	15	1	1	CD
38536	1577	16	)	)	-RRB-
38536	1577	17	y^(2n	y^(2n	CD
38536	1577	18	-	-	SYM
38536	1577	19	6	6	CD
38536	1577	20	)	)	-RRB-
38536	1577	21	a	a	DT
38536	1577	22	homogeneous	homogeneous	JJ
38536	1577	23	binomial	binomial	NN
38536	1577	24	?	?	.
38536	1578	1	5	5	CD
38536	1578	2	.	.	.
38536	1579	1	Extract	extract	VB
38536	1579	2	the	the	DT
38536	1579	3	square	square	JJ
38536	1579	4	root	root	NN
38536	1579	5	of	of	IN
38536	1579	6	x(x	x(x	NNP
38536	1579	7	-	-	HYPH
38536	1579	8	2^(1/2))(x	2^(1/2))(x	NNP
38536	1579	9	-	-	HYPH
38536	1579	10	8^(1/2))(x	8^(1/2))(x	NNP
38536	1579	11	-	-	HYPH
38536	1579	12	18^(1/2	18^(1/2	CD
38536	1579	13	)	)	-RRB-
38536	1579	14	)	)	-RRB-
38536	1579	15	+	+	CC
38536	1579	16	4	4	CD
38536	1579	17	.	.	.
38536	1580	1	6	6	CD
38536	1580	2	.	.	.
38536	1581	1	Two	two	CD
38536	1581	2	vessels	vessel	NNS
38536	1581	3	contain	contain	VBP
38536	1581	4	each	each	DT
38536	1581	5	a	a	DT
38536	1581	6	mixture	mixture	NN
38536	1581	7	of	of	IN
38536	1581	8	wine	wine	NN
38536	1581	9	and	and	CC
38536	1581	10	water	water	NN
38536	1581	11	.	.	.
38536	1582	1	In	in	IN
38536	1582	2	the	the	DT
38536	1582	3	first	first	JJ
38536	1582	4	vessel	vessel	NN
38536	1582	5	the	the	DT
38536	1582	6	quantity	quantity	NN
38536	1582	7	of	of	IN
38536	1582	8	wine	wine	NN
38536	1582	9	is	be	VBZ
38536	1582	10	to	to	IN
38536	1582	11	the	the	DT
38536	1582	12	quantity	quantity	NN
38536	1582	13	of	of	IN
38536	1582	14	water	water	NN
38536	1582	15	as	as	IN
38536	1582	16	1	1	CD
38536	1582	17	:	:	SYM
38536	1582	18	3	3	CD
38536	1582	19	,	,	,
38536	1582	20	and	and	CC
38536	1582	21	in	in	IN
38536	1582	22	the	the	DT
38536	1582	23	second	second	JJ
38536	1582	24	as	as	IN
38536	1582	25	3	3	CD
38536	1582	26	:	:	SYM
38536	1582	27	5	5	CD
38536	1582	28	.	.	.
38536	1583	1	What	what	WDT
38536	1583	2	quantity	quantity	NN
38536	1583	3	must	must	MD
38536	1583	4	be	be	VB
38536	1583	5	taken	take	VBN
38536	1583	6	from	from	IN
38536	1583	7	each	each	DT
38536	1583	8	,	,	,
38536	1583	9	so	so	IN
38536	1583	10	as	as	IN
38536	1583	11	to	to	TO
38536	1583	12	form	form	VB
38536	1583	13	a	a	DT
38536	1583	14	third	third	JJ
38536	1583	15	mixture	mixture	NN
38536	1583	16	which	which	WDT
38536	1583	17	shall	shall	MD
38536	1583	18	contain	contain	VB
38536	1583	19	5	5	CD
38536	1583	20	gallons	gallon	NNS
38536	1583	21	of	of	IN
38536	1583	22	wine	wine	NN
38536	1583	23	and	and	CC
38536	1583	24	9	9	CD
38536	1583	25	gallons	gallon	NNS
38536	1583	26	of	of	IN
38536	1583	27	water	water	NN
38536	1583	28	?	?	.
38536	1584	1	7	7	LS
38536	1584	2	.	.	.
38536	1585	1	Find	find	VB
38536	1585	2	a	a	DT
38536	1585	3	quantity	quantity	NN
38536	1585	4	such	such	JJ
38536	1585	5	that	that	IN
38536	1585	6	by	by	IN
38536	1585	7	adding	add	VBG
38536	1585	8	it	-PRON-	PRP
38536	1585	9	to	to	IN
38536	1585	10	each	each	DT
38536	1585	11	of	of	IN
38536	1585	12	the	the	DT
38536	1585	13	quantities	quantity	NNS
38536	1585	14	a	a	NNP
38536	1585	15	,	,	,
38536	1585	16	b	b	NN
38536	1585	17	,	,	,
38536	1585	18	c	c	NN
38536	1585	19	,	,	,
38536	1585	20	d	d	NNP
38536	1585	21	,	,	,
38536	1585	22	we	-PRON-	PRP
38536	1585	23	obtain	obtain	VBP
38536	1585	24	four	four	CD
38536	1585	25	quantities	quantity	NNS
38536	1585	26	in	in	IN
38536	1585	27	proportion	proportion	NN
38536	1585	28	.	.	.
38536	1586	1	8	8	LS
38536	1586	2	.	.	.
38536	1587	1	What	what	WDT
38536	1587	2	values	value	NNS
38536	1587	3	must	must	MD
38536	1587	4	be	be	VB
38536	1587	5	given	give	VBN
38536	1587	6	to	to	IN
38536	1587	7	a	a	DT
38536	1587	8	and	and	CC
38536	1587	9	b	b	NN
38536	1587	10	,	,	,
38536	1587	11	so	so	IN
38536	1587	12	that	that	IN
38536	1587	13	(	(	-LRB-
38536	1587	14	3a	3a	NN
38536	1587	15	+	+	SYM
38536	1587	16	2b	2b	CD
38536	1587	17	+	+	SYM
38536	1587	18	17)/2	17)/2	CD
38536	1587	19	,	,	,
38536	1587	20	(	(	-LRB-
38536	1587	21	2a	2a	CD
38536	1587	22	-	-	HYPH
38536	1587	23	3b	3b	NN
38536	1587	24	+	+	SYM
38536	1587	25	25)/3	25)/3	CD
38536	1587	26	,	,	,
38536	1587	27	4	4	CD
38536	1587	28	-	-	HYPH
38536	1587	29	5a	5a	CD
38536	1587	30	-	-	HYPH
38536	1587	31	13b	13b	NNS
38536	1587	32	may	may	MD
38536	1587	33	be	be	VB
38536	1587	34	equal	equal	JJ
38536	1587	35	?	?	.
38536	1588	1	~MOUNT	~MOUNT	NFP
38536	1588	2	HOLYOKE	HOLYOKE	NNP
38536	1588	3	COLLEGE~	COLLEGE~	NNP
38536	1588	4	ELEMENTARY	ELEMENTARY	NNP
38536	1588	5	ALGEBRA	ALGEBRA	NNP
38536	1588	6	TIME	time	NN
38536	1588	7	:	:	:
38536	1588	8	TWO	two	CD
38536	1588	9	HOURS	hours	NN
38536	1588	10	1	1	CD
38536	1588	11	.	.	.
38536	1589	1	Factor	factor	VB
38536	1589	2	the	the	DT
38536	1589	3	following	follow	VBG
38536	1589	4	expressions	expression	NNS
38536	1589	5	:	:	:
38536	1589	6	(	(	-LRB-
38536	1589	7	_	_	NNP
38536	1589	8	a	a	DT
38536	1589	9	_	_	NNP
38536	1589	10	)	)	-RRB-
38536	1589	11	a^(3/4	a^(3/4	NNP
38536	1589	12	)	)	-RRB-
38536	1589	13	-	-	HYPH
38536	1589	14	b^(3/4	b^(3/4	NN
38536	1589	15	)	)	-RRB-
38536	1589	16	.	.	.
38536	1590	1	(	(	-LRB-
38536	1590	2	_	_	NNP
38536	1590	3	b	b	NNP
38536	1590	4	_	_	NNP
38536	1590	5	)	)	-RRB-
38536	1590	6	x^2	x^2	NNP
38536	1590	7	y^2	y^2	NNP
38536	1590	8	z^2	z^2	FW
38536	1590	9	-	-	HYPH
38536	1590	10	x^2	x^2	NNP
38536	1590	11	z	z	NNP
38536	1590	12	-	-	HYPH
38536	1590	13	y^2	y^2	NNP
38536	1590	14	z	z	NN
38536	1590	15	+	+	SYM
38536	1590	16	1	1	CD
38536	1590	17	.	.	.
38536	1591	1	(	(	-LRB-
38536	1591	2	_	_	NNP
38536	1591	3	c	c	NNP
38536	1591	4	_	_	NNP
38536	1591	5	)	)	-RRB-
38536	1591	6	16(x	16(x	CD
38536	1591	7	+	+	SYM
38536	1591	8	y)^4	y)^4	NNS
38536	1591	9	-	-	:
38536	1591	10	(	(	-LRB-
38536	1591	11	2x	2x	NN
38536	1591	12	-	-	,
38536	1591	13	y)^4	y)^4	NNS
38536	1591	14	.	.	.
38536	1592	1	2	2	LS
38536	1592	2	.	.	.
38536	1593	1	(	(	-LRB-
38536	1593	2	_	_	NNP
38536	1593	3	a	a	DT
38536	1593	4	_	_	NNP
38536	1593	5	)	)	-RRB-
38536	1593	6	Simplify	Simplify	NNP
38536	1593	7	(	(	-LRB-
38536	1593	8	a^2	a^2	NNP
38536	1593	9	+	+	SYM
38536	1593	10	b^2){(b^4)/(b^2	b^2){(b^4)/(b^2	NNP
38536	1593	11	-	-	HYPH
38536	1593	12	a^2	a^2	NNP
38536	1593	13	)	)	-RRB-
38536	1593	14	-	-	:
38536	1593	15	a^2}/{a/(a	a^2}/{a/(a	NNS
38536	1593	16	+	+	SYM
38536	1593	17	b	b	NN
38536	1593	18	)	)	-RRB-
38536	1593	19	+	+	SYM
38536	1593	20	b/(a	b/(a	NN
38536	1593	21	-	-	HYPH
38536	1593	22	b	b	NN
38536	1593	23	)	)	-RRB-
38536	1593	24	}	}	-RRB-
38536	1593	25	}	}	-RRB-
38536	1593	26	.	.	.
38536	1594	1	(	(	-LRB-
38536	1594	2	_	_	NNP
38536	1594	3	b	b	NNP
38536	1594	4	_	_	NNP
38536	1594	5	)	)	-RRB-
38536	1594	6	Extract	extract	VB
38536	1594	7	the	the	DT
38536	1594	8	square	square	JJ
38536	1594	9	root	root	NN
38536	1594	10	of	of	IN
38536	1594	11	x^4	x^4	NNP
38536	1594	12	-	-	HYPH
38536	1594	13	2x^3	2x^3	CD
38536	1594	14	+	+	CC
38536	1594	15	5x^2	5x^2	CD
38536	1594	16	-	-	HYPH
38536	1594	17	4x	4x	NNS
38536	1594	18	+	+	SYM
38536	1594	19	4	4	CD
38536	1594	20	.	.	.
38536	1595	1	3	3	LS
38536	1595	2	.	.	.
38536	1596	1	Solve	solve	VB
38536	1596	2	the	the	DT
38536	1596	3	following	following	JJ
38536	1596	4	equations	equation	NNS
38536	1596	5	:	:	:
38536	1596	6	(	(	-LRB-
38536	1596	7	_	_	NNP
38536	1596	8	a	a	DT
38536	1596	9	_	_	NNP
38536	1596	10	)	)	-RRB-
38536	1596	11	1	1	CD
38536	1596	12	/	/	SYM
38536	1596	13	x	x	NNS
38536	1596	14	+	+	SYM
38536	1596	15	1	1	CD
38536	1596	16	/	/	SYM
38536	1596	17	y	y	NN
38536	1596	18	=	=	SYM
38536	1596	19	5	5	CD
38536	1596	20	,	,	,
38536	1596	21	1/(x^2	1/(x^2	CD
38536	1596	22	)	)	-RRB-
38536	1596	23	+	+	SYM
38536	1596	24	1/(y^2	1/(y^2	CD
38536	1596	25	)	)	-RRB-
38536	1596	26	=	=	SYM
38536	1596	27	13	13	CD
38536	1596	28	.	.	.
38536	1597	1	(	(	-LRB-
38536	1597	2	_	_	NNP
38536	1597	3	b	b	NNP
38536	1597	4	_	_	NNP
38536	1597	5	)	)	-RRB-
38536	1597	6	x^2	x^2	NNP
38536	1597	7	-	-	HYPH
38536	1597	8	5x	5x	CD
38536	1597	9	+	+	SYM
38536	1597	10	2	2	CD
38536	1597	11	=	=	SYM
38536	1597	12	0	0	CD
38536	1597	13	.	.	.
38536	1598	1	(	(	-LRB-
38536	1598	2	_	_	NNP
38536	1598	3	c	c	NNP
38536	1598	4	_	_	NNP
38536	1598	5	)	)	-RRB-
38536	1598	6	[	[	-LRB-
38536	1598	7	27x	27x	NNS
38536	1598	8	+	+	SYM
38536	1598	9	1]^(1/2	1]^(1/2	NN
38536	1598	10	)	)	-RRB-
38536	1598	11	=	=	SYM
38536	1598	12	2	2	CD
38536	1598	13	-	-	SYM
38536	1598	14	3[3x^(1/2	3[3x^(1/2	CD
38536	1598	15	)	)	-RRB-
38536	1598	16	]	]	-RRB-
38536	1598	17	.	.	.
38536	1599	1	4	4	LS
38536	1599	2	.	.	.
38536	1600	1	Simplify	simplify	NN
38536	1600	2	:	:	:
38536	1600	3	(	(	-LRB-
38536	1600	4	_	_	NNP
38536	1600	5	a	a	DT
38536	1600	6	_	_	NNP
38536	1600	7	)	)	-RRB-
38536	1600	8	7[54]^(1/3	7[54]^(1/3	CD
38536	1600	9	)	)	-RRB-
38536	1600	10	+	+	SYM
38536	1600	11	256^(1/6	256^(1/6	NNS
38536	1600	12	)	)	-RRB-
38536	1600	13	+	+	CC
38536	1600	14	[	[	-LRB-
38536	1600	15	432/(-250)]^(1/3	432/(-250)]^(1/3	CD
38536	1600	16	)	)	-RRB-
38536	1600	17	.	.	.
38536	1601	1	(	(	-LRB-
38536	1601	2	_	_	NNP
38536	1601	3	b	b	NNP
38536	1601	4	_	_	NNP
38536	1601	5	)	)	-RRB-
38536	1601	6	1/[(a	1/[(a	CD
38536	1601	7	-	-	HYPH
38536	1601	8	b)(b	b)(b	NNP
38536	1601	9	-	-	HYPH
38536	1601	10	c	c	NNP
38536	1601	11	)	)	-RRB-
38536	1601	12	]	]	-RRB-
38536	1601	13	+	+	CC
38536	1601	14	1/[(c	1/[(c	CD
38536	1601	15	-	-	HYPH
38536	1601	16	a)(b	a)(b	NNP
38536	1601	17	-	-	HYPH
38536	1601	18	a	a	NNP
38536	1601	19	)	)	-RRB-
38536	1601	20	]	]	-RRB-
38536	1601	21	.	.	.
38536	1602	1	(	(	-LRB-
38536	1602	2	_	_	NNP
38536	1602	3	c	c	NNP
38536	1602	4	_	_	NNP
38536	1602	5	)	)	-RRB-
38536	1602	6	Find	find	VB
38536	1602	7	[	[	-LRB-
38536	1602	8	19	19	CD
38536	1602	9	-	-	HYPH
38536	1602	10	8[3^(1/2)]]^(1/2	8[3^(1/2)]]^(1/2	NNS
38536	1602	11	)	)	-RRB-
38536	1602	12	.	.	.
38536	1603	1	5	5	CD
38536	1603	2	.	.	.
38536	1604	1	Plot	plot	VB
38536	1604	2	the	the	DT
38536	1604	3	graphs	graphs	NN
38536	1604	4	of	of	IN
38536	1604	5	the	the	DT
38536	1604	6	following	follow	VBG
38536	1604	7	system	system	NN
38536	1604	8	,	,	,
38536	1604	9	and	and	CC
38536	1604	10	determine	determine	VB
38536	1604	11	the	the	DT
38536	1604	12	solution	solution	NN
38536	1604	13	from	from	IN
38536	1604	14	the	the	DT
38536	1604	15	point	point	NN
38536	1604	16	of	of	IN
38536	1604	17	intersection	intersection	NN
38536	1604	18	:	:	:
38536	1604	19	{	{	-LRB-
38536	1604	20	x	x	NN
38536	1604	21	-	-	SYM
38536	1604	22	2y	2y	NNP
38536	1604	23	=	=	SYM
38536	1604	24	0	0	CD
38536	1604	25	,	,	,
38536	1604	26	{	{	-LRB-
38536	1604	27	2x	2x	CD
38536	1604	28	-	-	HYPH
38536	1604	29	3y	3y	NNS
38536	1604	30	=	=	SYM
38536	1604	31	4	4	CD
38536	1604	32	.	.	.
38536	1605	1	6	6	CD
38536	1605	2	.	.	.
38536	1606	1	(	(	-LRB-
38536	1606	2	_	_	NNP
38536	1606	3	a	a	DT
38536	1606	4	_	_	NNP
38536	1606	5	)	)	-RRB-
38536	1606	6	Derive	derive	VB
38536	1606	7	the	the	DT
38536	1606	8	formula	formula	NN
38536	1606	9	for	for	IN
38536	1606	10	the	the	DT
38536	1606	11	solution	solution	NN
38536	1606	12	of	of	IN
38536	1606	13	ax^2	ax^2	NN
38536	1606	14	+	+	SYM
38536	1606	15	bx	bx	NN
38536	1606	16	+	+	NNS
38536	1606	17	c	c	NN
38536	1606	18	=	=	SYM
38536	1606	19	0	0	NFP
38536	1606	20	.	.	.
38536	1607	1	(	(	-LRB-
38536	1607	2	_	_	NNP
38536	1607	3	b	b	NNP
38536	1607	4	_	_	NNP
38536	1607	5	)	)	-RRB-
38536	1607	6	Determine	determine	VB
38536	1607	7	the	the	DT
38536	1607	8	value	value	NN
38536	1607	9	of	of	IN
38536	1607	10	m	m	NN
38536	1607	11	for	for	IN
38536	1607	12	which	which	WDT
38536	1607	13	the	the	DT
38536	1607	14	roots	root	NNS
38536	1607	15	of	of	IN
38536	1607	16	2x^2	2x^2	CD
38536	1607	17	+	+	SYM
38536	1607	18	4x	4x	NNS
38536	1607	19	+	+	SYM
38536	1607	20	m	m	NN
38536	1607	21	=	=	SYM
38536	1607	22	0	0	CD
38536	1607	23	are	be	VBP
38536	1607	24	(	(	-LRB-
38536	1607	25	i	i	PRP
38536	1607	26	)	)	-RRB-
38536	1607	27	equal	equal	JJ
38536	1607	28	,	,	,
38536	1607	29	(	(	-LRB-
38536	1607	30	ii	ii	LS
38536	1607	31	)	)	-RRB-
38536	1607	32	real	real	JJ
38536	1607	33	,	,	,
38536	1607	34	(	(	-LRB-
38536	1607	35	iii	iii	NN
38536	1607	36	)	)	-RRB-
38536	1607	37	imaginary	imaginary	NN
38536	1607	38	.	.	.
38536	1608	1	(	(	-LRB-
38536	1608	2	_	_	NNP
38536	1608	3	c	c	NNP
38536	1608	4	_	_	NNP
38536	1608	5	)	)	-RRB-
38536	1608	6	Form	form	VB
38536	1608	7	the	the	DT
38536	1608	8	quadratic	quadratic	JJ
38536	1608	9	equation	equation	NN
38536	1608	10	whose	whose	WP$
38536	1608	11	roots	root	NNS
38536	1608	12	are	be	VBP
38536	1608	13	2	2	CD
38536	1608	14	+	+	SYM
38536	1608	15	3^(1/2	3^(1/2	CD
38536	1608	16	)	)	-RRB-
38536	1608	17	and	and	CC
38536	1608	18	2	2	CD
38536	1608	19	-	-	SYM
38536	1608	20	3^(1/2	3^(1/2	CD
38536	1608	21	)	)	-RRB-
38536	1608	22	.	.	.
38536	1609	1	7	7	LS
38536	1609	2	.	.	.
38536	1610	1	A	a	DT
38536	1610	2	page	page	NN
38536	1610	3	is	be	VBZ
38536	1610	4	to	to	TO
38536	1610	5	have	have	VB
38536	1610	6	a	a	DT
38536	1610	7	margin	margin	NN
38536	1610	8	of	of	IN
38536	1610	9	1	1	CD
38536	1610	10	inch	inch	NN
38536	1610	11	,	,	,
38536	1610	12	and	and	CC
38536	1610	13	is	be	VBZ
38536	1610	14	to	to	TO
38536	1610	15	contain	contain	VB
38536	1610	16	35	35	CD
38536	1610	17	square	square	JJ
38536	1610	18	inches	inch	NNS
38536	1610	19	of	of	IN
38536	1610	20	printing	printing	NN
38536	1610	21	.	.	.
38536	1611	1	How	how	WRB
38536	1611	2	large	large	JJ
38536	1611	3	must	must	MD
38536	1611	4	the	the	DT
38536	1611	5	page	page	NN
38536	1611	6	be	be	VB
38536	1611	7	,	,	,
38536	1611	8	if	if	IN
38536	1611	9	the	the	DT
38536	1611	10	length	length	NN
38536	1611	11	is	be	VBZ
38536	1611	12	to	to	TO
38536	1611	13	exceed	exceed	VB
38536	1611	14	the	the	DT
38536	1611	15	width	width	NN
38536	1611	16	by	by	IN
38536	1611	17	2	2	CD
38536	1611	18	inches	inch	NNS
38536	1611	19	?	?	.
38536	1612	1	8	8	LS
38536	1612	2	.	.	.
38536	1613	1	(	(	-LRB-
38536	1613	2	_	_	NNP
38536	1613	3	a	a	DT
38536	1613	4	_	_	NNP
38536	1613	5	)	)	-RRB-
38536	1613	6	In	in	IN
38536	1613	7	an	an	DT
38536	1613	8	arithmetical	arithmetical	JJ
38536	1613	9	progression	progression	NN
38536	1613	10	the	the	DT
38536	1613	11	sum	sum	NN
38536	1613	12	of	of	IN
38536	1613	13	the	the	DT
38536	1613	14	first	first	JJ
38536	1613	15	six	six	CD
38536	1613	16	terms	term	NNS
38536	1613	17	is	be	VBZ
38536	1613	18	261	261	CD
38536	1613	19	,	,	,
38536	1613	20	and	and	CC
38536	1613	21	the	the	DT
38536	1613	22	sum	sum	NN
38536	1613	23	of	of	IN
38536	1613	24	the	the	DT
38536	1613	25	first	first	JJ
38536	1613	26	nine	nine	CD
38536	1613	27	terms	term	NNS
38536	1613	28	is	be	VBZ
38536	1613	29	297	297	CD
38536	1613	30	.	.	.
38536	1614	1	Find	find	VB
38536	1614	2	the	the	DT
38536	1614	3	common	common	JJ
38536	1614	4	difference	difference	NN
38536	1614	5	.	.	.
38536	1615	1	(	(	-LRB-
38536	1615	2	_	_	NNP
38536	1615	3	b	b	NNP
38536	1615	4	_	_	NNP
38536	1615	5	)	)	-RRB-
38536	1615	6	Three	three	CD
38536	1615	7	numbers	number	NNS
38536	1615	8	whose	whose	WP$
38536	1615	9	sum	sum	NN
38536	1615	10	is	be	VBZ
38536	1615	11	27	27	CD
38536	1615	12	are	be	VBP
38536	1615	13	in	in	IN
38536	1615	14	arithmetical	arithmetical	JJ
38536	1615	15	progression	progression	NN
38536	1615	16	.	.	.
38536	1616	1	If	if	IN
38536	1616	2	1	1	CD
38536	1616	3	is	be	VBZ
38536	1616	4	added	add	VBN
38536	1616	5	to	to	IN
38536	1616	6	the	the	DT
38536	1616	7	first	first	JJ
38536	1616	8	,	,	,
38536	1616	9	3	3	CD
38536	1616	10	to	to	IN
38536	1616	11	the	the	DT
38536	1616	12	second	second	JJ
38536	1616	13	,	,	,
38536	1616	14	and	and	CC
38536	1616	15	11	11	CD
38536	1616	16	to	to	IN
38536	1616	17	the	the	DT
38536	1616	18	third	third	JJ
38536	1616	19	,	,	,
38536	1616	20	the	the	DT
38536	1616	21	sums	sum	NNS
38536	1616	22	will	will	MD
38536	1616	23	be	be	VB
38536	1616	24	in	in	IN
38536	1616	25	geometrical	geometrical	JJ
38536	1616	26	progression	progression	NN
38536	1616	27	.	.	.
38536	1617	1	Find	find	VB
38536	1617	2	the	the	DT
38536	1617	3	numbers	number	NNS
38536	1617	4	.	.	.
38536	1618	1	(	(	-LRB-
38536	1618	2	_	_	NNP
38536	1618	3	c	c	NNP
38536	1618	4	_	_	NNP
38536	1618	5	)	)	-RRB-
38536	1618	6	Derive	derive	VB
38536	1618	7	the	the	DT
38536	1618	8	formula	formula	NN
38536	1618	9	for	for	IN
38536	1618	10	the	the	DT
38536	1618	11	sum	sum	NN
38536	1618	12	of	of	IN
38536	1618	13	_	_	NNP
38536	1618	14	n	n	CC
38536	1618	15	_	_	NNP
38536	1618	16	terms	term	NNS
38536	1618	17	of	of	IN
38536	1618	18	a	a	DT
38536	1618	19	geometrical	geometrical	JJ
38536	1618	20	progression	progression	NN
38536	1618	21	.	.	.
38536	1619	1	9	9	CD
38536	1619	2	.	.	.
38536	1620	1	(	(	-LRB-
38536	1620	2	_	_	NNP
38536	1620	3	a	a	DT
38536	1620	4	_	_	NNP
38536	1620	5	)	)	-RRB-
38536	1620	6	Expand	expand	VB
38536	1620	7	and	and	CC
38536	1620	8	simplify	simplify	VB
38536	1620	9	(	(	-LRB-
38536	1620	10	2a^2	2a^2	CD
38536	1620	11	-	-	SYM
38536	1620	12	3x^3)^7	3x^3)^7	CD
38536	1620	13	.	.	.
38536	1621	1	(	(	-LRB-
38536	1621	2	_	_	NNP
38536	1621	3	b	b	NNP
38536	1621	4	_	_	NNP
38536	1621	5	)	)	-RRB-
38536	1621	6	For	for	IN
38536	1621	7	what	what	WDT
38536	1621	8	value	value	NN
38536	1621	9	of	of	IN
38536	1621	10	x	x	NN
38536	1621	11	will	will	MD
38536	1621	12	the	the	DT
38536	1621	13	ratio	ratio	NN
38536	1621	14	7	7	CD
38536	1621	15	+	+	SYM
38536	1621	16	x	x	SYM
38536	1621	17	:	:	:
38536	1621	18	12	12	CD
38536	1621	19	+	+	SYM
38536	1621	20	x	x	TO
38536	1621	21	be	be	VB
38536	1621	22	equal	equal	JJ
38536	1621	23	to	to	IN
38536	1621	24	the	the	DT
38536	1621	25	ratio	ratio	NN
38536	1621	26	5	5	CD
38536	1621	27	:	:	SYM
38536	1621	28	6	6	CD
38536	1621	29	?	?	.
38536	1622	1	~UNIVERSITY	~UNIVERSITY	NNP
38536	1622	2	OF	of	IN
38536	1622	3	PENNSYLVANIA~	PENNSYLVANIA~	NNP
38536	1622	4	ELEMENTARY	ELEMENTARY	NNP
38536	1622	5	ALGEBRA	ALGEBRA	NNP
38536	1622	6	TIME	time	NN
38536	1622	7	:	:	:
38536	1622	8	THREE	three	CD
38536	1622	9	HOURS	hours	NN
38536	1622	10	1	1	CD
38536	1622	11	.	.	.
38536	1623	1	Simplify	simplify	NN
38536	1623	2	:	:	:
38536	1623	3	[	[	-LRB-
38536	1623	4	(	(	-LRB-
38536	1623	5	a	a	DT
38536	1623	6	+	+	SYM
38536	1623	7	x)/(a	x)/(a	NNP
38536	1623	8	-	-	HYPH
38536	1623	9	x	x	NNP
38536	1623	10	)	)	-RRB-
38536	1623	11	-	-	:
38536	1623	12	(	(	-LRB-
38536	1623	13	a	a	NN
38536	1623	14	-	-	HYPH
38536	1623	15	x)/(a	x)/(a	NN
38536	1623	16	+	+	SYM
38536	1623	17	x	x	SYM
38536	1623	18	)	)	-RRB-
38536	1623	19	]	]	-RRB-
38536	1623	20	÷	÷	NNP
38536	1623	21	(	(	-LRB-
38536	1623	22	4ax)/(a^2	4ax)/(a^2	CD
38536	1623	23	-	-	HYPH
38536	1623	24	x^2	x^2	JJ
38536	1623	25	)	)	-RRB-
38536	1623	26	.	.	.
38536	1624	1	2	2	LS
38536	1624	2	.	.	.
38536	1625	1	Find	find	VB
38536	1625	2	the	the	DT
38536	1625	3	H.	H.	NNP
38536	1625	4	C.	C.	NNP
38536	1625	5	F.	F.	NNP
38536	1625	6	and	and	CC
38536	1625	7	L.	L.	NNP
38536	1625	8	C.	C.	NNP
38536	1625	9	M.	M.	NNP
38536	1625	10	of	of	IN
38536	1625	11	10ab^2(x^2	10ab^2(x^2	NNP
38536	1625	12	-	-	HYPH
38536	1625	13	2ax	2ax	NN
38536	1625	14	)	)	-RRB-
38536	1625	15	,	,	,
38536	1625	16	15a^3b(x^2	15a^3b(x^2	CD
38536	1625	17	-	-	HYPH
38536	1625	18	ax	ax	CD
38536	1625	19	-	-	HYPH
38536	1625	20	2a^2	2a^2	CD
38536	1625	21	)	)	-RRB-
38536	1625	22	,	,	,
38536	1625	23	25b^3(x^2	25b^3(x^2	CD
38536	1625	24	-	-	HYPH
38536	1625	25	a^2)^2	a^2)^2	RB
38536	1625	26	.	.	.
38536	1626	1	3	3	LS
38536	1626	2	.	.	.
38536	1627	1	A	a	DT
38536	1627	2	grocer	grocer	NN
38536	1627	3	buys	buy	VBZ
38536	1627	4	eggs	egg	NNS
38536	1627	5	at	at	IN
38536	1627	6	4	4	CD
38536	1627	7	for	for	IN
38536	1627	8	7¢.	7¢.	CD
38536	1628	1	He	-PRON-	PRP
38536	1628	2	sells	sell	VBZ
38536	1628	3	1/4	1/4	CD
38536	1628	4	of	of	IN
38536	1628	5	them	-PRON-	PRP
38536	1628	6	at	at	IN
38536	1628	7	5	5	CD
38536	1628	8	for	for	IN
38536	1628	9	12¢	12¢	CD
38536	1628	10	,	,	,
38536	1628	11	and	and	CC
38536	1628	12	the	the	DT
38536	1628	13	rest	rest	NN
38536	1628	14	at	at	IN
38536	1628	15	6	6	CD
38536	1628	16	for	for	IN
38536	1628	17	11¢	11¢	CD
38536	1628	18	,	,	,
38536	1628	19	making	make	VBG
38536	1628	20	27¢	27¢	CD
38536	1628	21	by	by	IN
38536	1628	22	the	the	DT
38536	1628	23	transaction	transaction	NN
38536	1628	24	.	.	.
38536	1629	1	How	how	WRB
38536	1629	2	many	many	JJ
38536	1629	3	eggs	egg	NNS
38536	1629	4	does	do	VBZ
38536	1629	5	he	-PRON-	PRP
38536	1629	6	buy	buy	VB
38536	1629	7	?	?	.
38536	1630	1	4	4	LS
38536	1630	2	.	.	.
38536	1631	1	Solve	solve	VB
38536	1631	2	for	for	IN
38536	1631	3	t	t	NN
38536	1631	4	:	:	:
38536	1631	5	(	(	-LRB-
38536	1631	6	t	t	NN
38536	1631	7	+	+	CC
38536	1631	8	4a	4a	NN
38536	1631	9	+	+	SYM
38536	1631	10	b)/(t	b)/(t	NN
38536	1631	11	+	+	CC
38536	1631	12	a	a	NN
38536	1631	13	+	+	SYM
38536	1631	14	b	b	NN
38536	1631	15	)	)	-RRB-
38536	1631	16	-	-	:
38536	1631	17	(	(	-LRB-
38536	1631	18	4	4	CD
38536	1631	19	t	t	NN
38536	1631	20	-	-	HYPH
38536	1631	21	a	a	DT
38536	1631	22	-	-	HYPH
38536	1631	23	2b)/(t	2b)/(t	CD
38536	1631	24	+	+	CC
38536	1631	25	a	a	NN
38536	1631	26	-	-	HYPH
38536	1631	27	b	b	NN
38536	1631	28	)	)	-RRB-
38536	1631	29	=	=	NFP
38536	1631	30	-3	-3	:
38536	1631	31	.	.	.
38536	1632	1	5	5	CD
38536	1632	2	.	.	.
38536	1633	1	Find	find	VB
38536	1633	2	the	the	DT
38536	1633	3	square	square	JJ
38536	1633	4	root	root	NN
38536	1633	5	of	of	IN
38536	1633	6	a^2	a^2	NNP
38536	1633	7	-	-	HYPH
38536	1633	8	(	(	-LRB-
38536	1633	9	3/2)a^(3/2	3/2)a^(3/2	NNP
38536	1633	10	)	)	-RRB-
38536	1633	11	-	-	:
38536	1633	12	(	(	-LRB-
38536	1633	13	3/2)a^(1/2	3/2)a^(1/2	NNP
38536	1633	14	)	)	-RRB-
38536	1633	15	+	+	CC
38536	1633	16	(	(	-LRB-
38536	1633	17	41/16)a	41/16)a	CD
38536	1633	18	+	+	SYM
38536	1633	19	1	1	CD
38536	1633	20	.	.	.
38536	1634	1	6	6	CD
38536	1634	2	.	.	.
38536	1635	1	(	(	-LRB-
38536	1635	2	_	_	NNP
38536	1635	3	a	a	DT
38536	1635	4	_	_	NNP
38536	1635	5	)	)	-RRB-
38536	1635	6	For	for	IN
38536	1635	7	what	what	WDT
38536	1635	8	values	value	NNS
38536	1635	9	of	of	IN
38536	1635	10	m	m	NN
38536	1635	11	will	will	MD
38536	1635	12	the	the	DT
38536	1635	13	roots	root	NNS
38536	1635	14	of	of	IN
38536	1635	15	2x^2	2x^2	CD
38536	1635	16	+	+	CD
38536	1635	17	3mx	3mx	NN
38536	1635	18	=	=	SYM
38536	1635	19	-2	-2	:
38536	1635	20	be	be	VB
38536	1635	21	equal	equal	JJ
38536	1635	22	?	?	.
38536	1636	1	(	(	-LRB-
38536	1636	2	_	_	NNP
38536	1636	3	b	b	NNP
38536	1636	4	_	_	NNP
38536	1636	5	)	)	-RRB-
38536	1636	6	If	if	IN
38536	1636	7	2a	2a	CD
38536	1636	8	+	+	SYM
38536	1636	9	3b	3b	NN
38536	1636	10	is	be	VBZ
38536	1636	11	a	a	DT
38536	1636	12	root	root	NN
38536	1636	13	of	of	IN
38536	1636	14	x^2	x^2	JJ
38536	1636	15	-	-	JJ
38536	1636	16	6bx	6bx	JJ
38536	1636	17	-	-	HYPH
38536	1636	18	4a^2	4a^2	CD
38536	1636	19	+	+	SYM
38536	1636	20	9b^2	9b^2	CD
38536	1636	21	=	=	SYM
38536	1636	22	0	0	CD
38536	1636	23	,	,	,
38536	1636	24	find	find	VB
38536	1636	25	the	the	DT
38536	1636	26	other	other	JJ
38536	1636	27	root	root	NN
38536	1636	28	without	without	IN
38536	1636	29	solving	solve	VBG
38536	1636	30	the	the	DT
38536	1636	31	equation	equation	NN
38536	1636	32	.	.	.
38536	1637	1	7	7	LS
38536	1637	2	.	.	.
38536	1638	1	(	(	-LRB-
38536	1638	2	_	_	NNP
38536	1638	3	a	a	DT
38536	1638	4	_	_	NNP
38536	1638	5	)	)	-RRB-
38536	1638	6	Solve	solve	VB
38536	1638	7	for	for	IN
38536	1638	8	x	x	NNS
38536	1638	9	:	:	:
38536	1638	10	[	[	-LRB-
38536	1638	11	2x	2x	CD
38536	1638	12	-	-	SYM
38536	1638	13	3a]^(1/2	3a]^(1/2	CD
38536	1638	14	)	)	-RRB-
38536	1638	15	+	+	CC
38536	1638	16	[	[	-LRB-
38536	1638	17	3x	3x	CD
38536	1638	18	-	-	HYPH
38536	1638	19	2a]^(1/2	2a]^(1/2	CD
38536	1638	20	)	)	-RRB-
38536	1638	21	=	=	NFP
38536	1638	22	3[a^(1/2	3[a^(1/2	CD
38536	1638	23	)	)	-RRB-
38536	1638	24	]	]	-RRB-
38536	1638	25	.	.	.
38536	1639	1	(	(	-LRB-
38536	1639	2	_	_	NNP
38536	1639	3	b	b	NNP
38536	1639	4	_	_	NNP
38536	1639	5	)	)	-RRB-
38536	1639	6	Solve	solve	VB
38536	1639	7	for	for	IN
38536	1639	8	m	m	NN
38536	1639	9	:	:	:
38536	1639	10	1	1	CD
38536	1639	11	-	-	HYPH
38536	1639	12	(	(	-LRB-
38536	1639	13	1)/(2	1)/(2	NN
38536	1639	14	-	-	HYPH
38536	1639	15	m	m	NN
38536	1639	16	)	)	-RRB-
38536	1639	17	=	=	NFP
38536	1639	18	1/(m	1/(m	CD
38536	1639	19	+	+	SYM
38536	1639	20	2	2	CD
38536	1639	21	)	)	-RRB-
38536	1639	22	+	+	CC
38536	1639	23	(	(	-LRB-
38536	1639	24	m	m	NNP
38536	1639	25	-	-	HYPH
38536	1639	26	6)/(4	6)/(4	NNP
38536	1639	27	-	-	HYPH
38536	1639	28	m^2	m^2	CD
38536	1639	29	)	)	-RRB-
38536	1639	30	.	.	.
38536	1640	1	8	8	LS
38536	1640	2	.	.	.
38536	1641	1	Solve	solve	VB
38536	1641	2	the	the	DT
38536	1641	3	system	system	NN
38536	1641	4	:	:	:
38536	1641	5	x^2	x^2	NNS
38536	1641	6	+	+	SYM
38536	1641	7	2y^2	2y^2	CD
38536	1641	8	=	=	SYM
38536	1641	9	17	17	CD
38536	1641	10	;	;	:
38536	1641	11	xy	xy	NNP
38536	1641	12	-	-	HYPH
38536	1641	13	y^2	y^2	NNP
38536	1641	14	=	=	SYM
38536	1641	15	2	2	CD
38536	1641	16	.	.	.
38536	1642	1	9	9	CD
38536	1642	2	.	.	.
38536	1643	1	Two	two	CD
38536	1643	2	boats	boat	NNS
38536	1643	3	leave	leave	VBP
38536	1643	4	simultaneously	simultaneously	RB
38536	1643	5	opposite	opposite	JJ
38536	1643	6	shores	shore	NNS
38536	1643	7	of	of	IN
38536	1643	8	a	a	DT
38536	1643	9	river	river	NN
38536	1643	10	2	2	CD
38536	1643	11	-	-	SYM
38536	1643	12	1/4	1/4	CD
38536	1643	13	mi	mi	NNP
38536	1643	14	.	.	.
38536	1644	1	wide	wide	JJ
38536	1644	2	and	and	CC
38536	1644	3	pass	pass	VB
38536	1644	4	each	each	DT
38536	1644	5	other	other	JJ
38536	1644	6	in	in	IN
38536	1644	7	15	15	CD
38536	1644	8	min	min	NN
38536	1644	9	.	.	.
38536	1645	1	The	the	DT
38536	1645	2	faster	fast	JJR
38536	1645	3	boat	boat	NN
38536	1645	4	completes	complete	VBZ
38536	1645	5	the	the	DT
38536	1645	6	trip	trip	NN
38536	1645	7	6	6	CD
38536	1645	8	-	-	HYPH
38536	1645	9	3/4	3/4	CD
38536	1645	10	min	min	NN
38536	1645	11	.	.	.
38536	1646	1	before	before	IN
38536	1646	2	the	the	DT
38536	1646	3	other	other	JJ
38536	1646	4	reaches	reach	VBZ
38536	1646	5	the	the	DT
38536	1646	6	opposite	opposite	JJ
38536	1646	7	shore	shore	NN
38536	1646	8	.	.	.
38536	1647	1	Find	find	VB
38536	1647	2	the	the	DT
38536	1647	3	rates	rate	NNS
38536	1647	4	of	of	IN
38536	1647	5	the	the	DT
38536	1647	6	boats	boat	NNS
38536	1647	7	in	in	IN
38536	1647	8	miles	mile	NNS
38536	1647	9	per	per	IN
38536	1647	10	hour	hour	NN
38536	1647	11	.	.	.
38536	1648	1	10	10	CD
38536	1648	2	.	.	.
38536	1649	1	Write	write	VB
38536	1649	2	the	the	DT
38536	1649	3	sixth	sixth	JJ
38536	1649	4	term	term	NN
38536	1649	5	of	of	IN
38536	1649	6	[	[	-LRB-
38536	1649	7	x/(2[y^2]^(1/3	x/(2[y^2]^(1/3	NFP
38536	1649	8	)	)	-RRB-
38536	1649	9	)	)	-RRB-
38536	1649	10	-	-	:
38536	1649	11	(	(	-LRB-
38536	1649	12	y^(1/2))/x]^9	y^(1/2))/x]^9	NNP
38536	1649	13	without	without	IN
38536	1649	14	writing	write	VBG
38536	1649	15	the	the	DT
38536	1649	16	preceding	precede	VBG
38536	1649	17	terms	term	NNS
38536	1649	18	.	.	.
38536	1650	1	11	11	CD
38536	1650	2	.	.	.
38536	1651	1	The	the	DT
38536	1651	2	sum	sum	NN
38536	1651	3	of	of	IN
38536	1651	4	the	the	DT
38536	1651	5	2d	2d	JJ
38536	1651	6	and	and	CC
38536	1651	7	20th	20th	JJ
38536	1651	8	terms	term	NNS
38536	1651	9	of	of	IN
38536	1651	10	an	an	DT
38536	1651	11	A.	a.	NN
38536	1651	12	P.	p.	NN
38536	1651	13	is	be	VBZ
38536	1651	14	10	10	CD
38536	1651	15	,	,	,
38536	1651	16	and	and	CC
38536	1651	17	their	-PRON-	PRP$
38536	1651	18	product	product	NN
38536	1651	19	is	be	VBZ
38536	1651	20	23	23	CD
38536	1651	21	-	-	SYM
38536	1651	22	47/64	47/64	CD
38536	1651	23	.	.	.
38536	1652	1	What	what	WP
38536	1652	2	is	be	VBZ
38536	1652	3	the	the	DT
38536	1652	4	sum	sum	NN
38536	1652	5	of	of	IN
38536	1652	6	sixteen	sixteen	CD
38536	1652	7	terms	term	NNS
38536	1652	8	?	?	.
38536	1653	1	~PRINCETON	~PRINCETON	NFP
38536	1653	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1653	3	ALGEBRA	ALGEBRA	NNP
38536	1653	4	A	a	DT
38536	1653	5	TIME	time	NN
38536	1653	6	:	:	:
38536	1653	7	TWO	two	CD
38536	1653	8	HOURS	hours	JJ
38536	1653	9	Candidates	candidate	NNS
38536	1653	10	who	who	WP
38536	1653	11	are	be	VBP
38536	1653	12	at	at	IN
38536	1653	13	this	this	DT
38536	1653	14	time	time	NN
38536	1653	15	taking	take	VBG
38536	1653	16	_	_	NNP
38536	1653	17	both	both	DT
38536	1653	18	_	_	NNP
38536	1653	19	Algebra	Algebra	NNP
38536	1653	20	A	A	NNP
38536	1653	21	and	and	CC
38536	1653	22	Algebra	Algebra	NNP
38536	1653	23	B	B	NNP
38536	1653	24	may	may	MD
38536	1653	25	omit	omit	VB
38536	1653	26	from	from	IN
38536	1653	27	Algebra	Algebra	NNP
38536	1653	28	A	A	NNP
38536	1653	29	questions	question	NNS
38536	1653	30	4	4	CD
38536	1653	31	,	,	,
38536	1653	32	5	5	CD
38536	1653	33	,	,	,
38536	1653	34	and	and	CC
38536	1653	35	6	6	CD
38536	1653	36	,	,	,
38536	1653	37	and	and	CC
38536	1653	38	from	from	IN
38536	1653	39	Algebra	Algebra	NNP
38536	1653	40	B	B	NNP
38536	1653	41	questions	question	NNS
38536	1653	42	1	1	CD
38536	1653	43	(	(	-LRB-
38536	1653	44	_	_	NNP
38536	1653	45	a	a	DT
38536	1653	46	_	_	NNP
38536	1653	47	)	)	-RRB-
38536	1653	48	,	,	,
38536	1653	49	3	3	CD
38536	1653	50	,	,	,
38536	1653	51	and	and	CC
38536	1653	52	4	4	CD
38536	1653	53	.	.	.
38536	1654	1	1	1	LS
38536	1654	2	.	.	.
38536	1655	1	Simplify	Simplify	NNP
38536	1655	2	(	(	-LRB-
38536	1655	3	a^3	a^3	NNP
38536	1655	4	+	+	SYM
38536	1655	5	a^2b	a^2b	NN
38536	1655	6	+	+	SYM
38536	1655	7	ab^2)/(a^2	ab^2)/(a^2	JJ
38536	1655	8	-	-	HYPH
38536	1655	9	3ab	3ab	JJ
38536	1655	10	-	-	HYPH
38536	1655	11	4b^2	4b^2	CD
38536	1655	12	)	)	-RRB-
38536	1655	13	÷	÷	NNP
38536	1655	14	{	{	-LRB-
38536	1655	15	(	(	-LRB-
38536	1655	16	a^2	a^2	NNP
38536	1655	17	+	+	CC
38536	1655	18	6ab	6ab	JJ
38536	1655	19	-	-	HYPH
38536	1655	20	7b^2)/(a^2	7b^2)/(a^2	CD
38536	1655	21	+	+	CC
38536	1655	22	8ab	8ab	JJ
38536	1655	23	-	-	SYM
38536	1655	24	9b^2	9b^2	CD
38536	1655	25	)	)	-RRB-
38536	1655	26	·	·	NFP
38536	1655	27	(	(	-LRB-
38536	1655	28	a^3	a^3	NNP
38536	1655	29	-	-	HYPH
38536	1655	30	b^3)/(a^2	b^3)/(a^2	NNP
38536	1655	31	-	-	HYPH
38536	1655	32	7ab	7ab	NN
38536	1655	33	+	+	SYM
38536	1655	34	12b^2	12b^2	NN
38536	1655	35	)	)	-RRB-
38536	1655	36	}	}	-RRB-
38536	1655	37	.	.	.
38536	1656	1	2	2	LS
38536	1656	2	.	.	.
38536	1657	1	(	(	-LRB-
38536	1657	2	_	_	NNP
38536	1657	3	a	a	DT
38536	1657	4	_	_	NNP
38536	1657	5	)	)	-RRB-
38536	1657	6	Divide	Divide	NNP
38536	1657	7	a^(5/2	a^(5/2	NNP
38536	1657	8	)	)	-RRB-
38536	1657	9	+	+	SYM
38536	1657	10	ab^(3/2	ab^(3/2	FW
38536	1657	11	)	)	-RRB-
38536	1657	12	+	+	NFP
38536	1657	13	b^(5/2	b^(5/2	NNP
38536	1657	14	)	)	-RRB-
38536	1657	15	-	-	HYPH
38536	1657	16	2a^(1/2)b^2	2a^(1/2)b^2	CD
38536	1657	17	-	-	HYPH
38536	1657	18	a^(3/2)b	a^(3/2)b	NNP
38536	1657	19	by	by	IN
38536	1657	20	a^(3/2	a^(3/2	NNP
38536	1657	21	)	)	-RRB-
38536	1657	22	-	-	:
38536	1657	23	b^(3/2	b^(3/2	NNP
38536	1657	24	)	)	-RRB-
38536	1657	25	+	+	SYM
38536	1657	26	a^(1/2)b	a^(1/2)b	NN
38536	1657	27	-	-	HYPH
38536	1657	28	ab^(1/2	ab^(1/2	NN
38536	1657	29	)	)	-RRB-
38536	1657	30	.	.	.
38536	1658	1	(	(	-LRB-
38536	1658	2	_	_	NNP
38536	1658	3	b	b	NNP
38536	1658	4	_	_	NNP
38536	1658	5	)	)	-RRB-
38536	1658	6	Simplify	Simplify	NNP
38536	1658	7	(	(	-LRB-
38536	1658	8	1)/(x^(-1	1)/(x^(-1	CD
38536	1658	9	)	)	-RRB-
38536	1658	10	+	+	NFP
38536	1658	11	y^(-1	y^(-1	NNP
38536	1658	12	)	)	-RRB-
38536	1658	13	}	}	-RRB-
38536	1658	14	·	·	NFP
38536	1658	15	(	(	-LRB-
38536	1658	16	x^(1/4)y^(1/2))^3	x^(1/4)y^(1/2))^3	NNP
38536	1658	17	+	+	SYM
38536	1658	18	1	1	CD
38536	1658	19	.	.	.
38536	1659	1	3	3	LS
38536	1659	2	.	.	.
38536	1660	1	Factor	factor	NN
38536	1660	2	:	:	:
38536	1660	3	(	(	-LRB-
38536	1660	4	_	_	NNP
38536	1660	5	a	a	DT
38536	1660	6	_	_	NNP
38536	1660	7	)	)	-RRB-
38536	1660	8	(	(	-LRB-
38536	1660	9	x^2	x^2	NNS
38536	1660	10	+	+	SYM
38536	1660	11	3x)^2	3x)^2	CD
38536	1660	12	-	-	HYPH
38536	1660	13	(	(	-LRB-
38536	1660	14	2x	2x	NN
38536	1660	15	-	-	.
38536	1660	16	6)^2	6)^2	JJ
38536	1660	17	.	.	.
38536	1661	1	(	(	-LRB-
38536	1661	2	_	_	NNP
38536	1661	3	b	b	NNP
38536	1661	4	_	_	NNP
38536	1661	5	)	)	-RRB-
38536	1661	6	a^2	a^2	CD
38536	1661	7	+	+	CC
38536	1661	8	ac	ac	RB
38536	1661	9	-	-	HYPH
38536	1661	10	4b^2	4b^2	CD
38536	1661	11	-	-	HYPH
38536	1661	12	2bc	2bc	NN
38536	1661	13	.	.	.
38536	1662	1	4	4	LS
38536	1662	2	.	.	.
38536	1663	1	Solve	solve	VB
38536	1663	2	1/(x	1/(x	CD
38536	1663	3	+	+	SYM
38536	1663	4	1	1	CD
38536	1663	5	)	)	-RRB-
38536	1663	6	-	-	,
38536	1663	7	(	(	-LRB-
38536	1663	8	1)/(x	1)/(x	CD
38536	1663	9	-	-	HYPH
38536	1663	10	1	1	CD
38536	1663	11	)	)	-RRB-
38536	1663	12	-	-	:
38536	1663	13	(	(	-LRB-
38536	1663	14	1)/(x	1)/(x	CD
38536	1663	15	-	-	HYPH
38536	1663	16	3	3	CD
38536	1663	17	)	)	-RRB-
38536	1663	18	+	+	NFP
38536	1663	19	(	(	-LRB-
38536	1663	20	1)/(x	1)/(x	CD
38536	1663	21	-	-	HYPH
38536	1663	22	5	5	CD
38536	1663	23	)	)	-RRB-
38536	1663	24	=	=	SYM
38536	1663	25	0	0	NFP
38536	1663	26	.	.	.
38536	1664	1	5	5	CD
38536	1664	2	.	.	.
38536	1665	1	Solve	solve	VB
38536	1665	2	for	for	IN
38536	1665	3	x	x	NN
38536	1665	4	and	and	CC
38536	1665	5	y	y	NNP
38536	1665	6	:	:	:
38536	1665	7	mx	mx	NNP
38536	1665	8	+	+	CC
38536	1665	9	ax	ax	NN
38536	1665	10	=	=	NFP
38536	1665	11	my	-PRON-	PRP$
38536	1665	12	-	-	HYPH
38536	1665	13	by	by	NN
38536	1665	14	,	,	,
38536	1665	15	x	x	NNP
38536	1665	16	-	-	NN
38536	1665	17	y	y	NNP
38536	1665	18	=	=	NN
38536	1665	19	a	a	NN
38536	1665	20	+	+	SYM
38536	1665	21	b.	b.	NN
38536	1666	1	6	6	CD
38536	1666	2	.	.	.
38536	1667	1	The	the	DT
38536	1667	2	road	road	NN
38536	1667	3	from	from	IN
38536	1667	4	A	a	NN
38536	1667	5	to	to	IN
38536	1667	6	B	b	NN
38536	1667	7	is	be	VBZ
38536	1667	8	uphill	uphill	JJ
38536	1667	9	for	for	IN
38536	1667	10	5	5	CD
38536	1667	11	mi	mi	NNP
38536	1667	12	.	.	NNP
38536	1667	13	,	,	,
38536	1667	14	level	level	NN
38536	1667	15	for	for	IN
38536	1667	16	4	4	CD
38536	1667	17	mi	mi	NNP
38536	1667	18	.	.	NNP
38536	1667	19	,	,	,
38536	1667	20	and	and	CC
38536	1667	21	then	then	RB
38536	1667	22	downhill	downhill	RB
38536	1667	23	for	for	IN
38536	1667	24	6	6	CD
38536	1667	25	mi	mi	NNS
38536	1667	26	.	.	.
38536	1668	1	A	a	DT
38536	1668	2	man	man	NN
38536	1668	3	walks	walk	VBZ
38536	1668	4	from	from	IN
38536	1668	5	B	b	NN
38536	1668	6	to	to	IN
38536	1668	7	A	a	NN
38536	1668	8	in	in	IN
38536	1668	9	4	4	CD
38536	1668	10	hr	hr	NN
38536	1668	11	.	.	.
38536	1669	1	;	;	:
38536	1669	2	later	later	RB
38536	1669	3	he	-PRON-	PRP
38536	1669	4	walks	walk	VBZ
38536	1669	5	halfway	halfway	RB
38536	1669	6	from	from	IN
38536	1669	7	A	a	NN
38536	1669	8	to	to	IN
38536	1669	9	B	b	NN
38536	1669	10	and	and	CC
38536	1669	11	back	back	RB
38536	1669	12	again	again	RB
38536	1669	13	to	to	IN
38536	1669	14	A	a	NN
38536	1669	15	in	in	IN
38536	1669	16	3	3	CD
38536	1669	17	hr	hr	NN
38536	1669	18	.	.	.
38536	1670	1	and	and	CC
38536	1670	2	55	55	CD
38536	1670	3	min	min	NN
38536	1670	4	.	.	.
38536	1671	1	;	;	:
38536	1671	2	and	and	CC
38536	1671	3	later	later	RBR
38536	1671	4	he	-PRON-	PRP
38536	1671	5	walks	walk	VBZ
38536	1671	6	from	from	IN
38536	1671	7	A	a	NN
38536	1671	8	to	to	IN
38536	1671	9	B	b	NN
38536	1671	10	in	in	IN
38536	1671	11	3	3	CD
38536	1671	12	hr	hr	NN
38536	1671	13	.	.	.
38536	1672	1	and	and	CC
38536	1672	2	52	52	CD
38536	1672	3	min	min	NN
38536	1672	4	.	.	.
38536	1673	1	What	what	WP
38536	1673	2	are	be	VBP
38536	1673	3	his	-PRON-	PRP$
38536	1673	4	rates	rate	NNS
38536	1673	5	of	of	IN
38536	1673	6	walking	walk	VBG
38536	1673	7	uphill	uphill	RB
38536	1673	8	,	,	,
38536	1673	9	downhill	downhill	RB
38536	1673	10	,	,	,
38536	1673	11	and	and	CC
38536	1673	12	on	on	IN
38536	1673	13	the	the	DT
38536	1673	14	level	level	NN
38536	1673	15	,	,	,
38536	1673	16	if	if	IN
38536	1673	17	these	these	DT
38536	1673	18	do	do	VBP
38536	1673	19	not	not	RB
38536	1673	20	vary	vary	VB
38536	1673	21	?	?	.
38536	1674	1	ALGEBRA	ALGEBRA	NNP
38536	1674	2	B	B	NNP
38536	1674	3	1	1	CD
38536	1674	4	.	.	.
38536	1675	1	Solve	solve	VB
38536	1675	2	(	(	-LRB-
38536	1675	3	_	_	NNP
38536	1675	4	a	a	DT
38536	1675	5	_	_	NNP
38536	1675	6	)	)	-RRB-
38536	1675	7	(	(	-LRB-
38536	1675	8	x	x	SYM
38536	1675	9	+	+	CD
38536	1675	10	1)/(x	1)/(x	CD
38536	1675	11	-	-	HYPH
38536	1675	12	2	2	CD
38536	1675	13	)	)	-RRB-
38536	1675	14	+	+	NFP
38536	1675	15	(	(	-LRB-
38536	1675	16	2x	2x	CD
38536	1675	17	+	+	CD
38536	1675	18	1)/(x	1)/(x	CD
38536	1675	19	+	+	SYM
38536	1675	20	1	1	CD
38536	1675	21	)	)	-RRB-
38536	1675	22	+	+	CC
38536	1675	23	(	(	-LRB-
38536	1675	24	3x	3x	CD
38536	1675	25	+	+	SYM
38536	1675	26	3)/(1	3)/(1	NNP
38536	1675	27	-	-	HYPH
38536	1675	28	x	x	NNP
38536	1675	29	)	)	-RRB-
38536	1675	30	=	=	SYM
38536	1675	31	0	0	NFP
38536	1675	32	.	.	.
38536	1676	1	(	(	-LRB-
38536	1676	2	_	_	NNP
38536	1676	3	b	b	NNP
38536	1676	4	_	_	NNP
38536	1676	5	)	)	-RRB-
38536	1676	6	[	[	-LRB-
38536	1676	7	2x	2x	CD
38536	1676	8	+	+	SYM
38536	1676	9	7]^(1/2	7]^(1/2	NN
38536	1676	10	)	)	-RRB-
38536	1676	11	+	+	CC
38536	1676	12	[	[	-LRB-
38536	1676	13	3x	3x	CD
38536	1676	14	-	-	HYPH
38536	1676	15	18]^(1/2	18]^(1/2	CD
38536	1676	16	)	)	-RRB-
38536	1676	17	-	-	:
38536	1676	18	[	[	-LRB-
38536	1676	19	7x	7x	CD
38536	1676	20	+	+	CD
38536	1676	21	1]^(1/2	1]^(1/2	NN
38536	1676	22	)	)	-RRB-
38536	1676	23	=	=	SYM
38536	1676	24	0	0	NFP
38536	1676	25	.	.	.
38536	1677	1	(	(	-LRB-
38536	1677	2	_	_	NNP
38536	1677	3	c	c	NNP
38536	1677	4	_	_	NNP
38536	1677	5	)	)	-RRB-
38536	1677	6	6/(x^2	6/(x^2	CD
38536	1677	7	+	+	SYM
38536	1677	8	2x	2x	NN
38536	1677	9	)	)	-RRB-
38536	1677	10	=	=	SYM
38536	1677	11	5	5	CD
38536	1677	12	-	-	HYPH
38536	1677	13	2x	2x	CD
38536	1677	14	-	-	HYPH
38536	1677	15	x^2	x^2	NNP
38536	1677	16	.	.	.
38536	1678	1	2	2	LS
38536	1678	2	.	.	.
38536	1679	1	Solve	solve	VB
38536	1679	2	for	for	IN
38536	1679	3	x	x	NN
38536	1679	4	and	and	CC
38536	1679	5	y	y	NNP
38536	1679	6	,	,	,
38536	1679	7	checking	check	VBG
38536	1679	8	one	one	CD
38536	1679	9	solution	solution	NN
38536	1679	10	in	in	IN
38536	1679	11	each	each	DT
38536	1679	12	problem	problem	NN
38536	1679	13	:	:	:
38536	1679	14	(	(	-LRB-
38536	1679	15	_	_	NNP
38536	1679	16	a	a	DT
38536	1679	17	_	_	NNP
38536	1679	18	)	)	-RRB-
38536	1679	19	2x	2x	NN
38536	1679	20	+	+	SYM
38536	1679	21	3y	3y	NNS
38536	1679	22	=	=	SYM
38536	1679	23	1	1	CD
38536	1679	24	,	,	,
38536	1679	25	6	6	CD
38536	1679	26	/	/	SYM
38536	1679	27	x	x	NNS
38536	1679	28	+	+	SYM
38536	1679	29	1	1	CD
38536	1679	30	/	/	SYM
38536	1679	31	y	y	NN
38536	1679	32	=	=	SYM
38536	1679	33	2	2	CD
38536	1679	34	.	.	.
38536	1680	1	(	(	-LRB-
38536	1680	2	_	_	NNP
38536	1680	3	b	b	NNP
38536	1680	4	_	_	NNP
38536	1680	5	)	)	-RRB-
38536	1680	6	x^2	x^2	NNP
38536	1680	7	=	=	SYM
38536	1680	8	x	x	SYM
38536	1680	9	+	+	SYM
38536	1680	10	y	y	NN
38536	1680	11	,	,	,
38536	1680	12	y^2	y^2	NNP
38536	1680	13	=	=	SYM
38536	1680	14	3y	3y	NNP
38536	1680	15	-	-	HYPH
38536	1680	16	x.	x.	NNP
38536	1681	1	3	3	LS
38536	1681	2	.	.	.
38536	1682	1	A	a	DT
38536	1682	2	man	man	NN
38536	1682	3	arranges	arrange	VBZ
38536	1682	4	to	to	TO
38536	1682	5	pay	pay	VB
38536	1682	6	a	a	DT
38536	1682	7	debt	debt	NN
38536	1682	8	of	of	IN
38536	1682	9	$	$	$
38536	1682	10	3600	3600	CD
38536	1682	11	in	in	IN
38536	1682	12	40	40	CD
38536	1682	13	monthly	monthly	JJ
38536	1682	14	payments	payment	NNS
38536	1682	15	which	which	WDT
38536	1682	16	form	form	VBP
38536	1682	17	an	an	DT
38536	1682	18	A.	a.	NN
38536	1682	19	P.	P.	NNP
38536	1682	20	After	after	IN
38536	1682	21	paying	pay	VBG
38536	1682	22	30	30	CD
38536	1682	23	of	of	IN
38536	1682	24	them	-PRON-	PRP
38536	1682	25	he	-PRON-	PRP
38536	1682	26	still	still	RB
38536	1682	27	owes	owe	VBZ
38536	1682	28	1/3	1/3	CD
38536	1682	29	of	of	IN
38536	1682	30	his	-PRON-	PRP$
38536	1682	31	debt	debt	NN
38536	1682	32	.	.	.
38536	1683	1	What	what	WP
38536	1683	2	was	be	VBD
38536	1683	3	his	-PRON-	PRP$
38536	1683	4	first	first	JJ
38536	1683	5	payment	payment	NN
38536	1683	6	?	?	.
38536	1684	1	4	4	LS
38536	1684	2	.	.	.
38536	1685	1	If	if	IN
38536	1685	2	4	4	CD
38536	1685	3	quantities	quantity	NNS
38536	1685	4	are	be	VBP
38536	1685	5	in	in	IN
38536	1685	6	proportion	proportion	NN
38536	1685	7	and	and	CC
38536	1685	8	the	the	DT
38536	1685	9	second	second	JJ
38536	1685	10	is	be	VBZ
38536	1685	11	a	a	DT
38536	1685	12	mean	mean	JJ
38536	1685	13	proportional	proportional	JJ
38536	1685	14	between	between	IN
38536	1685	15	the	the	DT
38536	1685	16	third	third	JJ
38536	1685	17	and	and	CC
38536	1685	18	fourth	fourth	JJ
38536	1685	19	,	,	,
38536	1685	20	prove	prove	VBP
38536	1685	21	that	that	IN
38536	1685	22	the	the	DT
38536	1685	23	third	third	NN
38536	1685	24	will	will	MD
38536	1685	25	be	be	VB
38536	1685	26	a	a	DT
38536	1685	27	mean	mean	JJ
38536	1685	28	prop	prop	NN
38536	1685	29	.	.	.
38536	1686	1	between	between	IN
38536	1686	2	the	the	DT
38536	1686	3	first	first	JJ
38536	1686	4	and	and	CC
38536	1686	5	second	second	JJ
38536	1686	6	.	.	.
38536	1687	1	5	5	CD
38536	1687	2	.	.	.
38536	1688	1	In	in	IN
38536	1688	2	the	the	DT
38536	1688	3	expansion	expansion	NN
38536	1688	4	of	of	IN
38536	1688	5	[	[	-LRB-
38536	1688	6	2x	2x	CD
38536	1688	7	+	+	SYM
38536	1688	8	1/3x]^6	1/3x]^6	CD
38536	1688	9	the	the	DT
38536	1688	10	ratio	ratio	NN
38536	1688	11	of	of	IN
38536	1688	12	the	the	DT
38536	1688	13	fourth	fourth	JJ
38536	1688	14	term	term	NN
38536	1688	15	to	to	IN
38536	1688	16	the	the	DT
38536	1688	17	fifth	fifth	NN
38536	1688	18	is	be	VBZ
38536	1688	19	2	2	CD
38536	1688	20	:	:	SYM
38536	1688	21	1	1	CD
38536	1688	22	.	.	.
38536	1689	1	Find	find	VB
38536	1689	2	x.	x.	NN
38536	1690	1	6	6	CD
38536	1690	2	.	.	.
38536	1691	1	Two	two	CD
38536	1691	2	men	man	NNS
38536	1691	3	A	a	NN
38536	1691	4	and	and	CC
38536	1691	5	B	b	NN
38536	1691	6	can	can	MD
38536	1691	7	together	together	RB
38536	1691	8	do	do	VB
38536	1691	9	a	a	DT
38536	1691	10	piece	piece	NN
38536	1691	11	of	of	IN
38536	1691	12	work	work	NN
38536	1691	13	in	in	IN
38536	1691	14	12	12	CD
38536	1691	15	days	day	NNS
38536	1691	16	;	;	:
38536	1691	17	B	b	NN
38536	1691	18	would	would	MD
38536	1691	19	need	need	VB
38536	1691	20	10	10	CD
38536	1691	21	days	day	NNS
38536	1691	22	more	more	JJR
38536	1691	23	than	than	IN
38536	1691	24	A	a	DT
38536	1691	25	to	to	TO
38536	1691	26	do	do	VB
38536	1691	27	the	the	DT
38536	1691	28	whole	whole	JJ
38536	1691	29	work	work	NN
38536	1691	30	.	.	.
38536	1692	1	How	how	WRB
38536	1692	2	many	many	JJ
38536	1692	3	days	day	NNS
38536	1692	4	would	would	MD
38536	1692	5	it	-PRON-	PRP
38536	1692	6	take	take	VB
38536	1692	7	A	a	DT
38536	1692	8	alone	alone	JJ
38536	1692	9	to	to	TO
38536	1692	10	do	do	VB
38536	1692	11	the	the	DT
38536	1692	12	work	work	NN
38536	1692	13	?	?	.
38536	1693	1	ALGEBRA	ALGEBRA	NNP
38536	1693	2	TO	to	IN
38536	1693	3	QUADRATICS	QUADRATICS	NNP
38536	1693	4	1	1	CD
38536	1693	5	.	.	.
38536	1694	1	Simplify	Simplify	NNP
38536	1694	2	(	(	-LRB-
38536	1694	3	ab^(-2)c^2)^(1/2	ab^(-2)c^2)^(1/2	NNP
38536	1694	4	)	)	-RRB-
38536	1694	5	·	·	NFP
38536	1694	6	(	(	-LRB-
38536	1694	7	a^3b^2c^(-3))^(1/3	a^3b^2c^(-3))^(1/3	NN
38536	1694	8	)	)	-RRB-
38536	1694	9	+	+	CC
38536	1694	10	[	[	-LRB-
38536	1694	11	(	(	-LRB-
38536	1694	12	a^6)/(b)]^(1/3	a^6)/(b)]^(1/3	NN
38536	1694	13	)	)	-RRB-
38536	1694	14	.	.	.
38536	1695	1	2	2	LS
38536	1695	2	.	.	.
38536	1696	1	Simplify	Simplify	NNP
38536	1696	2	a/[(a	a/[(a	NNP
38536	1696	3	-	-	HYPH
38536	1696	4	b)(a	b)(a	NNP
38536	1696	5	-	-	HYPH
38536	1696	6	c	c	NNP
38536	1696	7	)	)	-RRB-
38536	1696	8	]	]	-RRB-
38536	1696	9	+	+	NFP
38536	1696	10	b/[(b	b/[(b	NNP
38536	1696	11	-	-	HYPH
38536	1696	12	c)(b	c)(b	FW
38536	1696	13	-	-	HYPH
38536	1696	14	a	a	NN
38536	1696	15	)	)	-RRB-
38536	1696	16	]	]	-RRB-
38536	1696	17	+	+	NFP
38536	1696	18	c/[(c	c/[(c	NNP
38536	1696	19	-	-	HYPH
38536	1696	20	a)(c	a)(c	NNP
38536	1696	21	-	-	HYPH
38536	1696	22	b	b	NNP
38536	1696	23	)	)	-RRB-
38536	1696	24	]	]	-RRB-
38536	1696	25	.	.	.
38536	1697	1	3	3	LS
38536	1697	2	.	.	.
38536	1698	1	Factor	Factor	NNP
38536	1698	2	(	(	-LRB-
38536	1698	3	_	_	NNP
38536	1698	4	a	a	DT
38536	1698	5	_	_	NNP
38536	1698	6	)	)	-RRB-
38536	1698	7	x^4	x^4	NNP
38536	1698	8	-	-	HYPH
38536	1698	9	10x^2	10x^2	CD
38536	1698	10	+	+	SYM
38536	1698	11	9	9	CD
38536	1698	12	.	.	.
38536	1699	1	(	(	-LRB-
38536	1699	2	_	_	NNP
38536	1699	3	b	b	NNP
38536	1699	4	_	_	NNP
38536	1699	5	)	)	-RRB-
38536	1699	6	x^2	x^2	NNP
38536	1699	7	+	+	SYM
38536	1699	8	2xy	2xy	JJ
38536	1699	9	-	-	HYPH
38536	1699	10	a^2	a^2	JJ
38536	1699	11	-	-	HYPH
38536	1699	12	2ay	2ay	NN
38536	1699	13	.	.	.
38536	1700	1	(	(	-LRB-
38536	1700	2	_	_	NNP
38536	1700	3	c	c	NNP
38536	1700	4	_	_	NNP
38536	1700	5	)	)	-RRB-
38536	1700	6	(	(	-LRB-
38536	1700	7	a	a	DT
38536	1700	8	+	+	SYM
38536	1700	9	b)^2	b)^2	NN
38536	1700	10	+	+	CC
38536	1700	11	(	(	-LRB-
38536	1700	12	a	a	DT
38536	1700	13	+	+	CD
38536	1700	14	c)^2	c)^2	RB
38536	1700	15	-	-	:
38536	1700	16	(	(	-LRB-
38536	1700	17	c	c	NN
38536	1700	18	+	+	SYM
38536	1700	19	d)^2	d)^2	NNP
38536	1700	20	-	-	,
38536	1700	21	(	(	-LRB-
38536	1700	22	b	b	NN
38536	1700	23	+	+	SYM
38536	1700	24	d)^2	d)^2	NN
38536	1700	25	.	.	.
38536	1701	1	4	4	LS
38536	1701	2	.	.	.
38536	1702	1	Find	find	VB
38536	1702	2	H.	H.	NNP
38536	1702	3	C.	C.	NNP
38536	1702	4	F.	F.	NNP
38536	1702	5	of	of	IN
38536	1702	6	x^4	x^4	NNP
38536	1702	7	-	-	HYPH
38536	1702	8	x^3	x^3	NNP
38536	1702	9	+	+	CC
38536	1702	10	2x^2	2x^2	CD
38536	1702	11	+	+	SYM
38536	1702	12	x	x	SYM
38536	1702	13	+	+	SYM
38536	1702	14	3	3	CD
38536	1702	15	and	and	CC
38536	1702	16	(	(	-LRB-
38536	1702	17	x	x	NNS
38536	1702	18	+	+	SYM
38536	1702	19	2)(x^3	2)(x^3	CD
38536	1702	20	-	-	SYM
38536	1702	21	1	1	CD
38536	1702	22	)	)	-RRB-
38536	1702	23	.	.	.
38536	1703	1	5	5	CD
38536	1703	2	.	.	.
38536	1704	1	Solve	solve	VB
38536	1704	2	x/(x	x/(x	NNS
38536	1704	3	-	-	:
38536	1704	4	2	2	CD
38536	1704	5	)	)	-RRB-
38536	1704	6	+	+	CC
38536	1704	7	(	(	-LRB-
38536	1704	8	x	x	SYM
38536	1704	9	-	-	NNP
38536	1704	10	9)/(x	9)/(x	NNP
38536	1704	11	-	-	HYPH
38536	1704	12	7	7	CD
38536	1704	13	)	)	-RRB-
38536	1704	14	=	=	NFP
38536	1704	15	(	(	-LRB-
38536	1704	16	x	x	SYM
38536	1704	17	+	+	CD
38536	1704	18	1)/(x	1)/(x	CD
38536	1704	19	-	-	HYPH
38536	1704	20	1	1	CD
38536	1704	21	)	)	-RRB-
38536	1704	22	+	+	CC
38536	1704	23	(	(	-LRB-
38536	1704	24	x	x	SYM
38536	1704	25	-	-	NNP
38536	1704	26	8)/(x	8)/(x	CD
38536	1704	27	-	-	HYPH
38536	1704	28	6	6	CD
38536	1704	29	)	)	-RRB-
38536	1704	30	.	.	.
38536	1705	1	6	6	CD
38536	1705	2	.	.	.
38536	1706	1	The	the	DT
38536	1706	2	sum	sum	NN
38536	1706	3	of	of	IN
38536	1706	4	three	three	CD
38536	1706	5	numbers	number	NNS
38536	1706	6	is	be	VBZ
38536	1706	7	51	51	CD
38536	1706	8	;	;	:
38536	1706	9	if	if	IN
38536	1706	10	the	the	DT
38536	1706	11	first	first	JJ
38536	1706	12	number	number	NN
38536	1706	13	be	be	VB
38536	1706	14	divided	divide	VBN
38536	1706	15	by	by	IN
38536	1706	16	the	the	DT
38536	1706	17	second	second	JJ
38536	1706	18	,	,	,
38536	1706	19	the	the	DT
38536	1706	20	quotient	quotient	NN
38536	1706	21	is	be	VBZ
38536	1706	22	2	2	CD
38536	1706	23	and	and	CC
38536	1706	24	the	the	DT
38536	1706	25	remainder	remainder	NN
38536	1706	26	5	5	CD
38536	1706	27	;	;	:
38536	1706	28	if	if	IN
38536	1706	29	the	the	DT
38536	1706	30	second	second	JJ
38536	1706	31	number	number	NN
38536	1706	32	be	be	VB
38536	1706	33	divided	divide	VBN
38536	1706	34	by	by	IN
38536	1706	35	the	the	DT
38536	1706	36	third	third	JJ
38536	1706	37	,	,	,
38536	1706	38	the	the	DT
38536	1706	39	quotient	quotient	NN
38536	1706	40	is	be	VBZ
38536	1706	41	3	3	CD
38536	1706	42	and	and	CC
38536	1706	43	the	the	DT
38536	1706	44	remainder	remainder	NN
38536	1706	45	2	2	CD
38536	1706	46	.	.	.
38536	1707	1	What	what	WP
38536	1707	2	are	be	VBP
38536	1707	3	the	the	DT
38536	1707	4	numbers	number	NNS
38536	1707	5	?	?	.
38536	1708	1	~SMITH	~SMITH	NFP
38536	1708	2	COLLEGE~	COLLEGE~	NNP
38536	1708	3	ELEMENTARY	ELEMENTARY	NNP
38536	1708	4	ALGEBRA	ALGEBRA	NNP
38536	1708	5	1	1	CD
38536	1708	6	.	.	.
38536	1709	1	Factor	Factor	NNP
38536	1709	2	e^(2x	e^(2x	NNP
38536	1709	3	)	)	-RRB-
38536	1709	4	-	-	:
38536	1709	5	2	2	CD
38536	1709	6	+	+	CD
38536	1709	7	e^(-2x	e^(-2x	CD
38536	1709	8	)	)	-RRB-
38536	1709	9	,	,	,
38536	1709	10	x^(12	x^(12	NNP
38536	1709	11	)	)	-RRB-
38536	1709	12	-	-	:
38536	1709	13	8	8	CD
38536	1709	14	,	,	,
38536	1709	15	x^2	x^2	JJ
38536	1709	16	-	-	HYPH
38536	1709	17	x	x	IN
38536	1709	18	-	-	HYPH
38536	1709	19	y^2	y^2	NNP
38536	1709	20	-	-	HYPH
38536	1709	21	y	y	NNP
38536	1709	22	,	,	,
38536	1709	23	18a^2x^2	18a^2x^2	CD
38536	1709	24	-24axy	-24axy	:
38536	1709	25	-	-	HYPH
38536	1709	26	10y^2	10y^2	CD
38536	1709	27	.	.	.
38536	1710	1	2	2	LS
38536	1710	2	.	.	.
38536	1711	1	Solve	solve	VB
38536	1711	2	[	[	-LRB-
38536	1711	3	7	7	CD
38536	1711	4	+	+	SYM
38536	1711	5	4x	4x	NNS
38536	1711	6	+	+	SYM
38536	1711	7	3[2x^2	3[2x^2	,
38536	1711	8	+	+	SYM
38536	1711	9	5x	5x	CD
38536	1711	10	+	+	SYM
38536	1711	11	7]^(1/2)]^(1/2	7]^(1/2)]^(1/2	CD
38536	1711	12	)	)	-RRB-
38536	1711	13	-	-	SYM
38536	1711	14	3	3	CD
38536	1711	15	=	=	SYM
38536	1711	16	0	0	CD
38536	1711	17	.	.	.
38536	1712	1	3	3	LS
38536	1712	2	.	.	.
38536	1713	1	The	the	DT
38536	1713	2	second	second	JJ
38536	1713	3	term	term	NN
38536	1713	4	of	of	IN
38536	1713	5	a	a	DT
38536	1713	6	geometrical	geometrical	JJ
38536	1713	7	progression	progression	NN
38536	1713	8	is	be	VBZ
38536	1713	9	3[2^(1/2	3[2^(1/2	CD
38536	1713	10	)	)	-RRB-
38536	1713	11	]	]	-RRB-
38536	1713	12	,	,	,
38536	1713	13	and	and	CC
38536	1713	14	the	the	DT
38536	1713	15	fifth	fifth	JJ
38536	1713	16	term	term	NN
38536	1713	17	is	be	VBZ
38536	1713	18	3/16	3/16	CD
38536	1713	19	.	.	.
38536	1714	1	Find	find	VB
38536	1714	2	the	the	DT
38536	1714	3	first	first	JJ
38536	1714	4	term	term	NN
38536	1714	5	and	and	CC
38536	1714	6	the	the	DT
38536	1714	7	ratio	ratio	NN
38536	1714	8	.	.	.
38536	1715	1	4	4	LS
38536	1715	2	.	.	.
38536	1716	1	Solve	solve	VB
38536	1716	2	the	the	DT
38536	1716	3	following	following	JJ
38536	1716	4	equations	equation	NNS
38536	1716	5	and	and	CC
38536	1716	6	check	check	VB
38536	1716	7	your	-PRON-	PRP$
38536	1716	8	results	result	NNS
38536	1716	9	by	by	IN
38536	1716	10	plotting	plot	VBG
38536	1716	11	:	:	:
38536	1716	12	{	{	-LRB-
38536	1716	13	x^2	x^2	NNS
38536	1716	14	+	+	SYM
38536	1716	15	y^2	y^2	NNP
38536	1716	16	-	-	HYPH
38536	1716	17	xy	xy	NN
38536	1716	18	=	=	NNS
38536	1716	19	7	7	CD
38536	1716	20	,	,	,
38536	1716	21	{	{	-LRB-
38536	1716	22	x	x	SYM
38536	1716	23	+	+	SYM
38536	1716	24	y	y	NN
38536	1716	25	=	=	SYM
38536	1716	26	4	4	CD
38536	1716	27	.	.	.
38536	1717	1	5	5	CD
38536	1717	2	.	.	.
38536	1718	1	Solve	solve	VB
38536	1718	2	1/(x^3	1/(x^3	CD
38536	1718	3	)	)	-RRB-
38536	1718	4	+	+	SYM
38536	1718	5	1/(y^3	1/(y^3	CD
38536	1718	6	)	)	-RRB-
38536	1718	7	=	=	NFP
38536	1718	8	243/8	243/8	CD
38536	1718	9	,	,	,
38536	1718	10	1	1	CD
38536	1718	11	/	/	SYM
38536	1718	12	x	x	NNS
38536	1718	13	+	+	SYM
38536	1718	14	1	1	CD
38536	1718	15	/	/	SYM
38536	1718	16	y	y	NN
38536	1718	17	=	=	SYM
38536	1718	18	9/2	9/2	CD
38536	1718	19	.	.	.
38536	1719	1	6	6	CD
38536	1719	2	.	.	.
38536	1720	1	In	in	IN
38536	1720	2	an	an	DT
38536	1720	3	arithmetical	arithmetical	JJ
38536	1720	4	progression	progression	NN
38536	1720	5	d	d	NN
38536	1720	6	=	=	SYM
38536	1720	7	-11	-11	NNP
38536	1720	8	,	,	,
38536	1720	9	n	n	NN
38536	1720	10	=	=	SYM
38536	1720	11	13	13	CD
38536	1720	12	,	,	,
38536	1720	13	s	s	NNP
38536	1720	14	=	=	SYM
38536	1720	15	0	0	NFP
38536	1720	16	.	.	.
38536	1721	1	Find	find	VB
38536	1721	2	a	a	NN
38536	1721	3	and	and	CC
38536	1721	4	l.	l.	NN
38536	1721	5	7	7	CD
38536	1721	6	.	.	.
38536	1722	1	Expand	expand	VB
38536	1722	2	by	by	IN
38536	1722	3	the	the	DT
38536	1722	4	binomial	binomial	NN
38536	1722	5	theorem	theorem	NN
38536	1722	6	and	and	CC
38536	1722	7	simplify	simplify	VB
38536	1722	8	:	:	:
38536	1722	9	[	[	-LRB-
38536	1722	10	(	(	-LRB-
38536	1722	11	2x)/(y^3	2x)/(y^3	CD
38536	1722	12	)	)	-RRB-
38536	1722	13	-	-	:
38536	1722	14	(	(	-LRB-
38536	1722	15	y^4)/(x^5	y^4)/(x^5	NN
38536	1722	16	[	[	-LRB-
38536	1722	17	-6]^(1/2))]^5	-6]^(1/2))]^5	NFP
38536	1722	18	.	.	.
38536	1723	1	8	8	LS
38536	1723	2	.	.	.
38536	1724	1	The	the	DT
38536	1724	2	diagonal	diagonal	JJ
38536	1724	3	of	of	IN
38536	1724	4	a	a	DT
38536	1724	5	rectangle	rectangle	NN
38536	1724	6	is	be	VBZ
38536	1724	7	13	13	CD
38536	1724	8	ft	ft	NN
38536	1724	9	.	.	NN
38536	1724	10	long	long	RB
38536	1724	11	.	.	.
38536	1725	1	If	if	IN
38536	1725	2	each	each	DT
38536	1725	3	side	side	NN
38536	1725	4	were	be	VBD
38536	1725	5	longer	long	JJR
38536	1725	6	by	by	IN
38536	1725	7	2	2	CD
38536	1725	8	ft	ft	NN
38536	1725	9	.	.	NNP
38536	1725	10	,	,	,
38536	1725	11	the	the	DT
38536	1725	12	area	area	NN
38536	1725	13	would	would	MD
38536	1725	14	be	be	VB
38536	1725	15	increased	increase	VBN
38536	1725	16	by	by	IN
38536	1725	17	38	38	CD
38536	1725	18	sq	sq	NN
38536	1725	19	.	.	.
38536	1726	1	ft	ft	UH
38536	1726	2	.	.	.
38536	1726	3	Find	find	VB
38536	1726	4	the	the	DT
38536	1726	5	lengths	length	NNS
38536	1726	6	of	of	IN
38536	1726	7	the	the	DT
38536	1726	8	sides	side	NNS
38536	1726	9	.	.	.
38536	1727	1	~SMITH	~SMITH	NFP
38536	1727	2	COLLEGE~	COLLEGE~	NNP
38536	1727	3	ELEMENTARY	ELEMENTARY	NNP
38536	1727	4	ALGEBRA	ALGEBRA	NNP
38536	1727	5	1	1	CD
38536	1727	6	.	.	.
38536	1728	1	Find	find	VB
38536	1728	2	the	the	DT
38536	1728	3	H.	H.	NNP
38536	1728	4	C.	C.	NNP
38536	1728	5	F.	F.	NNP
38536	1728	6	of	of	IN
38536	1728	7	8x^3	8x^3	CD
38536	1728	8	-	-	SYM
38536	1728	9	27	27	CD
38536	1728	10	,	,	,
38536	1728	11	32x^5	32x^5	CD
38536	1728	12	-	-	SYM
38536	1728	13	243	243	CD
38536	1728	14	,	,	,
38536	1728	15	and	and	CC
38536	1728	16	6x^3	6x^3	CD
38536	1728	17	-	-	SYM
38536	1728	18	9x^2	9x^2	CD
38536	1728	19	+	+	CC
38536	1728	20	4x	4x	NNP
38536	1728	21	-	-	SYM
38536	1728	22	6	6	CD
38536	1728	23	.	.	.
38536	1729	1	2	2	LS
38536	1729	2	.	.	.
38536	1730	1	Solve	solve	VB
38536	1730	2	:	:	:
38536	1730	3	(	(	-LRB-
38536	1730	4	_	_	NNP
38536	1730	5	a	a	DT
38536	1730	6	_	_	NNP
38536	1730	7	)	)	-RRB-
38536	1730	8	(	(	-LRB-
38536	1730	9	2x	2x	CD
38536	1730	10	+	+	NNP
38536	1730	11	5)^(-5	5)^(-5	NNP
38536	1730	12	)	)	-RRB-
38536	1730	13	+	+	SYM
38536	1730	14	31(2x	31(2x	CD
38536	1730	15	+	+	CC
38536	1730	16	5)^(-5/2	5)^(-5/2	CD
38536	1730	17	)	)	-RRB-
38536	1730	18	=	=	SYM
38536	1730	19	32	32	CD
38536	1730	20	.	.	.
38536	1731	1	(	(	-LRB-
38536	1731	2	_	_	NNP
38536	1731	3	b	b	NNP
38536	1731	4	_	_	NNP
38536	1731	5	)	)	-RRB-
38536	1731	6	(	(	-LRB-
38536	1731	7	x	x	NNP
38536	1731	8	-	-	SYM
38536	1731	9	1)^(1/2	1)^(1/2	CD
38536	1731	10	)	)	-RRB-
38536	1731	11	+	+	CC
38536	1731	12	(	(	-LRB-
38536	1731	13	3x	3x	CD
38536	1731	14	+	+	SYM
38536	1731	15	1)^(1/2	1)^(1/2	CD
38536	1731	16	)	)	-RRB-
38536	1731	17	=	=	SYM
38536	1731	18	4	4	CD
38536	1731	19	.	.	.
38536	1732	1	3	3	LS
38536	1732	2	.	.	.
38536	1733	1	A	a	DT
38536	1733	2	farmer	farmer	NN
38536	1733	3	sold	sell	VBD
38536	1733	4	a	a	DT
38536	1733	5	horse	horse	NN
38536	1733	6	at	at	IN
38536	1733	7	$	$	$
38536	1733	8	75	75	CD
38536	1733	9	for	for	IN
38536	1733	10	which	which	WDT
38536	1733	11	he	-PRON-	PRP
38536	1733	12	had	have	VBD
38536	1733	13	paid	pay	VBN
38536	1733	14	x	x	SYM
38536	1733	15	dollars	dollar	NNS
38536	1733	16	.	.	.
38536	1734	1	He	-PRON-	PRP
38536	1734	2	realized	realize	VBD
38536	1734	3	x	x	NNS
38536	1734	4	per	per	IN
38536	1734	5	cent	cent	NN
38536	1734	6	profit	profit	NN
38536	1734	7	by	by	IN
38536	1734	8	his	-PRON-	PRP$
38536	1734	9	sale	sale	NN
38536	1734	10	.	.	.
38536	1735	1	Find	find	VB
38536	1735	2	x.	x.	NN
38536	1736	1	4	4	LS
38536	1736	2	.	.	.
38536	1737	1	Find	find	VB
38536	1737	2	the	the	DT
38536	1737	3	13th	13th	JJ
38536	1737	4	term	term	NN
38536	1737	5	and	and	CC
38536	1737	6	the	the	DT
38536	1737	7	sum	sum	NN
38536	1737	8	of	of	IN
38536	1737	9	13	13	CD
38536	1737	10	terms	term	NNS
38536	1737	11	of	of	IN
38536	1737	12	the	the	DT
38536	1737	13	arithmetical	arithmetical	JJ
38536	1737	14	progression	progression	NN
38536	1737	15	(	(	-LRB-
38536	1737	16	2^(1/2	2^(1/2	CD
38536	1737	17	)	)	-RRB-
38536	1737	18	-	-	:
38536	1737	19	1)/2	1)/2	CD
38536	1737	20	,	,	,
38536	1737	21	(	(	-LRB-
38536	1737	22	2^(1/2))/2	2^(1/2))/2	CD
38536	1737	23	,	,	,
38536	1737	24	(	(	-LRB-
38536	1737	25	1)/[2([2]^(1/2	1)/[2([2]^(1/2	CD
38536	1737	26	)	)	-RRB-
38536	1737	27	-	-	:
38536	1737	28	1	1	CD
38536	1737	29	)	)	-RRB-
38536	1737	30	]	]	-RRB-
38536	1737	31	,	,	,
38536	1737	32	·	·	NFP
38536	1737	33	·	·	NFP
38536	1737	34	·	·	NFP
38536	1737	35	.	.	.
38536	1738	1	5	5	CD
38536	1738	2	.	.	.
38536	1739	1	The	the	DT
38536	1739	2	difference	difference	NN
38536	1739	3	between	between	IN
38536	1739	4	two	two	CD
38536	1739	5	numbers	number	NNS
38536	1739	6	is	be	VBZ
38536	1739	7	48	48	CD
38536	1739	8	.	.	.
38536	1740	1	Their	-PRON-	PRP$
38536	1740	2	arithmetical	arithmetical	JJ
38536	1740	3	mean	mean	NN
38536	1740	4	exceeds	exceed	VBZ
38536	1740	5	their	-PRON-	PRP$
38536	1740	6	geometrical	geometrical	JJ
38536	1740	7	mean	mean	NN
38536	1740	8	by	by	IN
38536	1740	9	18	18	CD
38536	1740	10	.	.	.
38536	1741	1	Find	find	VB
38536	1741	2	the	the	DT
38536	1741	3	numbers	number	NNS
38536	1741	4	.	.	.
38536	1742	1	6	6	CD
38536	1742	2	.	.	.
38536	1743	1	Expand	expand	VB
38536	1743	2	by	by	IN
38536	1743	3	the	the	DT
38536	1743	4	binomial	binomial	JJ
38536	1743	5	theorem	theorem	NN
38536	1743	6	and	and	CC
38536	1743	7	simplify	simplify	VB
38536	1743	8	[	[	-LRB-
38536	1743	9	3a^(-2	3a^(-2	CD
38536	1743	10	)	)	-RRB-
38536	1743	11	-	-	:
38536	1743	12	a/[-2]^(1/2)]^5	a/[-2]^(1/2)]^5	NNP
38536	1743	13	.	.	.
38536	1744	1	7	7	LS
38536	1744	2	.	.	.
38536	1745	1	Solve	solve	NN
38536	1745	2	:	:	:
38536	1745	3	1	1	CD
38536	1745	4	/	/	SYM
38536	1745	5	x	x	NNS
38536	1745	6	+	+	SYM
38536	1745	7	1	1	CD
38536	1745	8	/	/	SYM
38536	1745	9	y	y	NN
38536	1745	10	=	=	SYM
38536	1745	11	3/2	3/2	CD
38536	1745	12	,	,	,
38536	1745	13	1/(x^2	1/(x^2	CD
38536	1745	14	)	)	-RRB-
38536	1745	15	+	+	SYM
38536	1745	16	1/(y^2	1/(y^2	CD
38536	1745	17	)	)	-RRB-
38536	1745	18	=	=	SYM
38536	1745	19	5/4	5/4	CD
38536	1745	20	.	.	.
38536	1746	1	8	8	LS
38536	1746	2	.	.	.
38536	1747	1	Solve	solve	VB
38536	1747	2	the	the	DT
38536	1747	3	following	following	JJ
38536	1747	4	equations	equation	NNS
38536	1747	5	and	and	CC
38536	1747	6	check	check	VB
38536	1747	7	the	the	DT
38536	1747	8	results	result	NNS
38536	1747	9	by	by	IN
38536	1747	10	finding	find	VBG
38536	1747	11	the	the	DT
38536	1747	12	intersections	intersection	NNS
38536	1747	13	of	of	IN
38536	1747	14	the	the	DT
38536	1747	15	graphs	graphs	NN
38536	1747	16	of	of	IN
38536	1747	17	the	the	DT
38536	1747	18	two	two	CD
38536	1747	19	equations	equation	NNS
38536	1747	20	:	:	:
38536	1747	21	{	{	-LRB-
38536	1747	22	x^2	x^2	NNP
38536	1747	23	=	=	SYM
38536	1747	24	4y	4y	CD
38536	1747	25	,	,	,
38536	1747	26	{	{	-LRB-
38536	1747	27	x	x	SYM
38536	1747	28	+	+	CD
38536	1747	29	2y	2y	CD
38536	1747	30	=	=	SYM
38536	1747	31	4	4	CD
38536	1747	32	.	.	.
38536	1748	1	~VASSAR	~VASSAR	NFP
38536	1748	2	COLLEGE~	COLLEGE~	NNP
38536	1748	3	ELEMENTARY	ELEMENTARY	NNP
38536	1748	4	AND	and	CC
38536	1748	5	INTERMEDIATE	INTERMEDIATE	NNP
38536	1748	6	ALGEBRA	ALGEBRA	NNP
38536	1748	7	Answer	answer	VB
38536	1748	8	any	any	DT
38536	1748	9	six	six	CD
38536	1748	10	questions	question	NNS
38536	1748	11	.	.	.
38536	1749	1	1	1	LS
38536	1749	2	.	.	.
38536	1750	1	Find	find	VB
38536	1750	2	the	the	DT
38536	1750	3	product	product	NN
38536	1750	4	of	of	IN
38536	1750	5	[	[	-LRB-
38536	1750	6	1	1	CD
38536	1750	7	+	+	SYM
38536	1750	8	2a/3	2a/3	CD
38536	1750	9	-	-	HYPH
38536	1750	10	(	(	-LRB-
38536	1750	11	5a^2)/(6	5a^2)/(6	NN
38536	1750	12	)	)	-RRB-
38536	1750	13	]	]	-RRB-
38536	1750	14	and	and	CC
38536	1750	15	[	[	-LRB-
38536	1750	16	2	2	CD
38536	1750	17	-	-	SYM
38536	1750	18	3a/4	3a/4	CD
38536	1750	19	+	+	NFP
38536	1750	20	(	(	-LRB-
38536	1750	21	a^2)/(3	a^2)/(3	JJ
38536	1750	22	)	)	-RRB-
38536	1750	23	]	]	-RRB-
38536	1750	24	.	.	.
38536	1751	1	2	2	LS
38536	1751	2	.	.	.
38536	1752	1	Resolve	resolve	VB
38536	1752	2	into	into	IN
38536	1752	3	linear	linear	JJ
38536	1752	4	factors	factor	NNS
38536	1752	5	:	:	:
38536	1752	6	(	(	-LRB-
38536	1752	7	_	_	NNP
38536	1752	8	a	a	DT
38536	1752	9	_	_	NNP
38536	1752	10	)	)	-RRB-
38536	1752	11	4x^2	4x^2	CD
38536	1752	12	-	-	SYM
38536	1752	13	25	25	CD
38536	1752	14	;	;	:
38536	1752	15	(	(	-LRB-
38536	1752	16	_	_	NNP
38536	1752	17	b	b	NNP
38536	1752	18	_	_	NNP
38536	1752	19	)	)	-RRB-
38536	1752	20	6x^2	6x^2	CD
38536	1752	21	-	-	HYPH
38536	1752	22	x	x	NN
38536	1752	23	-	-	CD
38536	1752	24	12	12	CD
38536	1752	25	;	;	:
38536	1752	26	(	(	-LRB-
38536	1752	27	_	_	NNP
38536	1752	28	c	c	NNP
38536	1752	29	_	_	NNP
38536	1752	30	)	)	-RRB-
38536	1752	31	a^2b^2	a^2b^2	CD
38536	1752	32	+	+	CC
38536	1752	33	1	1	CD
38536	1752	34	-	-	HYPH
38536	1752	35	a^2	a^2	CD
38536	1752	36	-	-	HYPH
38536	1752	37	b^2	b^2	NNS
38536	1752	38	;	;	:
38536	1752	39	(	(	-LRB-
38536	1752	40	_	_	NNP
38536	1752	41	d	d	NNP
38536	1752	42	_	_	NNP
38536	1752	43	)	)	-RRB-
38536	1752	44	y^3	y^3	NN
38536	1752	45	+	+	CC
38536	1752	46	(	(	-LRB-
38536	1752	47	x	x	SYM
38536	1752	48	-	-	CD
38536	1752	49	3)y^2	3)y^2	CD
38536	1752	50	-	-	HYPH
38536	1752	51	(	(	-LRB-
38536	1752	52	3x	3x	CD
38536	1752	53	-	-	HYPH
38536	1752	54	2)y	2)y	CD
38536	1752	55	+	+	SYM
38536	1752	56	2x	2x	CD
38536	1752	57	.	.	.
38536	1753	1	3	3	LS
38536	1753	2	.	.	.
38536	1754	1	Reduce	reduce	VB
38536	1754	2	to	to	TO
38536	1754	3	simplest	simple	JJS
38536	1754	4	form	form	NN
38536	1754	5	:	:	:
38536	1754	6	(	(	-LRB-
38536	1754	7	_	_	NNP
38536	1754	8	a	a	DT
38536	1754	9	_	_	NNP
38536	1754	10	)	)	-RRB-
38536	1754	11	z/(1	z/(1	NNP
38536	1754	12	/	/	SYM
38536	1754	13	x	x	NNP
38536	1754	14	-	-	SYM
38536	1754	15	1	1	CD
38536	1754	16	/	/	SYM
38536	1754	17	y	y	NN
38536	1754	18	)	)	-RRB-
38536	1754	19	+	+	SYM
38536	1754	20	y/(1	y/(1	NN
38536	1754	21	-	-	HYPH
38536	1754	22	y	y	NN
38536	1754	23	/	/	SYM
38536	1754	24	x	x	NNP
38536	1754	25	)	)	-RRB-
38536	1754	26	-	-	HYPH
38536	1754	27	x/(1	x/(1	NN
38536	1754	28	-	-	HYPH
38536	1754	29	x	x	NNP
38536	1754	30	/	/	SYM
38536	1754	31	y	y	NN
38536	1754	32	)	)	-RRB-
38536	1754	33	.	.	.
38536	1755	1	(	(	-LRB-
38536	1755	2	_	_	NNP
38536	1755	3	b	b	NNP
38536	1755	4	_	_	NNP
38536	1755	5	)	)	-RRB-
38536	1755	6	[	[	-LRB-
38536	1755	7	-(x^3)^(1/2)]^(1/3	-(x^3)^(1/2)]^(1/3	NN
38536	1755	8	)	)	-RRB-
38536	1755	9	×	×	NFP
38536	1755	10	(	(	-LRB-
38536	1755	11	4y^(-3))^(1/2	4y^(-3))^(1/2	NN
38536	1755	12	)	)	-RRB-
38536	1755	13	.	.	.
38536	1756	1	4	4	LS
38536	1756	2	.	.	.
38536	1757	1	(	(	-LRB-
38536	1757	2	_	_	NNP
38536	1757	3	a	a	DT
38536	1757	4	_	_	NNP
38536	1757	5	)	)	-RRB-
38536	1757	6	Divide	Divide	NNP
38536	1757	7	x^(3/2	x^(3/2	NNP
38536	1757	8	)	)	-RRB-
38536	1757	9	-	-	:
38536	1757	10	x^(-3/2	x^(-3/2	NNP
38536	1757	11	)	)	-RRB-
38536	1757	12	by	by	IN
38536	1757	13	x^(1/2	x^(1/2	NNP
38536	1757	14	)	)	-RRB-
38536	1757	15	-	-	:
38536	1757	16	x^(-1/2	x^(-1/2	NN
38536	1757	17	)	)	-RRB-
38536	1757	18	.	.	.
38536	1758	1	(	(	-LRB-
38536	1758	2	_	_	NNP
38536	1758	3	b	b	NNP
38536	1758	4	_	_	NNP
38536	1758	5	)	)	-RRB-
38536	1758	6	Find	find	VB
38536	1758	7	correct	correct	JJ
38536	1758	8	to	to	IN
38536	1758	9	one	one	CD
38536	1758	10	place	place	NN
38536	1758	11	of	of	IN
38536	1758	12	decimals	decimal	NNS
38536	1758	13	the	the	DT
38536	1758	14	value	value	NN
38536	1758	15	of	of	IN
38536	1758	16	[	[	-LRB-
38536	1758	17	5^(1/2	5^(1/2	CD
38536	1758	18	)	)	-RRB-
38536	1758	19	+	+	CC
38536	1758	20	7^(1/2)]/[2	7^(1/2)]/[2	CD
38536	1758	21	-	-	SYM
38536	1758	22	3^(1/2	3^(1/2	CD
38536	1758	23	)	)	-RRB-
38536	1758	24	]	]	-RRB-
38536	1758	25	.	.	.
38536	1759	1	5	5	CD
38536	1759	2	.	.	.
38536	1760	1	(	(	-LRB-
38536	1760	2	_	_	NNP
38536	1760	3	a	a	DT
38536	1760	4	_	_	NNP
38536	1760	5	)	)	-RRB-
38536	1760	6	If	if	IN
38536	1760	7	a	a	NN
38536	1760	8	/	/	SYM
38536	1760	9	b	b	NN
38536	1760	10	=	=	SYM
38536	1760	11	c	c	NN
38536	1760	12	/	/	SYM
38536	1760	13	d	d	NNP
38536	1760	14	,	,	,
38536	1760	15	show	show	VBP
38536	1760	16	that	that	IN
38536	1760	17	(	(	-LRB-
38536	1760	18	a^2	a^2	NNS
38536	1760	19	+	+	SYM
38536	1760	20	c^2)/(b^2	c^2)/(b^2	NN
38536	1760	21	+	+	CC
38536	1760	22	d^2	d^2	NN
38536	1760	23	)	)	-RRB-
38536	1760	24	=	=	NFP
38536	1760	25	ac	ac	NNP
38536	1760	26	/	/	SYM
38536	1760	27	bd	bd	NNP
38536	1760	28	.	.	.
38536	1761	1	(	(	-LRB-
38536	1761	2	_	_	NNP
38536	1761	3	b	b	NNP
38536	1761	4	_	_	NNP
38536	1761	5	)	)	-RRB-
38536	1761	6	Two	two	CD
38536	1761	7	numbers	number	NNS
38536	1761	8	are	be	VBP
38536	1761	9	in	in	IN
38536	1761	10	the	the	DT
38536	1761	11	ratio	ratio	NN
38536	1761	12	3	3	CD
38536	1761	13	:	:	SYM
38536	1761	14	4	4	CD
38536	1761	15	,	,	,
38536	1761	16	and	and	CC
38536	1761	17	if	if	IN
38536	1761	18	7	7	CD
38536	1761	19	be	be	VB
38536	1761	20	subtracted	subtract	VBN
38536	1761	21	from	from	IN
38536	1761	22	each	each	DT
38536	1761	23	the	the	DT
38536	1761	24	remainders	remainder	NNS
38536	1761	25	are	be	VBP
38536	1761	26	in	in	IN
38536	1761	27	the	the	DT
38536	1761	28	ratio	ratio	NN
38536	1761	29	2	2	CD
38536	1761	30	:	:	SYM
38536	1761	31	3	3	CD
38536	1761	32	.	.	.
38536	1762	1	Find	find	VB
38536	1762	2	the	the	DT
38536	1762	3	numbers	number	NNS
38536	1762	4	.	.	.
38536	1763	1	6	6	CD
38536	1763	2	.	.	.
38536	1764	1	Solve	solve	VB
38536	1764	2	the	the	DT
38536	1764	3	equations	equation	NNS
38536	1764	4	:	:	:
38536	1764	5	(	(	-LRB-
38536	1764	6	_	_	NNP
38536	1764	7	a	a	DT
38536	1764	8	_	_	NNP
38536	1764	9	)	)	-RRB-
38536	1764	10	(	(	-LRB-
38536	1764	11	x	x	NN
38536	1764	12	+	+	SYM
38536	1764	13	1)/(2	1)/(2	JJ
38536	1764	14	)	)	-RRB-
38536	1764	15	-	-	SYM
38536	1764	16	3	3	CD
38536	1764	17	/	/	SYM
38536	1764	18	x	x	SYM
38536	1764	19	=	=	SYM
38536	1764	20	x/3	x/3	NN
38536	1764	21	-	-	HYPH
38536	1764	22	(	(	-LRB-
38536	1764	23	5	5	CD
38536	1764	24	-	-	HYPH
38536	1764	25	x)/(6	x)/(6	NNP
38536	1764	26	)	)	-RRB-
38536	1764	27	.	.	.
38536	1765	1	(	(	-LRB-
38536	1765	2	_	_	NNP
38536	1765	3	b	b	NNP
38536	1765	4	_	_	NNP
38536	1765	5	)	)	-RRB-
38536	1765	6	11x^2	11x^2	CD
38536	1765	7	-	-	HYPH
38536	1765	8	11	11	CD
38536	1765	9	-	-	HYPH
38536	1765	10	1/4	1/4	CD
38536	1765	11	=	=	SYM
38536	1765	12	9x	9x	NN
38536	1765	13	.	.	.
38536	1766	1	(	(	-LRB-
38536	1766	2	_	_	NNP
38536	1766	3	c	c	NNP
38536	1766	4	_	_	NNP
38536	1766	5	)	)	-RRB-
38536	1766	6	{	{	-LRB-
38536	1766	7	x^2	x^2	NNP
38536	1766	8	-	-	HYPH
38536	1766	9	2y^2	2y^2	CD
38536	1766	10	=	=	SYM
38536	1766	11	71	71	CD
38536	1766	12	,	,	,
38536	1766	13	{	{	-LRB-
38536	1766	14	x	x	SYM
38536	1766	15	+	+	SYM
38536	1766	16	y	y	NN
38536	1766	17	=	=	SYM
38536	1766	18	20	20	CD
38536	1766	19	.	.	.
38536	1767	1	7	7	LS
38536	1767	2	.	.	.
38536	1768	1	A	a	DT
38536	1768	2	field	field	NN
38536	1768	3	could	could	MD
38536	1768	4	be	be	VB
38536	1768	5	made	make	VBN
38536	1768	6	into	into	IN
38536	1768	7	a	a	DT
38536	1768	8	square	square	NN
38536	1768	9	by	by	IN
38536	1768	10	diminishing	diminish	VBG
38536	1768	11	the	the	DT
38536	1768	12	length	length	NN
38536	1768	13	by	by	IN
38536	1768	14	10	10	CD
38536	1768	15	feet	foot	NNS
38536	1768	16	and	and	CC
38536	1768	17	increasing	increase	VBG
38536	1768	18	the	the	DT
38536	1768	19	breadth	breadth	NN
38536	1768	20	by	by	IN
38536	1768	21	5	5	CD
38536	1768	22	feet	foot	NNS
38536	1768	23	,	,	,
38536	1768	24	but	but	CC
38536	1768	25	its	-PRON-	PRP$
38536	1768	26	area	area	NN
38536	1768	27	would	would	MD
38536	1768	28	then	then	RB
38536	1768	29	be	be	VB
38536	1768	30	diminished	diminish	VBN
38536	1768	31	by	by	IN
38536	1768	32	210	210	CD
38536	1768	33	square	square	JJ
38536	1768	34	feet	foot	NNS
38536	1768	35	.	.	.
38536	1769	1	Find	find	VB
38536	1769	2	the	the	DT
38536	1769	3	length	length	NN
38536	1769	4	and	and	CC
38536	1769	5	the	the	DT
38536	1769	6	breadth	breadth	NN
38536	1769	7	of	of	IN
38536	1769	8	the	the	DT
38536	1769	9	field	field	NN
38536	1769	10	.	.	.
38536	1770	1	~VASSAR	~VASSAR	NFP
38536	1770	2	COLLEGE~	COLLEGE~	NNP
38536	1770	3	ELEMENTARY	ELEMENTARY	NNP
38536	1770	4	AND	and	CC
38536	1770	5	INTERMEDIATE	INTERMEDIATE	NNP
38536	1770	6	ALGEBRA	ALGEBRA	NNP
38536	1770	7	Answer	answer	NN
38536	1770	8	six	six	CD
38536	1770	9	questions	question	NNS
38536	1770	10	,	,	,
38536	1770	11	including	include	VBG
38536	1770	12	No	no	UH
38536	1770	13	.	.	.
38536	1771	1	5	5	CD
38536	1771	2	and	and	CC
38536	1771	3	No	no	UH
38536	1771	4	.	.	.
38536	1772	1	7	7	CD
38536	1772	2	or	or	CC
38536	1772	3	8	8	CD
38536	1772	4	.	.	.
38536	1773	1	Candidates	candidate	NNS
38536	1773	2	in	in	IN
38536	1773	3	Intermediate	Intermediate	NNP
38536	1773	4	Algebra	Algebra	NNP
38536	1773	5	will	will	MD
38536	1773	6	answer	answer	VB
38536	1773	7	Nos	Nos	NNP
38536	1773	8	.	.	.
38536	1774	1	5	5	CD
38536	1774	2	-	-	SYM
38536	1774	3	9	9	CD
38536	1774	4	.	.	.
38536	1775	1	1	1	LS
38536	1775	2	.	.	.
38536	1776	1	Find	find	VB
38536	1776	2	two	two	CD
38536	1776	3	numbers	number	NNS
38536	1776	4	whose	whose	WP$
38536	1776	5	ratio	ratio	NN
38536	1776	6	is	be	VBZ
38536	1776	7	3	3	CD
38536	1776	8	and	and	CC
38536	1776	9	such	such	JJ
38536	1776	10	that	that	IN
38536	1776	11	two	two	CD
38536	1776	12	sevenths	seventh	NNS
38536	1776	13	of	of	IN
38536	1776	14	the	the	DT
38536	1776	15	larger	large	JJR
38536	1776	16	is	be	VBZ
38536	1776	17	15	15	CD
38536	1776	18	more	more	JJR
38536	1776	19	than	than	IN
38536	1776	20	one	one	CD
38536	1776	21	half	half	NN
38536	1776	22	the	the	DT
38536	1776	23	smaller	small	JJR
38536	1776	24	.	.	.
38536	1777	1	2	2	LS
38536	1777	2	.	.	.
38536	1778	1	Determine	determine	VB
38536	1778	2	the	the	DT
38536	1778	3	factors	factor	NNS
38536	1778	4	of	of	IN
38536	1778	5	the	the	DT
38536	1778	6	lowest	low	JJS
38536	1778	7	common	common	JJ
38536	1778	8	multiple	multiple	NN
38536	1778	9	of	of	IN
38536	1778	10	3x^4	3x^4	CD
38536	1778	11	(	(	-LRB-
38536	1778	12	x^3	x^3	NNP
38536	1778	13	-	-	HYPH
38536	1778	14	y^3	y^3	NNP
38536	1778	15	)	)	-RRB-
38536	1778	16	,	,	,
38536	1778	17	15	15	CD
38536	1778	18	(	(	-LRB-
38536	1778	19	x^4	x^4	NNP
38536	1778	20	-	-	HYPH
38536	1778	21	2x^2y^2	2x^2y^2	CD
38536	1778	22	+	+	SYM
38536	1778	23	y^4	y^4	UH
38536	1778	24	)	)	-RRB-
38536	1778	25	,	,	,
38536	1778	26	and	and	CC
38536	1778	27	10y	10y	NNS
38536	1778	28	(	(	-LRB-
38536	1778	29	x^4	x^4	NNP
38536	1778	30	+	+	SYM
38536	1778	31	x^2y^2	x^2y^2	XX
38536	1778	32	+	+	SYM
38536	1778	33	y^4	y^4	UH
38536	1778	34	)	)	-RRB-
38536	1778	35	.	.	.
38536	1779	1	3	3	LS
38536	1779	2	.	.	.
38536	1780	1	Find	find	VB
38536	1780	2	to	to	IN
38536	1780	3	two	two	CD
38536	1780	4	decimal	decimal	JJ
38536	1780	5	places	place	NNS
38536	1780	6	the	the	DT
38536	1780	7	value	value	NN
38536	1780	8	of	of	IN
38536	1780	9	[	[	-LRB-
38536	1780	10	4a^(-2/5	4a^(-2/5	CD
38536	1780	11	)	)	-RRB-
38536	1780	12	+	+	CC
38536	1780	13	b^0[ab^(-1)]^(1/2)]^(1/2	b^0[ab^(-1)]^(1/2)]^(1/2	NNP
38536	1780	14	)	)	-RRB-
38536	1780	15	,	,	,
38536	1780	16	when	when	WRB
38536	1780	17	a	a	DT
38536	1780	18	=	=	SYM
38536	1780	19	-32	-32	NNP
38536	1780	20	and	and	CC
38536	1780	21	b	b	LS
38536	1780	22	=	=	-RRB-
38536	1780	23	-8	-8	:
38536	1780	24	.	.	.
38536	1781	1	4	4	LS
38536	1781	2	.	.	.
38536	1782	1	Solve	solve	VB
38536	1782	2	the	the	DT
38536	1782	3	equations	equation	NNS
38536	1782	4	:	:	:
38536	1782	5	2x	2x	CD
38536	1782	6	+	+	SYM
38536	1782	7	5y	5y	CD
38536	1782	8	=	=	SYM
38536	1782	9	85	85	CD
38536	1782	10	,	,	,
38536	1782	11	2y	2y	CD
38536	1782	12	+	+	SYM
38536	1782	13	5z	5z	NNP
38536	1782	14	=	=	SYM
38536	1782	15	103	103	CD
38536	1782	16	,	,	,
38536	1782	17	2z	2z	CD
38536	1782	18	+	+	SYM
38536	1782	19	5x	5x	CD
38536	1782	20	=	=	SYM
38536	1782	21	57	57	CD
38536	1782	22	.	.	.
38536	1783	1	5	5	CD
38536	1783	2	.	.	.
38536	1784	1	Solve	solve	VB
38536	1784	2	any	any	DT
38536	1784	3	3	3	CD
38536	1784	4	of	of	IN
38536	1784	5	these	these	DT
38536	1784	6	equations	equation	NNS
38536	1784	7	:	:	:
38536	1784	8	(	(	-LRB-
38536	1784	9	_	_	NNP
38536	1784	10	a	a	DT
38536	1784	11	_	_	NNP
38536	1784	12	)	)	-RRB-
38536	1784	13	x^2	x^2	NNP
38536	1784	14	+	+	SYM
38536	1784	15	44	44	CD
38536	1784	16	-	-	HYPH
38536	1784	17	15x	15x	NNS
38536	1784	18	=	=	SYM
38536	1784	19	0	0	NFP
38536	1784	20	.	.	.
38536	1785	1	(	(	-LRB-
38536	1785	2	_	_	NNP
38536	1785	3	b	b	NNP
38536	1785	4	_	_	NNP
38536	1785	5	)	)	-RRB-
38536	1785	6	2	2	CD
38536	1785	7	/	/	SYM
38536	1785	8	x	x	NNP
38536	1785	9	-	-	HYPH
38536	1785	10	x/5	x/5	NNP
38536	1785	11	=	=	SYM
38536	1785	12	x/20	x/20	NNP
38536	1785	13	-	-	HYPH
38536	1785	14	223/30	223/30	CD
38536	1785	15	.	.	.
38536	1786	1	(	(	-LRB-
38536	1786	2	_	_	NNP
38536	1786	3	c	c	NNP
38536	1786	4	_	_	NNP
38536	1786	5	)	)	-RRB-
38536	1786	6	x^2	x^2	NNP
38536	1786	7	+	+	SYM
38536	1786	8	8x	8x	NN
38536	1786	9	-	-	HYPH
38536	1786	10	[	[	-LRB-
38536	1786	11	4x^2	4x^2	CD
38536	1786	12	+	+	CD
38536	1786	13	32x	32x	NNS
38536	1786	14	+	+	SYM
38536	1786	15	12]^(1/2	12]^(1/2	CD
38536	1786	16	)	)	-RRB-
38536	1786	17	=	=	SYM
38536	1786	18	21	21	CD
38536	1786	19	.	.	.
38536	1787	1	(	(	-LRB-
38536	1787	2	_	_	NNP
38536	1787	3	d	d	NNP
38536	1787	4	_	_	NNP
38536	1787	5	)	)	-RRB-
38536	1787	6	5/(x	5/(x	CD
38536	1787	7	+	+	SYM
38536	1787	8	1	1	CD
38536	1787	9	)	)	-RRB-
38536	1787	10	+	+	CC
38536	1787	11	8/(x	8/(x	CD
38536	1787	12	-	-	SYM
38536	1787	13	2	2	CD
38536	1787	14	)	)	-RRB-
38536	1787	15	=	=	SYM
38536	1787	16	12/(40	12/(40	CD
38536	1787	17	-	-	HYPH
38536	1787	18	2x	2x	NN
38536	1787	19	)	)	-RRB-
38536	1787	20	.	.	.
38536	1788	1	6	6	CD
38536	1788	2	.	.	.
38536	1789	1	The	the	DT
38536	1789	2	sum	sum	NN
38536	1789	3	of	of	IN
38536	1789	4	two	two	CD
38536	1789	5	numbers	number	NNS
38536	1789	6	is	be	VBZ
38536	1789	7	13	13	CD
38536	1789	8	,	,	,
38536	1789	9	and	and	CC
38536	1789	10	the	the	DT
38536	1789	11	sum	sum	NN
38536	1789	12	of	of	IN
38536	1789	13	their	-PRON-	PRP$
38536	1789	14	cubes	cube	NNS
38536	1789	15	is	be	VBZ
38536	1789	16	910	910	CD
38536	1789	17	.	.	.
38536	1790	1	Find	find	VB
38536	1790	2	the	the	DT
38536	1790	3	smaller	small	JJR
38536	1790	4	number	number	NN
38536	1790	5	,	,	,
38536	1790	6	correct	correct	JJ
38536	1790	7	to	to	IN
38536	1790	8	the	the	DT
38536	1790	9	second	second	JJ
38536	1790	10	decimal	decimal	JJ
38536	1790	11	place	place	NN
38536	1790	12	.	.	.
38536	1791	1	7	7	LS
38536	1791	2	.	.	.
38536	1792	1	The	the	DT
38536	1792	2	sum	sum	NN
38536	1792	3	of	of	IN
38536	1792	4	9	9	CD
38536	1792	5	terms	term	NNS
38536	1792	6	of	of	IN
38536	1792	7	an	an	DT
38536	1792	8	arithmetical	arithmetical	JJ
38536	1792	9	progression	progression	NN
38536	1792	10	is	be	VBZ
38536	1792	11	46	46	CD
38536	1792	12	;	;	:
38536	1792	13	the	the	DT
38536	1792	14	sum	sum	NN
38536	1792	15	of	of	IN
38536	1792	16	the	the	DT
38536	1792	17	first	first	JJ
38536	1792	18	5	5	CD
38536	1792	19	terms	term	NNS
38536	1792	20	is	be	VBZ
38536	1792	21	25	25	CD
38536	1792	22	.	.	.
38536	1793	1	Find	find	VB
38536	1793	2	the	the	DT
38536	1793	3	common	common	JJ
38536	1793	4	difference	difference	NN
38536	1793	5	.	.	.
38536	1794	1	8	8	LS
38536	1794	2	.	.	.
38536	1795	1	Explain	explain	VB
38536	1795	2	the	the	DT
38536	1795	3	terms	term	NNS
38536	1795	4	,	,	,
38536	1795	5	and	and	CC
38536	1795	6	prove	prove	VBP
38536	1795	7	that	that	IN
38536	1795	8	if	if	IN
38536	1795	9	four	four	CD
38536	1795	10	numbers	number	NNS
38536	1795	11	are	be	VBP
38536	1795	12	in	in	IN
38536	1795	13	proportion	proportion	NN
38536	1795	14	,	,	,
38536	1795	15	they	-PRON-	PRP
38536	1795	16	are	be	VBP
38536	1795	17	in	in	IN
38536	1795	18	proportion	proportion	NN
38536	1795	19	by	by	IN
38536	1795	20	_	_	NNP
38536	1795	21	alternation	alternation	NN
38536	1795	22	_	_	NNP
38536	1795	23	,	,	,
38536	1795	24	by	by	IN
38536	1795	25	_	_	NNP
38536	1795	26	inversion	inversion	NN
38536	1795	27	_	_	NNP
38536	1795	28	,	,	,
38536	1795	29	and	and	CC
38536	1795	30	by	by	IN
38536	1795	31	_	_	NNP
38536	1795	32	composition	composition	NN
38536	1795	33	_	_	NNP
38536	1795	34	.	.	.
38536	1796	1	Find	find	VB
38536	1796	2	x	x	NNS
38536	1796	3	when	when	WRB
38536	1796	4	(	(	-LRB-
38536	1796	5	3	3	CD
38536	1796	6	+	+	SYM
38536	1796	7	x)/(3	x)/(3	NNP
38536	1796	8	-	-	HYPH
38536	1796	9	x	x	NNP
38536	1796	10	)	)	-RRB-
38536	1796	11	=	=	NFP
38536	1796	12	(	(	-LRB-
38536	1796	13	40	40	CD
38536	1796	14	+	+	SYM
38536	1796	15	x^3)/(40	x^3)/(40	CD
38536	1796	16	-	-	HYPH
38536	1796	17	x^3	x^3	NN
38536	1796	18	)	)	-RRB-
38536	1796	19	.	.	.
38536	1797	1	9	9	CD
38536	1797	2	.	.	.
38536	1798	1	Find	find	VB
38536	1798	2	the	the	DT
38536	1798	3	value	value	NN
38536	1798	4	of	of	IN
38536	1798	5	x	x	NNS
38536	1798	6	in	in	IN
38536	1798	7	each	each	DT
38536	1798	8	of	of	IN
38536	1798	9	these	these	DT
38536	1798	10	equations	equation	NNS
38536	1798	11	:	:	:
38536	1798	12	(	(	-LRB-
38536	1798	13	_	_	NNP
38536	1798	14	a	a	DT
38536	1798	15	_	_	NNP
38536	1798	16	)	)	-RRB-
38536	1798	17	7x^(1/4	7x^(1/4	CD
38536	1798	18	)	)	-RRB-
38536	1798	19	-	-	:
38536	1798	20	3x^(1/2	3x^(1/2	CD
38536	1798	21	)	)	-RRB-
38536	1798	22	=	=	SYM
38536	1798	23	2	2	CD
38536	1798	24	.	.	.
38536	1799	1	(	(	-LRB-
38536	1799	2	_	_	NNP
38536	1799	3	b	b	NNP
38536	1799	4	_	_	NNP
38536	1799	5	)	)	-RRB-
38536	1799	6	(	(	-LRB-
38536	1799	7	x^2	x^2	NNS
38536	1799	8	+	+	SYM
38536	1799	9	2)^(5/2	2)^(5/2	CD
38536	1799	10	)	)	-RRB-
38536	1799	11	+	+	CC
38536	1799	12	3/{[x^2	3/{[x^2	CD
38536	1799	13	+	+	SYM
38536	1799	14	2]^(1/2	2]^(1/2	CD
38536	1799	15	)	)	-RRB-
38536	1799	16	}	}	-RRB-
38536	1799	17	=	=	NFP
38536	1799	18	4x^2	4x^2	CD
38536	1799	19	+	+	SYM
38536	1799	20	8	8	CD
38536	1799	21	.	.	.
38536	1800	1	~YALE	~YALE	NFP
38536	1800	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1800	3	ALGEBRA	ALGEBRA	NNP
38536	1800	4	A	a	DT
38536	1800	5	TIME	time	NN
38536	1800	6	:	:	:
38536	1800	7	ONE	one	CD
38536	1800	8	HOUR	HOUR	NNP
38536	1800	9	Omit	Omit	NNP
38536	1800	10	one	one	CD
38536	1800	11	question	question	NN
38536	1800	12	in	in	IN
38536	1800	13	Group	Group	NNP
38536	1800	14	II	II	NNP
38536	1800	15	and	and	CC
38536	1800	16	one	one	CD
38536	1800	17	in	in	IN
38536	1800	18	Group	Group	NNP
38536	1800	19	III	III	NNP
38536	1800	20	.	.	.
38536	1801	1	Credit	credit	NN
38536	1801	2	will	will	MD
38536	1801	3	be	be	VB
38536	1801	4	given	give	VBN
38536	1801	5	for	for	IN
38536	1801	6	_	_	NNP
38536	1801	7	six	six	CD
38536	1801	8	_	_	NNP
38536	1801	9	questions	question	NNS
38536	1801	10	only	only	RB
38536	1801	11	.	.	.
38536	1802	1	_	_	NNP
38536	1802	2	Group	Group	NNP
38536	1802	3	I	I	NNP
38536	1802	4	_	_	NNP
38536	1802	5	1	1	CD
38536	1802	6	.	.	.
38536	1803	1	Resolve	resolve	VB
38536	1803	2	into	into	IN
38536	1803	3	prime	prime	JJ
38536	1803	4	factors	factor	NNS
38536	1803	5	:	:	:
38536	1803	6	(	(	-LRB-
38536	1803	7	_	_	NNP
38536	1803	8	a	a	DT
38536	1803	9	_	_	NNP
38536	1803	10	)	)	-RRB-
38536	1803	11	6x^2	6x^2	CD
38536	1803	12	-	-	HYPH
38536	1803	13	7x	7x	CD
38536	1803	14	-	-	HYPH
38536	1803	15	20	20	CD
38536	1803	16	;	;	:
38536	1803	17	(	(	-LRB-
38536	1803	18	_	_	NNP
38536	1803	19	b	b	NNP
38536	1803	20	_	_	NNP
38536	1803	21	)	)	-RRB-
38536	1803	22	(	(	-LRB-
38536	1803	23	x^2	x^2	NNP
38536	1803	24	-	-	HYPH
38536	1803	25	5x)^2	5x)^2	CD
38536	1803	26	-	-	HYPH
38536	1803	27	2(x^2	2(x^2	CD
38536	1803	28	-	-	HYPH
38536	1803	29	5x	5x	CD
38536	1803	30	)	)	-RRB-
38536	1803	31	-	-	:
38536	1803	32	24	24	CD
38536	1803	33	;	;	:
38536	1803	34	(	(	-LRB-
38536	1803	35	_	_	NNP
38536	1803	36	c	c	NNP
38536	1803	37	_	_	NNP
38536	1803	38	)	)	-RRB-
38536	1803	39	a^4	a^4	CD
38536	1803	40	+	+	SYM
38536	1803	41	4a^2	4a^2	CD
38536	1803	42	+	+	SYM
38536	1803	43	16	16	CD
38536	1803	44	.	.	.
38536	1804	1	2	2	LS
38536	1804	2	.	.	.
38536	1805	1	Simplify	Simplify	NNP
38536	1805	2	[	[	-LRB-
38536	1805	3	5	5	CD
38536	1805	4	-	-	:
38536	1805	5	(	(	-LRB-
38536	1805	6	a^2	a^2	CD
38536	1805	7	-	-	HYPH
38536	1805	8	19x^2)/(a^2	19x^2)/(a^2	CD
38536	1805	9	-	-	HYPH
38536	1805	10	4x^2	4x^2	CD
38536	1805	11	)	)	-RRB-
38536	1805	12	]	]	-RRB-
38536	1805	13	÷	÷	NNP
38536	1805	14	[	[	-LRB-
38536	1805	15	3	3	CD
38536	1805	16	-	-	SYM
38536	1805	17	(	(	-LRB-
38536	1805	18	a	a	DT
38536	1805	19	-	-	HYPH
38536	1805	20	5x)/(a	5x)/(a	CD
38536	1805	21	-	-	HYPH
38536	1805	22	2x	2x	CD
38536	1805	23	)	)	-RRB-
38536	1805	24	]	]	-RRB-
38536	1805	25	.	.	.
38536	1806	1	3	3	LS
38536	1806	2	.	.	.
38536	1807	1	Solve	solve	VB
38536	1807	2	[	[	-LRB-
38536	1807	3	2(x	2(x	CD
38536	1807	4	-	-	:
38536	1807	5	7)]/(x^2	7)]/(x^2	JJ
38536	1807	6	+	+	SYM
38536	1807	7	3x	3x	CD
38536	1807	8	-	-	SYM
38536	1807	9	28	28	CD
38536	1807	10	)	)	-RRB-
38536	1807	11	+	+	CC
38536	1807	12	(	(	-LRB-
38536	1807	13	2	2	CD
38536	1807	14	-	-	HYPH
38536	1807	15	x)/(4	x)/(4	NNP
38536	1807	16	-	-	HYPH
38536	1807	17	x	x	NNP
38536	1807	18	)	)	-RRB-
38536	1807	19	-	-	:
38536	1807	20	(	(	-LRB-
38536	1807	21	x	x	NN
38536	1807	22	+	+	NNP
38536	1807	23	3)/(x	3)/(x	CD
38536	1807	24	+	+	SYM
38536	1807	25	7	7	CD
38536	1807	26	)	)	-RRB-
38536	1807	27	=	=	SYM
38536	1807	28	0	0	NFP
38536	1807	29	.	.	.
38536	1808	1	_	_	NNP
38536	1808	2	Group	Group	NNP
38536	1808	3	II	II	NNP
38536	1808	4	_	_	NNP
38536	1808	5	4	4	CD
38536	1808	6	.	.	.
38536	1809	1	Simplify	simplify	NN
38536	1809	2	[	[	-LRB-
38536	1809	3	2^(1/2	2^(1/2	CD
38536	1809	4	)	)	-RRB-
38536	1809	5	+	+	CC
38536	1809	6	2[3^(1/2)]]/[2^(1/2	2[3^(1/2)]]/[2^(1/2	LS
38536	1809	7	)	)	-RRB-
38536	1809	8	-	-	:
38536	1809	9	12^(1/2	12^(1/2	CD
38536	1809	10	)	)	-RRB-
38536	1809	11	]	]	-RRB-
38536	1809	12	,	,	,
38536	1809	13	and	and	CC
38536	1809	14	compute	compute	VB
38536	1809	15	the	the	DT
38536	1809	16	value	value	NN
38536	1809	17	of	of	IN
38536	1809	18	the	the	DT
38536	1809	19	fraction	fraction	NN
38536	1809	20	to	to	IN
38536	1809	21	two	two	CD
38536	1809	22	decimal	decimal	JJ
38536	1809	23	places	place	NNS
38536	1809	24	.	.	.
38536	1810	1	5	5	CD
38536	1810	2	.	.	.
38536	1811	1	Solve	solve	VB
38536	1811	2	the	the	DT
38536	1811	3	simultaneous	simultaneous	JJ
38536	1811	4	equations	equation	NNS
38536	1811	5	{	{	-LRB-
38536	1811	6	x^(-1/2	x^(-1/2	NN
38536	1811	7	)	)	-RRB-
38536	1811	8	+	+	CC
38536	1811	9	2y^(-1/2	2y^(-1/2	CD
38536	1811	10	)	)	-RRB-
38536	1811	11	=	=	SYM
38536	1811	12	7/6	7/6	CD
38536	1811	13	,	,	,
38536	1811	14	{	{	-LRB-
38536	1811	15	2x^(-1/2	2x^(-1/2	CD
38536	1811	16	)	)	-RRB-
38536	1811	17	-	-	:
38536	1811	18	y^(-1/2	y^(-1/2	NN
38536	1811	19	)	)	-RRB-
38536	1811	20	=	=	NFP
38536	1811	21	2/3	2/3	CD
38536	1811	22	.	.	.
38536	1812	1	_	_	NNP
38536	1812	2	Group	Group	NNP
38536	1812	3	III	III	NNP
38536	1812	4	_	_	NNP
38536	1812	5	6	6	CD
38536	1812	6	.	.	.
38536	1813	1	Two	two	CD
38536	1813	2	numbers	number	NNS
38536	1813	3	are	be	VBP
38536	1813	4	in	in	IN
38536	1813	5	the	the	DT
38536	1813	6	ratio	ratio	NN
38536	1813	7	of	of	IN
38536	1813	8	c	c	NN
38536	1813	9	:	:	:
38536	1813	10	d.	d.	NNP
38536	1813	11	If	if	IN
38536	1813	12	a	a	DT
38536	1813	13	be	be	NN
38536	1813	14	added	add	VBN
38536	1813	15	to	to	IN
38536	1813	16	the	the	DT
38536	1813	17	first	first	JJ
38536	1813	18	and	and	CC
38536	1813	19	subtracted	subtract	VBD
38536	1813	20	from	from	IN
38536	1813	21	the	the	DT
38536	1813	22	second	second	JJ
38536	1813	23	,	,	,
38536	1813	24	the	the	DT
38536	1813	25	results	result	NNS
38536	1813	26	will	will	MD
38536	1813	27	be	be	VB
38536	1813	28	in	in	IN
38536	1813	29	the	the	DT
38536	1813	30	ratio	ratio	NN
38536	1813	31	of	of	IN
38536	1813	32	3	3	CD
38536	1813	33	:	:	SYM
38536	1813	34	2	2	CD
38536	1813	35	.	.	.
38536	1814	1	Find	find	VB
38536	1814	2	the	the	DT
38536	1814	3	numbers	number	NNS
38536	1814	4	.	.	.
38536	1815	1	7	7	LS
38536	1815	2	.	.	.
38536	1816	1	A	a	DT
38536	1816	2	dealer	dealer	NN
38536	1816	3	has	have	VBZ
38536	1816	4	two	two	CD
38536	1816	5	kinds	kind	NNS
38536	1816	6	of	of	IN
38536	1816	7	coffee	coffee	NN
38536	1816	8	,	,	,
38536	1816	9	worth	worth	JJ
38536	1816	10	30	30	CD
38536	1816	11	and	and	CC
38536	1816	12	40	40	CD
38536	1816	13	cents	cent	NNS
38536	1816	14	per	per	IN
38536	1816	15	pound	pound	NN
38536	1816	16	.	.	.
38536	1817	1	How	how	WRB
38536	1817	2	many	many	JJ
38536	1817	3	pounds	pound	NNS
38536	1817	4	of	of	IN
38536	1817	5	each	each	DT
38536	1817	6	must	must	MD
38536	1817	7	be	be	VB
38536	1817	8	taken	take	VBN
38536	1817	9	to	to	TO
38536	1817	10	make	make	VB
38536	1817	11	a	a	DT
38536	1817	12	mixture	mixture	NN
38536	1817	13	of	of	IN
38536	1817	14	70	70	CD
38536	1817	15	pounds	pound	NNS
38536	1817	16	,	,	,
38536	1817	17	worth	worth	JJ
38536	1817	18	36	36	CD
38536	1817	19	cents	cent	NNS
38536	1817	20	per	per	IN
38536	1817	21	pound	pound	NN
38536	1817	22	?	?	.
38536	1818	1	8	8	LS
38536	1818	2	.	.	.
38536	1819	1	A	a	DT
38536	1819	2	,	,	,
38536	1819	3	B	b	NN
38536	1819	4	,	,	,
38536	1819	5	and	and	CC
38536	1819	6	C	C	NNP
38536	1819	7	can	can	MD
38536	1819	8	do	do	VB
38536	1819	9	a	a	DT
38536	1819	10	piece	piece	NN
38536	1819	11	of	of	IN
38536	1819	12	work	work	NN
38536	1819	13	in	in	IN
38536	1819	14	30	30	CD
38536	1819	15	hours	hour	NNS
38536	1819	16	.	.	.
38536	1820	1	A	a	DT
38536	1820	2	can	can	MD
38536	1820	3	do	do	VB
38536	1820	4	half	half	NN
38536	1820	5	as	as	RB
38536	1820	6	much	much	JJ
38536	1820	7	again	again	RB
38536	1820	8	as	as	IN
38536	1820	9	B	b	NN
38536	1820	10	,	,	,
38536	1820	11	and	and	CC
38536	1820	12	B	b	NN
38536	1820	13	two	two	CD
38536	1820	14	thirds	third	NNS
38536	1820	15	as	as	RB
38536	1820	16	much	much	RB
38536	1820	17	again	again	RB
38536	1820	18	as	as	IN
38536	1820	19	C.	C.	NNP
38536	1820	20	How	how	WRB
38536	1820	21	long	long	RB
38536	1820	22	would	would	MD
38536	1820	23	each	each	DT
38536	1820	24	require	require	VB
38536	1820	25	to	to	TO
38536	1820	26	do	do	VB
38536	1820	27	the	the	DT
38536	1820	28	work	work	NN
38536	1820	29	alone	alone	RB
38536	1820	30	?	?	.
38536	1821	1	~YALE	~YALE	NFP
38536	1821	2	UNIVERSITY~	UNIVERSITY~	NNP
38536	1821	3	ALGEBRA	ALGEBRA	NNP
38536	1821	4	B	B	NNP
38536	1821	5	TIME	time	NN
38536	1821	6	:	:	:
38536	1821	7	ONE	one	CD
38536	1821	8	HOUR	HOUR	NNP
38536	1821	9	Omit	Omit	NNP
38536	1821	10	one	one	CD
38536	1821	11	question	question	NN
38536	1821	12	in	in	IN
38536	1821	13	Group	Group	NNP
38536	1821	14	I	I	NNP
38536	1821	15	and	and	CC
38536	1821	16	one	one	CD
38536	1821	17	in	in	IN
38536	1821	18	Group	Group	NNP
38536	1821	19	II	II	NNP
38536	1821	20	.	.	.
38536	1822	1	Credit	credit	NN
38536	1822	2	will	will	MD
38536	1822	3	be	be	VB
38536	1822	4	given	give	VBN
38536	1822	5	for	for	IN
38536	1822	6	_	_	NNP
38536	1822	7	five	five	CD
38536	1822	8	_	_	NNP
38536	1822	9	questions	question	NNS
38536	1822	10	only	only	RB
38536	1822	11	.	.	.
38536	1823	1	_	_	NNP
38536	1823	2	Group	Group	NNP
38536	1823	3	I	I	NNP
38536	1823	4	_	_	NNP
38536	1823	5	1	1	CD
38536	1823	6	.	.	.
38536	1824	1	Solve	solve	VB
38536	1824	2	(	(	-LRB-
38536	1824	3	x	x	NN
38536	1824	4	+	+	SYM
38536	1824	5	a)/(x	a)/(x	NN
38536	1824	6	+	+	SYM
38536	1824	7	b	b	NN
38536	1824	8	)	)	-RRB-
38536	1824	9	+	+	NFP
38536	1824	10	(	(	-LRB-
38536	1824	11	x	x	NN
38536	1824	12	+	+	CD
38536	1824	13	b)/(x	b)/(x	CD
38536	1824	14	+	+	SYM
38536	1824	15	a	a	NN
38536	1824	16	)	)	-RRB-
38536	1824	17	=	=	SYM
38536	1824	18	5/2	5/2	CD
38536	1824	19	.	.	.
38536	1825	1	2	2	LS
38536	1825	2	.	.	.
38536	1826	1	Solve	solve	VB
38536	1826	2	the	the	DT
38536	1826	3	simultaneous	simultaneous	JJ
38536	1826	4	equations	equation	NNS
38536	1826	5	{	{	-LRB-
38536	1826	6	x^2y^2	x^2y^2	NFP
38536	1826	7	+	+	CC
38536	1826	8	28xy	28xy	NNP
38536	1826	9	-	-	HYPH
38536	1826	10	480	480	CD
38536	1826	11	=	=	SYM
38536	1826	12	0	0	CD
38536	1826	13	,	,	,
38536	1826	14	{	{	-LRB-
38536	1826	15	2x	2x	CD
38536	1826	16	+	+	SYM
38536	1826	17	y	y	NN
38536	1826	18	=	=	SYM
38536	1826	19	11	11	CD
38536	1826	20	.	.	.
38536	1827	1	Arrange	arrange	VB
38536	1827	2	the	the	DT
38536	1827	3	roots	root	NNS
38536	1827	4	in	in	IN
38536	1827	5	corresponding	correspond	VBG
38536	1827	6	pairs	pair	NNS
38536	1827	7	.	.	.
38536	1828	1	3	3	LS
38536	1828	2	.	.	.
38536	1829	1	Solve	solve	VB
38536	1829	2	3x^(-3/2	3x^(-3/2	CD
38536	1829	3	)	)	-RRB-
38536	1829	4	+	+	CD
38536	1829	5	20x^(-3/4	20x^(-3/4	CD
38536	1829	6	)	)	-RRB-
38536	1829	7	=	=	SYM
38536	1829	8	32	32	CD
38536	1829	9	.	.	.
38536	1830	1	_	_	NNP
38536	1830	2	Group	Group	NNP
38536	1830	3	II	II	NNP
38536	1830	4	_	_	NNP
38536	1830	5	4	4	CD
38536	1830	6	.	.	.
38536	1831	1	In	in	IN
38536	1831	2	going	go	VBG
38536	1831	3	7500	7500	CD
38536	1831	4	yd	yd	NNP
38536	1831	5	.	.	.
38536	1832	1	a	a	DT
38536	1832	2	front	front	JJ
38536	1832	3	wheel	wheel	NN
38536	1832	4	of	of	IN
38536	1832	5	a	a	DT
38536	1832	6	wagon	wagon	NN
38536	1832	7	makes	make	VBZ
38536	1832	8	1000	1000	CD
38536	1832	9	more	more	JJR
38536	1832	10	revolutions	revolution	NNS
38536	1832	11	than	than	IN
38536	1832	12	a	a	DT
38536	1832	13	rear	rear	JJ
38536	1832	14	one	one	NN
38536	1832	15	.	.	.
38536	1833	1	If	if	IN
38536	1833	2	the	the	DT
38536	1833	3	wheels	wheel	NNS
38536	1833	4	were	be	VBD
38536	1833	5	each	each	DT
38536	1833	6	1	1	CD
38536	1833	7	yd	yd	NN
38536	1833	8	.	.	.
38536	1834	1	greater	great	JJR
38536	1834	2	in	in	IN
38536	1834	3	circumference	circumference	NNP
38536	1834	4	,	,	,
38536	1834	5	a	a	DT
38536	1834	6	front	front	JJ
38536	1834	7	wheel	wheel	NN
38536	1834	8	would	would	MD
38536	1834	9	make	make	VB
38536	1834	10	625	625	CD
38536	1834	11	more	more	JJR
38536	1834	12	revolutions	revolution	NNS
38536	1834	13	than	than	IN
38536	1834	14	a	a	DT
38536	1834	15	rear	rear	JJ
38536	1834	16	one	one	NN
38536	1834	17	.	.	.
38536	1835	1	Find	find	VB
38536	1835	2	the	the	DT
38536	1835	3	circumference	circumference	NN
38536	1835	4	of	of	IN
38536	1835	5	each	each	DT
38536	1835	6	.	.	.
38536	1836	1	5	5	CD
38536	1836	2	.	.	.
38536	1837	1	Two	two	CD
38536	1837	2	cars	car	NNS
38536	1837	3	of	of	IN
38536	1837	4	equal	equal	JJ
38536	1837	5	speed	speed	NN
38536	1837	6	leave	leave	NN
38536	1837	7	A	a	NN
38536	1837	8	and	and	CC
38536	1837	9	B	b	NN
38536	1837	10	,	,	,
38536	1837	11	20	20	CD
38536	1837	12	mi	mi	NNS
38536	1837	13	.	.	.
38536	1838	1	apart	apart	RB
38536	1838	2	,	,	,
38536	1838	3	at	at	IN
38536	1838	4	different	different	JJ
38536	1838	5	times	time	NNS
38536	1838	6	.	.	.
38536	1839	1	Just	just	RB
38536	1839	2	as	as	IN
38536	1839	3	the	the	DT
38536	1839	4	cars	car	NNS
38536	1839	5	pass	pass	VBP
38536	1839	6	each	each	DT
38536	1839	7	other	other	JJ
38536	1839	8	an	an	DT
38536	1839	9	accident	accident	NN
38536	1839	10	reduces	reduce	VBZ
38536	1839	11	the	the	DT
38536	1839	12	power	power	NN
38536	1839	13	and	and	CC
38536	1839	14	their	-PRON-	PRP$
38536	1839	15	speed	speed	NN
38536	1839	16	is	be	VBZ
38536	1839	17	decreased	decrease	VBN
38536	1839	18	10	10	CD
38536	1839	19	mi	mi	NNS
38536	1839	20	.	.	.
38536	1840	1	per	per	IN
38536	1840	2	hour	hour	NN
38536	1840	3	.	.	.
38536	1841	1	One	one	CD
38536	1841	2	car	car	NN
38536	1841	3	makes	make	VBZ
38536	1841	4	the	the	DT
38536	1841	5	journey	journey	NN
38536	1841	6	from	from	IN
38536	1841	7	A	a	NN
38536	1841	8	to	to	IN
38536	1841	9	B	b	NN
38536	1841	10	in	in	IN
38536	1841	11	56	56	CD
38536	1841	12	min	min	NN
38536	1841	13	.	.	NNP
38536	1841	14	,	,	,
38536	1841	15	and	and	CC
38536	1841	16	the	the	DT
38536	1841	17	other	other	JJ
38536	1841	18	from	from	IN
38536	1841	19	B	B	NNP
38536	1841	20	to	to	IN
38536	1841	21	A	a	NN
38536	1841	22	in	in	IN
38536	1841	23	72	72	CD
38536	1841	24	min	min	NN
38536	1841	25	.	.	.
38536	1842	1	What	what	WP
38536	1842	2	is	be	VBZ
38536	1842	3	their	-PRON-	PRP$
38536	1842	4	common	common	JJ
38536	1842	5	speed	speed	NN
38536	1842	6	?	?	.
38536	1843	1	_	_	NNP
38536	1843	2	Group	Group	NNP
38536	1843	3	III	III	NNP
38536	1843	4	_	_	NNP
38536	1843	5	6	6	CD
38536	1843	6	.	.	.
38536	1844	1	Write	write	VB
38536	1844	2	in	in	IN
38536	1844	3	the	the	DT
38536	1844	4	simplest	simple	JJS
38536	1844	5	form	form	NN
38536	1844	6	the	the	DT
38536	1844	7	last	last	JJ
38536	1844	8	three	three	CD
38536	1844	9	terms	term	NNS
38536	1844	10	of	of	IN
38536	1844	11	the	the	DT
38536	1844	12	expansion	expansion	NN
38536	1844	13	of	of	IN
38536	1844	14	(	(	-LRB-
38536	1844	15	4a^(3/2	4a^(3/2	CD
38536	1844	16	)	)	-RRB-
38536	1844	17	-	-	:
38536	1844	18	a^(1/2	a^(1/2	NNP
38536	1844	19	)	)	-RRB-
38536	1844	20	x^(1/3))^8	x^(1/3))^8	NNP
38536	1844	21	.	.	.
38536	1845	1	7	7	LS
38536	1845	2	.	.	.
38536	1846	1	(	(	-LRB-
38536	1846	2	_	_	NNP
38536	1846	3	a	a	DT
38536	1846	4	_	_	NNP
38536	1846	5	)	)	-RRB-
38536	1846	6	Derive	derive	VB
38536	1846	7	the	the	DT
38536	1846	8	formula	formula	NN
38536	1846	9	for	for	IN
38536	1846	10	the	the	DT
38536	1846	11	sum	sum	NN
38536	1846	12	of	of	IN
38536	1846	13	an	an	DT
38536	1846	14	A.	a.	NN
38536	1846	15	P.	P.	NNP
38536	1846	16	(	(	-LRB-
38536	1846	17	_	_	NNP
38536	1846	18	b	b	NNP
38536	1846	19	_	_	NNP
38536	1846	20	)	)	-RRB-
38536	1846	21	Find	find	VB
38536	1846	22	the	the	DT
38536	1846	23	sum	sum	NN
38536	1846	24	to	to	IN
38536	1846	25	infinity	infinity	NN
38536	1846	26	of	of	IN
38536	1846	27	the	the	DT
38536	1846	28	series	series	NN
38536	1846	29	1	1	CD
38536	1846	30	,	,	,
38536	1846	31	-1/2	-1/2	NN
38536	1846	32	,	,	,
38536	1846	33	1/4	1/4	CD
38536	1846	34	,	,	,
38536	1846	35	-1/8	-1/8	CD
38536	1846	36	,	,	,
38536	1846	37	·	·	NFP
38536	1846	38	·	·	NFP
38536	1846	39	·	·	NFP
38536	1846	40	.	.	.
38536	1847	1	Also	also	RB
38536	1847	2	find	find	VB
38536	1847	3	the	the	DT
38536	1847	4	sum	sum	NN
38536	1847	5	of	of	IN
38536	1847	6	the	the	DT
38536	1847	7	positive	positive	JJ
38536	1847	8	terms	term	NNS
38536	1847	9	.	.	.