collegealgebra lr

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College Algebra


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OpenStax College

Rice University

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Houston, Texas 77005

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Algebra. OpenStax College. 13 February 2015. < openstaxcollege.org/textbooks/college-algebra>.

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For questions regarding this license, please contact [email protected].

ISBN-10 1938168380

ISBN-13 978-1-938168-38-3

Revision CA-1-001-AS


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> C@@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

C 1: ;; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1 ; == = *= >>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.3 ; = >=; E?>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.4 >;H=>= >;H=>=; E?>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

C 2: E;@ @ @>;; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.1 + =; C>>= *H= = &= ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

2.3 >; = A??;>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

2.4 C>>= > F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3003.5 +=>= > F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

3.6 A>; ; F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

3.7 != F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

C 4: ;@ F@;@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

4.1 #= F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

4.2 >;= #= F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

4.3 F= #= >; > D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

C 5: >K@?;> @ ;@> F@;@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

5.1 F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

5.2 > F=>= = >;H=>= . . . . . . . . . . . . . . . . . . . . . . . . . . 499

5.3 G? > >;H=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

5.4 D= >;H=> > >;H=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5625.6 >=; F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

5.7 != = ; F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

5.8 >;= = >= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624

C 6: EJ@@;> @ ;?; F@;@ . . . . . . . . . . . . . . . . . . . . . . . . . 643

6.1 E?>==; F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

6.2 G? > E?>==; F=>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

6.3 #>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

6.4 G? > #>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

6.5 #>? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722

6.6 E?>==; = #>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737

6.7 E?>==; = #>; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753

6.8 F= E?>==; >; > D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774

C 7: K? E;@ @ @>;; . . . . . . . . . . . . . . . . . . . . . . . . . . 809

7.1 *H #= E@>=: +> ; . . . . . . . . . . . . . . . . . . . . . . . . . 810

7.2 *H #= E@>=: + ; . . . . . . . . . . . . . . . . . . . . . . . . . 831

7.3 *H >=;= E@>= = !=@;: +> ; . . . . . . . . . . . . . . . 844

7.4 ; F>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858

7.5 = &?>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 870

7.6 *>;= *H;= *H;= *H


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8.2 + H?>; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962

8.3 + >; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984

8.4 >>= > A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002

8.5 C>= *>= = >; C>>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020

C 9: @, ;>;K, @ C@;@ K . . . . . . . . . . . . . . . . . . . . . 10439.1 *@= = + >>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044

9.2 A>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10859.5 C>== =?; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1100

9.6 B=>< . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112

9.7 >;H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1120

@J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141

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I : QEB PBQ LC FKQBDBOP

#: QEB PBQ LC OQFLKI KRJBOP

#]: QEB PBQ LC FOOQFLKI KRJBOP

%?

EB PBQ LC natural numbers FK@IRABP QEB KRJBOP RPBA CLO @LRKQFKD: {1, 2, 3, ...}.

EB PBQ LC whole numbers FP QEB PBQ LC KQROI KRJBOP MIRP WBOL: {0, 1, 2, 3, ...}.

EB PBQ LC integers AAP QEB KBDQFSB KQROI KRJBOP QL QEB PBQ LC TELIB KRJBOP: {..., −3, −2, −1, 0, 1, 2, 3, ...}.

EB PBQ LC rational numbers FK@IRABP CO@QFLKP TOFQQBK P ⎧

⎩⎨mn |m and n are integers and n ≠ 0

⎫

⎭⎬.

EB PBQ LC irrational numbers FP QEB PBQ LC KRJBOP QEQ OB KLQ OQFLKI, OB KLKOBMBQFKD, KA OB KLKQBOJFKQFKD:

{h|h is not a rational number}.

E


22/1147

1.5

N W I Q Q

. 3 6 = 6

. 83 = 2 . 6̄

@. 73

A. b6

B. 3.2121121112...

IPPFCV B@E KRJBO P BFKD KQROI KRJBO ( ), TELIB KRJBO ( ), FKQBDBO ( I ), OQFLKI KRJBO (#),

KA/LO FOOQFLKI KRJBO (# ).

. −357

. 0

@. 169

A. 24

B. 4.763763763 …

?;@ C>>;@ ;@ & &;@

EBK TB JRIQFMIV KRJBO V FQPBIC, TB PNROB FQ LO OFPB FQ QL MLTBO LC 2. LO BUJMIB, 42 = 4 ⋅ 4 = 1 6 . B @K OFPBKV KRJBO QL KV MLTBO. K DBKBOI, QEB exponential notation an JBKP QEQ QEB KRJBO LO SOFIB a FP RPBA P C@QLO

n QFJBP.

an = a ⋅ a ⋅ a ⋅ … ⋅ an factors

K QEFP KLQQFLK, an FP OBA P QEB 9QE MLTBO LC a, TEBOB a FP @IIBA QEB base KA n FP @IIBA QEB exponent. QBOJ FK

BUMLKBKQFI KLQQFLK JV B MOQ LC JQEBJQF@I BUMOBPPFLK, TEF@E FP @LJFKQFLK LC KRJBOP KA LMBOQFLKP. LO

BUJMIB,

2 4 + 6 ⋅ 23

− 42

FP JQEBJQF@I BUMOBPPFLK.

L BSIRQB JQEBJQF@I BUMOBPPFLK, TB MBOCLOJ QEB SOFLRP LMBOQFLKP. LTBSBO, TB AL KLQ MBOCLOJ QEBJ FK KV OKALJ

LOABO. B RPB QEB order of operations. EFP FP PBNRBK@B LC ORIBP CLO BSIRQFKD PR@E BUMOBPPFLKP.

B@II QEQ FK JQEBJQF@P TB RPB MOBKQEBPBP ( ), O@HBQP , KA O@BP X Z QL DOLRM KRJBOP KA BUMOBPPFLKP PL QEQ

KVQEFKD MMBOFKD TFQEFK QEB PVJLIP FP QOBQBA P RKFQ. AAFQFLKIIV, CO@QFLK OP, OAF@IP, KA PLIRQB SIRB OP OB

QOBQBA P DOLRMFKD PVJLIP. EBK BSIRQFKD JQEBJQF@I BUMOBPPFLK, BDFK V PFJMIFCVFKD BUMOBPPFLKP TFQEFK DOLRMFKD

PVJLIP.

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23/1147

EB KBUQ PQBM FP QL AAOBPP KV BUMLKBKQP LO OAF@IP. CQBOTOA, MBOCLOJ JRIQFMIF@QFLK KA AFSFPFLK COLJ IBCQ QL OFDEQ KA

CFKIIV AAFQFLK KA PRQO@QFLK COLJ IBCQ QL OFDEQ.

BQdP QHB ILLH Q QEB BUMOBPPFLK MOLSFABA.

2 4 + 6 ⋅ 23

− 42

EBOB OB KL DOLRMFKD PVJLIP, PL TB JLSB LK QL BUMLKBKQP LO OAF@IP. EB KRJBO 4 FP OFPBA QL MLTBO LC 2, PL PFJMIFCV

42

P 16.

2 4 + 6 ⋅ 23

− 42

2 4 + 6 ⋅ 23

− 16

BUQ, MBOCLOJ JRIQFMIF@QFLK LO AFSFPFLK, IBCQ QL OFDEQ.

2 4 + 6 ⋅ 23

− 16

2 4 + 4 − 1 6

PQIV, MBOCLOJ AAFQFLK LO PRQO@QFLK, IBCQ QL OFDEQ.

2 4 + 4 − 1 6 28−16 12

EBOBCLOB, 2 4 + 6 ⋅ 23

− 42 = 12.

LO PLJB @LJMIF@QBA BUMOBPPFLKP, PBSBOI MPPBP QEOLRDE QEB LOABO LC LMBOQFLKP TFII B KBBABA. LO FKPQK@B, QEBOB JV

B OAF@I BUMOBPPFLK FKPFAB MOBKQEBPBP QEQ JRPQ B PFJMIFCFBA BCLOB QEB MOBKQEBPBP OB BSIRQBA. LIILTFKD QEB LOABO

LC LMBOQFLKP BKPROBP QEQ KVLKB PFJMIFCVFKD QEB PJB JQEBJQF@I BUMOBPPFLK TFII DBQ QEB PJB OBPRIQ.

& &;@

MBOQFLKP FK JQEBJQF@I BUMOBPPFLKP JRPQ B BSIRQBA FK PVPQBJQF@ LOABO, TEF@E @K B PFJMIFCFBA RPFKD QEB

@OLKVJ PEMDAS:P(OBKQEBPBP)

E(UMLKBKQP)

M(RIQFMIF@QFLK) KA D(FSFPFLK)

A(AAFQFLK) KA S(RQO@QFLK)

Given a mathematical expression, simplify it using the order of operations.

1. FJMIFCV KV BUMOBPPFLKP TFQEFK DOLRMFKD PVJLIP.

2. FJMIFCV KV BUMOBPPFLKP @LKQFKFKD BUMLKBKQP LO OAF@IP.

3. BOCLOJ KV JRIQFMIF@QFLK KA AFSFPFLK FK LOABO, COLJ IBCQ QL OFDEQ.

4. BOCLOJ KV AAFQFLK KA PRQO@QFLK FK LOABO, COLJ IBCQ QL OFDEQ.

E


24/1147

. (3 ⋅ 2)2 − 4(6 + 2)

. 52 − 47 − 11 − 2

@. 6 − |5 − 8| + 3(4 − 1)

A.

1 4 − 3 ⋅ 22 ⋅ 5 − 32

B. 7 ( 5 ⋅ 3 ) − 2⎡⎣( 6 − 3 ) − 42⎤⎦ + 1

>;@

.

( 3 ⋅ 2 )2 − 4(6 + 2) = (6)2 − 4(8) Simplify parentheses= 36 − 4(8) Simplify exponent= 36 − 32 Simplify multiplication= 4 Simplify subtraction

.527 − 11 − 2 =

52 − 47 − 9 Simplify grouping symbols (radical)

= 52 − 47 − 3 Simplify radical

= 2 5 − 47 − 3 Simplify exponent

= 217 − 3 Simplify subtraction in numerator

= 3 − 3 Simplify division= 0 Simplify subtraction

LQB QEQ FK QEB CFOPQ PQBM, QEB OAF@I FP QOBQBA P DOLRMFKD PVJLI, IFHB MOBKQEBPBP. IPL, FK QEB QEFOA

PQBM, QEB CO@QFLK O FP @LKPFABOBA DOLRMFKD PVJLI PL QEB KRJBOQLO FP @LKPFABOBA QL B DOLRMBA.

@.6 − |5 − 8| + 3(4 − 1) = 6 − |−3| + 3(3) Simplify inside grouping symbols

= 6 − 3 + 3(3) Simplify absolute value= 6 − 3 + 9 Simplify multiplication= 3 + 9 Simplify subtraction= 12 Simplify addition

A.

1 4 − 3 ⋅ 22 ⋅ 5 − 32

= 1 4 − 3 ⋅ 22 ⋅ 5 − 9

Simplify exponent

= 1 4 − 61 0 − 9 Simplify products

=81 Simplify diffe ences

= 8 Simplify quotient

K QEFP BUJMIB, QEB CO@QFLK O PBMOQBP QEB KRJBOQLO KA ABKLJFKQLO, TEF@E TB PFJMIFCV PBMOQBIV

RKQFI QEB IPQ PQBM.

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25/1147

1.6

B.

7(5 ⋅ 3) − 2⎡⎣(6 − 3) − 42⎤⎦ + 1 = 7(15) − 2

⎡⎣(3) − 4

2⎤⎦ + 1 Simplify inside parentheses

= 7(15) − 2(3 − 1 6) + 1 Simplify exponent= 7(15) − 2(−13) + 1 Subtract= 105 + 26 + 1 Multiply= 132 Add

PB QEB LOABO LC LMBOQFLKP QL BSIRQB B@E LC QEB CLIILTFKD BUMOBPPFLKP.

. 52 − 42 + 7(5 − 4)2

. 1 + 7 ⋅ 5 − 8 ⋅ 49 − 6

@. |1.8− 4.3| + 0.4 15 + 10

A. 12⎡⎣5 ⋅ 3

2 − 72⎤⎦ + 13 ⋅ 92

B. ⎡⎣(3 − 8)

2 − 4⎤⎦ − (3 − 8)

;@ ; > %?

LO PLJB @QFSFQFBP TB MBOCLOJ, QEB LOABO LC @BOQFK LMBOQFLKP ALBP KLQ JQQBO, RQ QEB LOABO LC LQEBO LMBOQFLKP ALBP. LO

BUJMIB, FQ ALBP KLQ JHB AFCCBOBK@B FC TB MRQ LK QEB OFDEQ PELB BCLOB QEB IBCQ LO SF@B-SBOP. LTBSBO, FQ ALBP JQQBO

TEBQEBO TB MRQ LK PELBP LO PL@HP CFOPQ. EB PJB QEFKD FP QORB CLO LMBOQFLKP FK JQEBJQF@P.

C''a$/ P+*+$,

EB commutative property of addition PQQBP QEQ KRJBOP JV B AABA FK KV LOABO TFQELRQ CCB@QFKD QEB PRJ.

a + b = b + a

B @K BQQBO PBB QEFP OBIQFLKPEFM TEBK RPFKD OBI KRJBOP.

(−2) + 7 = 5 and 7 + (−2) = 5

FJFIOIV, QEB commutative property of multiplication PQQBP QEQ KRJBOP JV B JRIQFMIFBA FK KV LOABO TFQELRQ

CCB@QFKD QEB MOLAR@Q.

a ⋅ b = b ⋅ a

DFK, @LKPFABO K BUJMIB TFQE OBI KRJBOP.

(−11) ⋅ (−4) = 44 and (−4) ⋅ (−11) = 44

Q FP FJMLOQKQ QL KLQB QEQ KBFQEBO PRQO@QFLK KLO AFSFPFLK FP @LJJRQQFSB. LO BUJMIB, 1 7 − 5 FP KLQ QEB PJBP 5−17. FJFIOIV, 2 0 ÷ 5 ≠ 5 ÷ 2 0 .

A,,c$a$/ P+*+$,

EB associative property of multiplication QBIIP RP QEQ FQ ALBP KLQ JQQBO ELT TB DOLRM KRJBOP TEBK JRIQFMIVFKD. B

@K JLSB QEB DOLRMFKD PVJLIP QL JHB QEB @I@RIQFLK BPFBO, KA QEB MOLAR@Q OBJFKP QEB PJB.

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26/1147

a(bc) = (ab)c

LKPFABO QEFP BUJMIB.

(3 ⋅ 4) ⋅ 5 = 60 and 3 ⋅ (4 ⋅ 5) = 60

EB associative property of addition QBIIP RP QEQ KRJBOP JV B DOLRMBA AFCCBOBKQIV TFQELRQ CCB@QFKD QEB PRJ.

a + (b + c) = (a + b) + c

EFP MOLMBOQV @K B BPMB@FIIV EBIMCRI TEBK ABIFKD TFQE KBDQFSB FKQBDBOP. LKPFABO QEFP BUJMIB.⎡⎣15 + (−9)⎤⎦ + 23 = 29 and 15 + ⎡⎣(−9) + 23⎤⎦ = 29

OB PRQO@QFLK KA AFSFPFLK PPL@FQFSB? BSFBT QEBPB BUJMIBP.

8 − (3 − 15) =? (8 − 3) − 15 64 ÷ (8 ÷ 4) =? ( 6 4 ÷ 8 ) ÷ 4

8 − ( − 12) = 5 − 15 64 ÷ 2 =? 8 ÷ 420 ≠ 20 − 10 32 ≠ 2

P TB @K PBB, KBFQEBO PRQO@QFLK KLO AFSFPFLK FP PPL@FQFSB.

D$,+$b$/ P+*+

EB distributive property PQQBP QEQ QEB MOLAR@Q LC C@QLO QFJBP PRJ FP QEB PRJ LC QEB C@QLO QFJBP B@E QBOJ FK QEB PRJ.

a ⋅ (b + c) = a ⋅ b + a ⋅ c

EFP MOLMBOQV @LJFKBP LQE AAFQFLK KA JRIQFMIF@QFLK (KA FP QEB LKIV MOLMBOQV QL AL PL). BQ RP @LKPFABO K BUJMIB.

LQB QEQ 4 FP LRQPFAB QEB DOLRMFKD PVJLIP, PL TB AFPQOFRQB QEB 4 V JRIQFMIVFKD FQ V 12, JRIQFMIVFKD FQ V b7, KA AAFKD

QEB MOLAR@QP.

L B JLOB MOB@FPB TEBK ABP@OFFKD QEFP MOLMBOQV, TB PV QEQ JRIQFMIF@QFLK AFPQOFRQBP LSBO AAFQFLK. EB OBSBOPB FP KLQ

QORB, P TB @K PBB FK QEFP BUJMIB.

6 + (3 ⋅ 5) =? ( 6 + 3 ) ⋅ ( 6 + 5 )

6 + (15) =? (9) ⋅ (11)21 ≠ 99

RIQFMIF@QFLK ALBP KLQ AFPQOFRQB LSBO PRQO@QFLK, KA AFSFPFLK AFPQOFRQBP LSBO KBFQEBO AAFQFLK KLO PRQO@QFLK.

PMB@FI @PB LC QEB AFPQOFRQFSB MOLMBOQV L@@ROP TEBK PRJ LC QBOJP FP PRQO@QBA.

a − b = a + (−b)

LO BUJMIB, @LKPFABO QEB AFCCBOBK@B 12 − (5 + 3). B @K OBTOFQB QEB AFCCBOBK@B LC QEB QTL QBOJP 12 KA (5 + 3) V

QROKFKD QEB PRQO@QFLK BUMOBPPFLK FKQL AAFQFLK LC QEB LMMLPFQB. L FKPQBA LC PRQO@QFKD (5 + 3), TB AA QEB LMMLPFQB.

12 + (−1) ⋅ (5 + 3)

LT, AFPQOFRQB −1 KA PFJMIFCV QEB OBPRIQ.

12 − (5 + 3) = 12 + (−1) ⋅ (5 + 3)= 12 + [(−1) ⋅ 5 + (−1) ⋅ 3]= 12 + (−8)= 4

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27/1147

EFP PBBJP IFHB ILQ LC QOLRIB CLO PFJMIB PRJ, RQ FQ FIIRPQOQBP MLTBOCRI OBPRIQ QEQ TFII B RPBCRI LK@B TB FKQOLAR@B

IDBOF@ QBOJP. L PRQO@Q PRJ LC QBOJP, @EKDB QEB PFDK LC B@E QBOJ KA AA QEB OBPRIQP. FQE QEFP FK JFKA, TB @K

OBTOFQB QEB IPQ BUJMIB.

12 − (5 + 3) = 12 + (−5 − 3)= 12 + (−8)= 4

Id$ P+*+$,

EB identity property of addition PQQBP QEQ QEBOB FP RKFNRB KRJBO, @IIBA QEB AAFQFSB FABKQFQV (0) QEQ, TEBK AABA QL

KRJBO, OBPRIQP FK QEB LOFDFKI KRJBO.

a + 0 = a

EB identity property of multiplication PQQBP QEQ QEBOB FP RKFNRB KRJBO, @IIBA QEB JRIQFMIF@QFSB FABKQFQV (1) QEQ,

TEBK JRIQFMIFBA V KRJBO, OBPRIQP FK QEB LOFDFKI KRJBO.

a ⋅ 1 = a

LO BUJMIB, TB ESB (−6) + 0 = − 6 KA 2 3 ⋅ 1 = 2 3 . EBOB OB KL BU@BMQFLKP CLO QEBPB MOLMBOQFBP; QEBV TLOH CLO BSBOV

OBI KRJBO, FK@IRAFKD 0 KA 1.

I/+, P+*+$,

EB inverse property of addition PQQBP QEQ, CLO BSBOV OBI KRJBO , QEBOB FP RKFNRB KRJBO, @IIBA QEB AAFQFSB FKSBOPB

(LO LMMLPFQB), ABKLQBAj, QEQ, TEBK AABA QL QEB LOFDFKI KRJBO, OBPRIQP FK QEB AAFQFSB FABKQFQV, 0.

a + (−a) = 0

LO BUJMIB, FC a = −8, QEB AAFQFSB FKSBOPB FP 8, PFK@B (−8) + 8 = 0 .

EB inverse property of multiplication ELIAP CLO II OBI KRJBOP BU@BMQ 0 B@RPB QEB OB@FMOL@I LC 0 FP KLQ ABCFKBA. EB

MOLMBOQV PQQBP QEQ, CLO BSBOV OBI KRJBO , QEBOB FP RKFNRB KRJBO, @IIBA QEB JRIQFMIF@QFSB FKSBOPB (LO OB@FMOL@I),

ABKLQBA 1a, QEQ, TEBK JRIQFMIFBA V QEB LOFDFKI KRJBO, OBPRIQP FK QEB JRIQFMIF@QFSB FABKQFQV, 1.

a ⋅ 1a = 1

LO BUJMIB, FC a = − 23

, QEB OB@FMOL@I, ABKLQBA 1a, FP −32

B@RPB

a ⋅ 1a = ⎛⎝−

23⎞⎠ ⋅

⎛⎝−

32⎞⎠ = 1

; > %?

EB CLIILTFKD MOLMBOQFBP ELIA CLO OBI KRJBOP , , KA .

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28/1147

A>= ;?;>=

C??;

Ka + b = b + a a ⋅ b = b ⋅ a

A;;K

a + (b + c) = (a + b) + c a(bc) = (ab)c

D;;;

Ka ⋅ (b + c) = a ⋅ b + a ⋅ c

@;K

K

+ =@ ; = =H ; = –a,

a + (−a) = 0

EH =>=I> ; =;,

=> 1a,

a ⋅ ⎛⎝1a⎞⎠ = 1

E


29/1147

1.7

.

(5 + 8) + ( −8) = 5 + [8 + (−8)] Associative property of addition= 5 + 0 Inverse property of addition= 5 Identity property of addition

@.

6 − ( 15 + 9) = 6 + [(−15) + (−9)] Distributive property= 6 + (−24) Simplify= −18 Simplify

A.

47 ⋅

⎛⎝23

⋅ 74⎞⎠ =

47 ⋅

⎛⎝74

⋅ 23⎞⎠ Commutative property of multiplication

= ⎛⎝47 ⋅

74⎞⎠ ⋅

23

Associative property of multiplication

= 1 ⋅ 23

Inverse property of multiplication

= 23

Identity property of multiplication

B.

100 ⋅ [0.75 + ( − 2.38)] = 100 ⋅ 0.75 + 100 ⋅ (−2.38) Distributive property= 75 + (−238) Simplify= −163 Simplify

PB QEB MOLMBOQFBP LC OBI KRJBOP QL OBTOFQB KA PFJMIFCV B@E BUMOBPPFLK. QQB TEF@E MOLMBOQFBP MMIV.

. ⎛⎝−

235⎞⎠ ⋅

⎡⎣11 ⋅

⎛⎝−

523

⎞⎠⎤⎦

.

5 ⋅ (6.2+ 0.4)@. 18 − (7−15)

A. 1718

+ ⋅ ⎡⎣49

+ ⎛⎝−1718

⎞⎠⎤⎦

B. 6 ⋅ (−3) + 6 ⋅ 3

E>;@ A>; EJ;@

L CO, QEB JQEBJQF@I BUMOBPPFLKP TB ESB PBBK ESB FKSLISBA OBI KRJBOP LKIV. K JQEBJQF@P, TB JV PBB

BUMOBPPFLKP PR@E P x + 5, 43

πr 3, LO 2m3 n2. K QEB BUMOBPPFLK x + 5, 5 FP @IIBA constant B@RPB FQ ALBP KLQ SOV

KA FP @IIBA variable B@RPB FQ ALBP. (K KJFKD QEB SOFIB, FDKLOB KV BUMLKBKQP LO OAF@IP @LKQFKFKD QEB SOFIB.)

K algebraic expression FP @LIIB@QFLK LC @LKPQKQP KA SOFIBP GLFKBA QLDBQEBO V QEB IDBOF@ LMBOQFLKP LC AAFQFLK,

PRQO@QFLK, JRIQFMIF@QFLK, KA AFSFPFLK.

B ESB IOBAV PBBK PLJB OBI KRJBO BUJMIBP LC BUMLKBKQFI KLQQFLK, PELOQEKA JBQELA LC TOFQFKD MOLAR@QP LC QEB

PJB C@QLO. EBK SOFIBP OB RPBA, QEB @LKPQKQP KA SOFIBP OB QOBQBA QEB PJB TV.

(−3)5 = (−3) ⋅ (−3) ⋅ (−3) ⋅ (−3) ⋅ (−3) x5 = x ⋅ x ⋅ x ⋅ x ⋅ x

(2 ⋅ 7)3 = (2 ⋅ 7) ⋅ (2 ⋅ 7) ⋅ (2 ⋅ 7) ( yz)3 = ( yz) ⋅ ( yz) ⋅ ( yz)

K B@E @PB, QEB BUMLKBKQ QBIIP RP ELT JKV C@QLOP LC QEB PB QL RPB, TEBQEBO QEB PB @LKPFPQP LC @LKPQKQP LO SOFIBP.

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30/1147

1.8

KV SOFIB FK K IDBOF@ BUMOBPPFLK JV QHB LK LO B PPFDKBA AFCCBOBKQ SIRBP. EBK QEQ EMMBKP, QEB SIRB LC QEB

IDBOF@ BUMOBPPFLK @EKDBP. L BSIRQB K IDBOF@ BUMOBPPFLK JBKP QL ABQBOJFKB QEB SIRB LC QEB BUMOBPPFLK CLO

DFSBK SIRB LC B@E SOFIB FK QEB BUMOBPPFLK. BMI@B B@E SOFIB FK QEB BUMOBPPFLK TFQE QEB DFSBK SIRB, QEBK PFJMIFCV

QEB OBPRIQFKD BUMOBPPFLK RPFKD QEB LOABO LC LMBOQFLKP. C QEB IDBOF@ BUMOBPPFLK @LKQFKP JLOB QEK LKB SOFIB, OBMI@B

B@E SOFIB TFQE FQP PPFDKBA SIRB KA PFJMIFCV QEB BUMOBPPFLK P BCLOB.

E


31/1147

1.9

A. x = −4

>;@

. RPQFQRQB 0 CLO x.

2 x − 7 = 2(0) − 7= 0 − 7= −7

. RPQFQRQB 1 CLO x.

2 x − 7 = 2(1) − 7= 2 − 7= −5

@. RPQFQRQB 12

CLO x.

2 x − 7 = 2⎛⎝12⎞⎠ − 7

= 1 − 7= −6

A. RPQFQRQB

−4

CLO

x.2 x − 7 = 2( − 4) − 7

= −8 − 7= −15

SIRQB QEB BUMOBPPFLK 1 1 − 3 y CLO B@E SIRB CLO .

. y = 2

. y = 0

@. y = 23

A. y = −5

E


32/1147

1.10

>;@

. RPQFQRQB −5 CLO x.

x + 5 = (−5) + 5= 0

. RPQFQRQB 10 CLO t .

t 2t − 1

= (10)2(10) − 1

= 102 0 − 1

= 1019

@. RPQFQRQB 5 CLO r .

43

πr 3 = 43

π (5)3

= 43

π (125)

= 5003 π

A. RPQFQRQB 11 CLO a KA b8 CLO b.

a + ab + b = (11) + (11)(−8) + (−8)= 11 − 88 − 8= −85

B. RPQFQRQB 2 CLO m KA 3 CLO n.

2m3 n2 = 2(2)3 (3)2

= 2(8)(9)

= 144= 12

SIRQB B@E BUMOBPPFLK CLO QEB DFSBK SIRBP.

. y + 3 y − 3

CLO y = 5

. 7 − 2t CLO t = −2

@. 13

πr 2 CLO r = 11

A. ⎛⎝ p

2 q⎞⎠3

CLO p = −2, q = 3

B. 4(m − n) − 5(n − m) CLO m = 2

3, n = 1

3

F?>

K equation FP JQEBJQF@I PQQBJBKQ FKAF@QFKD QEQ QTL BUMOBPPFLKP OB BNRI. EB BUMOBPPFLKP @K B KRJBOF@I LO

IDBOF@. EB BNRQFLK FP KLQ FKEBOBKQIV QORB LO CIPB, RQ LKIV MOLMLPFQFLK. EB SIRBP QEQ JHB QEB BNRQFLK QORB, QEB

PLIRQFLKP, OB CLRKA RPFKD QEB MOLMBOQFBP LC OBI KRJBOP KA LQEBO OBPRIQP. LO BUJMIB, QEB BNRQFLK 2 x + 1 = 7 EP QEBRKFNRB PLIRQFLK x = 3 B@RPB TEBK TB PRPQFQRQB 3 CLO x FK QEB BNRQFLK, TB LQFK QEB QORB PQQBJBKQ 2(3) + 1 = 7 .

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33/1147

formula FP K BNRQFLK BUMOBPPFKD OBIQFLKPEFM BQTBBK @LKPQKQ KA SOFIB NRKQFQFBP. BOV LCQBK, QEB BNRQFLK FP

JBKP LC CFKAFKD QEB SIRB LC LKB NRKQFQV (LCQBK PFKDIB SOFIB) FK QBOJP LC KLQEBO LO LQEBO NRKQFQFBP. KB LC QEB JLPQ

@LJJLK BUJMIBP FP QEB CLOJRI CLO CFKAFKD QEB OB A LC @FO@IB FK QBOJP LC QEB OAFRP r LC QEB @FO@IB: A = πr 2. LO KV

SIRB LC r , QEB OB A @K B CLRKA V BSIRQFKD QEB BUMOBPPFLK πr 2.

E


34/1147

1.11 MELQLDOME TFQE IBKDQE L KA TFAQE FP MI@BA FK JQQB LC TFAQE 8 @BKQFJBQBOP (@J). EB OB LC QEB

JQQB (FK PNROB @BKQFJBQBOP, LO @J2) FP CLRKA QL B A = ( L + 16)(W + 16) − L ⋅ W . BB F; 1.5. FKA QEB

OB LC JQQB CLO MELQLDOME TFQE IBKDQE 32 @J KA TFAQE 24 @J.

F; 1.5

;?>;K;@ A>; EJ;@

LJBQFJBP TB @K PFJMIFCV K IDBOF@ BUMOBPPFLK QL JHB FQ BPFBO QL BSIRQB LO QL RPB FK PLJB LQEBO TV. L AL PL, TB

RPB QEB MOLMBOQFBP LC OBI KRJBOP. B @K RPB QEB PJB MOLMBOQFBP FK CLOJRIP B@RPB QEBV @LKQFK IDBOF@ BUMOBPPFLKP.

E


35/1147

1.12

1.13

@.

4t − 4⎛⎝t −54

s⎞⎠ −

⎛⎝23

t + 2s⎞⎠ = 4t −54

s − 23t − 2s Distributive property

= 4t − 23

t − 54

s − 2s Commutative property of addition

= 10

3 t − 13

4 s Simplify

A.

mn − 5m + 3mn + n = 2mn + 3mn − 5m + n Commutative property of addition= 5mn − 5m + n Simplify

FJMIFCV B@E IDBOF@ BUMOBPPFLK.

. 23 y − 2⎛⎝

43 y + z⎞⎠

. 5t − 2 − 3t + 1

@. 4 p⎛⎝q − 1⎞⎠ + q⎛⎝1 − p⎞⎠

A. 9r − (s + 2r ) + (6 − s)

E


36/1147

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

1.1 EECE

>

P 2 K BUJMIB LC OQFLKI QBOJFKQFKD, OQFLKIOBMBQFKD, LO FOOQFLKI KRJBO? BII TEV FQ CFQP QEQ

@QBDLOV.

EQ FP QEB LOABO LC LMBOQFLKP? EQ @OLKVJ FP RPBA

QL ABP@OFB QEB LOABO LC LMBOQFLKP, KA TEQ ALBP FQ PQKA

CLO?

EQ AL QEB PPL@FQFSB OLMBOQFBP IILT RP QL AL TEBK

CLIILTFKD QEB LOABO LC LMBOQFLKP? UMIFK VLRO KPTBO.

%?;

LO QEB CLIILTFKD BUBO@FPBP, PFJMIFCV QEB DFSBK BUMOBPPFLK.

1 0 + 2 × (5 − 3)

6 ÷ 2 −

⎛

⎝8 1 ÷ 3

2⎞

⎠

18 + (6 − 8)3

−2 × ⎡⎣16 ÷ (8 − 4)2⎤⎦

2

4 − 6 + 2 × 7

3(5 − 8)

4 + 6 − 1 0 ÷ 2

12 ÷ (3 6 ÷ 9) + 6

(4+5)2 ÷ 3

3 − 1 2 × 2 + 1 9

2 + 8

×

7 ÷ 4

5 + ( 6 + 4 ) − 1 1

9 − 1 8 ÷ 32

14

×

3 ÷ 7 − 6

9− ( 3+ 11) × 2

6 + 2 × 2 − 1

6 4 ÷ ( 8 + 4

×

2)

9 + 4⎛⎝22⎞⎠

(1 2 ÷ 3 × 3)2

2 5 ÷ 52 − 7

(1 5 − 7) × (3 − 7)

2 × 4 − 9(−1)

42 − 25 × 15

12(3 − 1) ÷ 6

A>;

LO QEB CLIILTFKD BUBO@FPBP, PLISB CLO QEB SOFIB.

8( x + 3 ) = 6 4

4 y + 8 = 2 y

(11a + 3 ) − 1 8a = −4

4 z − 2 z(1 + 4) = 36

4 y(7 − 2)2 = −200

−(2 x)2 + 1 = − 3

8(2+4)−15b = b

2(11c − 4) = 36

4(3 − 1) x = 4

14⎛⎝8w − 4

2⎞⎠ = 0

LO QEB CLIILTFKD BUBO@FPBP, PFJMIFCV QEB BUMOBPPFLK.

4 x + x(1 3 − 7)

2 y − (4)2 y − 11

a

23(64) − 12a ÷ 6

8b − 4b(3)+1

5l ÷ 3l × (9 − 6)

7 z − 3 + z × 62

30 C? 1 @

+ >== ;; > ?://cnx.org/>==/>;11759/1.3


37/1147

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

4 × 3 + 1 8 x ÷ 9 − 1 2

9⎛⎝ y + 8⎞⎠ − 27

⎛⎝96

t − 4⎞⎠2

6 + 1 2b − 3 × 6b

18 y − 2⎛⎝1 + 7 y⎞⎠

⎛⎝49⎞⎠

2 × 27 x

8(3 − m) + 1(−8)

9 x + 4 x(2+3)−4(2 x + 3 x)

52 − 4(3 x)

>-> A>;;@

LO QEB CLIILTFKD BUBO@FPBP, @LKPFABO QEFP P@BKOFL: OBA

BOKP $40 JLTFKD ITKP. B PMBKAP $10 LK JM3P, MRQP EIC

LC TEQ FP IBCQ FK PSFKDP @@LRKQ, KA DBQP KLQEBO $5 CLO

TPEFKD EFP KBFDELOdP @O.

OFQB QEB BUMOBPPFLK QEQ OBMOBPBKQP QEB KRJBO LC

ALIIOP OBA HBBMP (KA ALBP KLQ MRQ FK EFP PSFKDP

@@LRKQ). BJBJBO QEB LOABO LC LMBOQFLKP.

LT JR@E JLKBV ALBP OBA HBBM?

LO QEB CLIILTFKD BUBO@FPBP, PLISB QEB DFSBK MOLIBJ.

@@LOAFKD QL QEB .. FKQ, QEB AFJBQBO LC NROQBO FP

0.955 FK@EBP. EB @FO@RJCBOBK@B LC QEB NROQBO TLRIA B

QEB AFJBQBO JRIQFMIFBA V π . P QEB @FO@RJCBOBK@B LC NROQBO TELIB KRJBO, OQFLKI KRJBO, LO K FOOQFLKI

KRJBO?

*BPPF@ KA EBO OLLJJQB, AOFK, ESB AB@FABA QL

PEOB @EKDB GO CLO GLFKQ BUMBKPBP. *BPPF@ MRQ EBO ILLPB

@EKDB FK QEB GO CFOPQ, KA QEBK AOFK MRQ EBO @EKDB FK

QEB GO. B HKLT QEQ FQ ALBP KLQ JQQBO FK TEF@E LOABO QEB

@EKDB TP AABA QL QEB GO. EQ MOLMBOQV LC AAFQFLK

ABP@OFBP QEFP C@Q?

LO QEB CLIILTFKD BUBO@FPBP, @LKPFABO QEFP P@BKOFL: EBOBFP JLRKA LC g MLRKAP LC DOSBI FK NROOV. EOLRDELRQ

QEB AV, 400 MLRKAP LC DOSBI FP AABA QL QEB JLRKA. TL

LOABOP LC 600 MLRKAP OB PLIA KA QEB DOSBI FP OBJLSBA

COLJ QEB JLRKA. Q QEB BKA LC QEB AV, QEB JLRKA EP 1,200

MLRKAP LC DOSBI.

OFQB QEB BNRQFLK QEQ ABP@OFBP QEB PFQRQFLK.

LISB CLO .

LO QEB CLIILTFKD BUBO@FPB, PLISB QEB DFSBK MOLIBJ.

JLK ORKP QEB JOHBQFKD ABMOQJBKQ Q EFP @LJMKV.

FP ABMOQJBKQ DBQP RADBQ BSBOV VBO, KA BSBOV VBO, EB

JRPQ PMBKA QEB BKQFOB RADBQ TFQELRQ DLFKD LSBO. C EB

PMBKAP IBPP QEK QEB RADBQ, QEBK EFP ABMOQJBKQ DBQP

PJIIBO RADBQ QEB CLIILTFKD VBO. Q QEB BDFKKFKD LC QEFP

VBO, JLK DLQ $2.5 JFIIFLK CLO QEB KKRI JOHBQFKD

RADBQ. B JRPQ PMBKA QEB RADBQ PR@E QEQ 2,500,000 − x = 0. EQ MOLMBOQV LC AAFQFLK QBIIP RP

TEQ QEB SIRB LC JRPQ B?

@>K

LO QEB CLIILTFKD BUBO@FPBP, RPB DOMEFKD @I@RIQLO QL

PLISB CLO . LRKA QEB KPTBOP QL QEB KBOBPQ ERKAOBAQE.

0.5(12.3)2 − 48 x = 35

(0.25 − 0.75)2 x − 7.2 = 9.9

EJ@;@

C TELIB KRJBO FP KLQ KQROI KRJBO, TEQ JRPQ

QEB KRJBO B?

BQBOJFKB TEBQEBO QEB PQQBJBKQ FP QORB LO CIPB: EB

JRIQFMIF@QFSB FKSBOPB LC OQFLKI KRJBO FP IPL OQFLKI.

BQBOJFKB TEBQEBO QEB PQQBJBKQ FP QORB LO CIPB: EB

MOLAR@Q LC OQFLKI KA FOOQFLKI KRJBO FP ITVP

FOOQFLKI.

BQBOJFKB TEBQEBO QEB PFJMIFCFBA BUMOBPPFLK FP

OQFLKI LO FOOQFLKI: −18−4(5)(−1).

BQBOJFKB TEBQEBO QEB PFJMIFCFBA BUMOBPPFLK FP

OQFLKI LO FOOQFLKI: −16+4(5) + 5.

EB AFSFPFLK LC QTL TELIB KRJBOP TFII ITVP OBPRIQ

FK TEQ QVMB LC KRJBO?

EQ MOLMBOQV LC OBI KRJBOP TLRIA PFJMIFCV QEB

CLIILTFKD BUMOBPPFLK: 4 + 7 ( x −1)?

C? 1 @ 31


38/1147

1.2 M EJ@@ @ ;@;; %;@

@;@ &= = ;;:

1.2.1 ?> ; > ?>==.1.2.2 @>= ; > ?>==.

1.2.3 ?> ; > ?>==.

1.2.4 I> ?>== ; > ?>==.

1.2.5 = ; > ?>==.

1.2.6 F= ?> > ?> = @>=.

1.2.7 *==; ?>=.

1.2.8 = =>>=.

QEBJQF@FKP, P@FBKQFPQP, KA B@LKLJFPQP @LJJLKIV BK@LRKQBO SBOV IODB KA SBOV PJII KRJBOP. RQ FQ JV KLQ B

LSFLRP ELT @LJJLK PR@E CFDROBP OB FK BSBOVAV IFCB. LO FKPQK@B, MFUBI FP QEB PJIIBPQ RKFQ LC IFDEQ QEQ @K B

MBO@BFSBA KA OB@LOABA V AFDFQI @JBO. MOQF@RIO @JBO JFDEQ OB@LOA K FJDB QEQ FP 2,048 MFUBIP V 1,536 MFUBIP,

TEF@E FP SBOV EFDE OBPLIRQFLK MF@QROB. Q @K IPL MBO@BFSB @LILO ABMQE (DOAQFLKP FK @LILOP) LC RM QL 48 FQP MBO COJB,KA @K PELLQ QEB BNRFSIBKQ LC 24 COJBP MBO PB@LKA. EB JUFJRJ MLPPFIB KRJBO LC FQP LC FKCLOJQFLK RPBA QL CFIJ

LKB-ELRO (3,600-PB@LKA) AFDFQI CFIJ FP QEBK K BUQOBJBIV IODB KRJBO.

PFKD @I@RIQLO, TB BKQBO 2,048 × 1,536 × 48 × 24 × 3,600 KA MOBPP . EB @I@RIQLO AFPMIVP

1.30459631613. EQ ALBP QEFP JBK? EB e13f MLOQFLK LC QEB OBPRIQ OBMOBPBKQP QEB BUMLKBKQ 13 LC QBK, PL QEBOB OB

JUFJRJ LC MMOLUFJQBIV 1.3 × 1013 FQP LC AQ FK QEQ LKB-ELRO CFIJ. K QEFP PB@QFLK, TB OBSFBT ORIBP LC BUMLKBKQP CFOPQKA QEBK MMIV QEBJ QL @I@RIQFLKP FKSLISFKD SBOV IODB LO PJII KRJBOP.

;@ > EJ@@

LKPFABO QEB MOLAR@Q x3 ⋅ x4. LQE QBOJP ESB QEB PJB PB, , RQ QEBV OB OFPBA QL AFCCBOBKQ BUMLKBKQP. UMKA B@EBUMOBPPFLK, KA QEBK OBTOFQB QEB OBPRIQFKD BUMOBPPFLK.

x3 ⋅ x4 = x ⋅ x ⋅ x3 factors

⋅ x ⋅ x ⋅ x ⋅ x4 factors

= x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x7 factors

= x7

EB OBPRIQ FP QEQ x3 ⋅ x4 = x3 + 4 = x7.

LQF@B QEQ QEB BUMLKBKQ LC QEB MOLAR@Q FP QEB PRJ LC QEB BUMLKBKQP LC QEB QBOJP. K LQEBO TLOAP, TEBK JRIQFMIVFKD

BUMLKBKQFI BUMOBPPFLKP TFQE QEB PJB PB, TB TOFQB QEB OBPRIQ TFQE QEB @LJJLK PB KA AA QEB BUMLKBKQP. EFP FP QEB

;/@? @ ;99?.

am ⋅ an = am + n

LT @LKPFABO K BUJMIB TFQE OBI KRJBOP.

23

⋅ 24

= 23 + 4

= 27

B @K ITVP @EB@H QEQ QEFP FP QORB V PFJMIFCVFKD B@E BUMLKBKQFI BUMOBPPFLK. B CFKA QEQ 23 FP 8, 24 FP 16, KA 27 FP 128. EB MOLAR@Q 8 ⋅ 1 6 BNRIP 128, PL QEB OBIQFLKPEFM FP QORB. B @K RPB QEB MOLAR@Q ORIB LC BUMLKBKQP QL PFJMIFCVBUMOBPPFLKP QEQ OB MOLAR@Q LC QTL KRJBOP LO BUMOBPPFLKP TFQE QEB PJB PB RQ AFCCBOBKQ BUMLKBKQP.

> EJ@@

LO KV OBI KRJBO a KA KQROI KRJBOP m KA n, QEB MOLAR@Q ORIB LC BUMLKBKQP PQQBP QEQ

32 C? 1 @

+ >== ;; > ?://cnx.org/>==/>;11759/1.3


39/1147

1.14

(1.1)am ⋅ an = am + n

E


40/1147

y9

y5= y ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y y ⋅ y ⋅ y ⋅ y ⋅ y

= y ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y ⋅ y

y ⋅ y ⋅ y ⋅ y ⋅ y

= y ⋅ y ⋅ y ⋅ y1

= y4

LQF@B QEQ QEB BUMLKBKQ LC QEB NRLQFBKQ FP QEB AFCCBOBK@B BQTBBK QEB BUMLKBKQP LC QEB AFSFPLO KA AFSFABKA.

am

an= am − n

K LQEBO TLOAP, TEBK AFSFAFKD BUMLKBKQFI BUMOBPPFLKP TFQE QEB PJB PB, TB TOFQB QEB OBPRIQ TFQE QEB @LJJLK PB KA

PRQO@Q QEB BUMLKBKQP.

y9

y5= y9 − 5 = y4

LO QEB QFJB BFKD, TB JRPQ B TOB LC QEB @LKAFQFLK m > n. QEBOTFPB, QEB AFCCBOBK@B m − n @LRIA B WBOL LO KBDQFSB.ELPB MLPPFFIFQFBP TFII B BUMILOBA PELOQIV. IPL, FKPQBA LC NRIFCVFKD SOFIBP P KLKWBOL B@E QFJB, TB TFII PFJMIFCV

JQQBOP KA PPRJB COLJ EBOB LK QEQ II SOFIBP OBMOBPBKQ KLKWBOL OBI KRJBOP.

;@ > EJ@@

LO KV OBI KRJBO a KA KQROI KRJBOP m KA n, PR@E QEQ m > n, QEB NRLQFBKQ ORIB LC BUMLKBKQP PQQBP QEQ

(1.2)am

an= am − n

E


41/1147

1.15

@.⎛⎝ z 2

⎞⎠5

z 2= ⎛⎝ z 2

⎞⎠5 − 1 = ⎛⎝ z 2

⎞⎠4

OFQB B@E LC QEB CLIILTFKD MOLAR@QP TFQE PFKDIB PB. L KLQ PFJMIFCV CROQEBO.

. s75

s68

.(−3)6

−3

@.

⎛⎝e f

2⎞⎠

5

⎛⎝e f

2⎞⎠

3

;@ I > EJ@@

RMMLPB K BUMLKBKQFI BUMOBPPFLK FP OFPBA QL PLJB MLTBO. K TB PFJMIFCV QEB OBPRIQ? BP. L AL QEFP, TB RPB QEB ;

@ ;99?. LKPFABO QEB BUMOBPPFLK ⎛⎝ x

2⎞⎠

3. EB BUMOBPPFLK FKPFAB QEB MOBKQEBPBP FP JRIQFMIFBA QTF@B B@RPB FQ EP

K BUMLKBKQ LC 2. EBK QEB OBPRIQ FP JRIQFMIFBA QEOBB QFJBP B@RPB QEB BKQFOB BUMOBPPFLK EP K BUMLKBKQ LC 3.

⎛⎝ x

2⎞⎠

3= ⎛⎝ x

2⎞⎠ ⋅

⎛⎝ x

2⎞⎠ ⋅

⎛⎝ x

2⎞⎠

3 factors

=⎛

⎝⎜ x ⋅ x

⎧ ⎩ ⎨ 2 factors⎞

⎠⎟ ⋅

⎛

⎝⎜ x ⋅ x


⎠⎟ ⋅

⎛

⎝⎜ x ⋅ x


⎠⎟

3 factors

= x

⋅ x

⋅ x

⋅ x

⋅ x

⋅ x

= x6

EB BUMLKBKQ LC QEB KPTBO FP QEB MOLAR@Q LC QEB BUMLKBKQP: ⎛⎝ x

2⎞⎠

3= x2 ⋅ 3 = x6. K LQEBO TLOAP, TEBK OFPFKD K

BUMLKBKQFI BUMOBPPFLK QL MLTBO, TB TOFQB QEB OBPRIQ TFQE QEB @LJJLK PB KA QEB MOLAR@Q LC QEB BUMLKBKQP.

(am)n = am ⋅ n

B @OBCRI QL AFPQFKDRFPE BQTBBK RPBP LC QEB MOLAR@Q ORIB KA QEB MLTBO ORIB. EBK RPFKD QEB MOLAR@Q ORIB, AFCCBOBKQ QBOJP

TFQE QEB PJB PBP OB OFPBA QL BUMLKBKQP. K QEFP @PB, VLR AA QEB BUMLKBKQP. EBK RPFKD QEB MLTBO ORIB, QBOJ FK

BUMLKBKQFI KLQQFLK FP OFPBA QL MLTBO. K QEFP @PB, VLR JRIQFMIV QEB BUMLKBKQP.

Product Rule Power Rule

53 ⋅ 54 = 53 + 4 = 57 but (53)4 = 53 ⋅ 4 = 512

x5 ⋅ x2 = x5 + 2 = x7 but ( x5)2 = x5 ⋅ 2 = x10

(3a)7 ⋅ (3a)10 = (3a)7 + 1 0 = (3a)17 but ((3a)7)10 = (3a)7 ⋅ 1 0 = (3a)70

I > EJ@@

LO KV OBI KRJBO a KA MLPFQFSB FKQBDBOP m KA n, QEB MLTBO ORIB LC BUMLKBKQP PQQBP QEQ

(1.3)(am)n = am ⋅ n

C? 1 @ 35


42/1147

1.16

E


43/1147

EB PLIB BU@BMQFLK FP QEB BUMOBPPFLK 00. EFP MMBOP IQBO FK JLOB ASK@BA @LROPBP, RQ CLO KLT, TB TFII @LKPFABO QEBSIRB QL B RKABCFKBA.

EJ@@ > EJ@@

LO KV KLKWBOL OBI KRJBO a, QEB WBOL BUMLKBKQ ORIB LC BUMLKBKQP PQQBP QEQ

(1.4)a

0

= 1

E


44/1147

1.17

@.

⎛⎝ j

2 k ⎞⎠

4

⎛⎝ j

2 k ⎞⎠ ⋅

⎛⎝ j

2 k ⎞⎠

3 =⎛⎝ j

2 k ⎞⎠

4

⎛⎝ j

2 k ⎞⎠1 + 3 Use the product rule in the denominator.

=⎛⎝ j2 k ⎞⎠

4

⎛⎝ j

2 k ⎞⎠

4 Simplify.

= ⎛⎝ j2 k

⎞⎠

4 − 4Use the quotient rule.

= ⎛⎝ j2 k ⎞⎠

0Simplify.

= 1

A.

5⎛⎝rs2⎞⎠

2

⎛⎝rs

2⎞⎠

2 = 5⎛⎝rs

2⎞⎠

2 − 2Use the quotient rule.

= 5⎛⎝rs2⎞⎠

0Simplify.

= 5 ⋅ 1 Use the zero exponent rule.= 5 Simplify.

FJMIFCV B@E BUMOBPPFLK RPFKD QEB WBOL BUMLKBKQ ORIB LC BUMLKBKQP.

. t 7

t 7

.

⎛⎝de

2⎞⎠11

2⎛⎝de2⎞⎠

11

@. w4 ⋅ w2

w6

A. t 3 ⋅ t 4

t 2 ⋅ t 5

;@ %; > EJ@@

KLQEBO RPBCRI OBPRIQ L@@ROP FC TB OBIU QEB @LKAFQFLK QEQ

m > n

FK QEB NRLQFBKQ ORIB BSBK CROQEBO. LO BUJMIB, @K TB

PFJMIFCV h3

h5? EBK m == ;; > ?://cnx.org/>==/>;11759/1.3


45/1147

h3

h5= h ⋅ h ⋅ h

h ⋅ h ⋅ h ⋅ h ⋅ h

= h ⋅ h ⋅ hh ⋅ h ⋅ h ⋅ h ⋅ h

= 1h ⋅ h

= 1

h

2

C TB TBOB QL PFJMIFCV QEB LOFDFKI BUMOBPPFLK RPFKD QEB NRLQFBKQ ORIB, TB TLRIA ESB

h3

h5= h3 − 5

= h−2

RQQFKD QEB KPTBOP QLDBQEBO, TB ESB h−2 = 1h2

. EFP FP QORB CLO KV KLKWBOL OBI KRJBO, LO KV SOFIB OBMOBPBKQFKD

KLKWBOL OBI KRJBO.

C@QLO TFQE KBDQFSB BUMLKBKQ B@LJBP QEB PJB C@QLO TFQE MLPFQFSB BUMLKBKQ FC FQ FP JLSBA @OLPP QEB CO@QFLK

OcCOLJ KRJBOQLO QL ABKLJFKQLO LO SF@B SBOP.

a−n

= 1an and an

= 1a−n

B ESB PELTK QEQ QEB BUMLKBKQFI BUMOBPPFLK an FP ABCFKBA TEBK n FP KQROI KRJBO, 0, LO QEB KBDQFSB LC KQROI

KRJBO. EQ JBKP QEQ an FP ABCFKBA CLO KV FKQBDBO n. IPL, QEB MOLAR@Q KA NRLQFBKQ ORIBP KA II LC QEB ORIBP TB TFIIILLH Q PLLK ELIA CLO KV FKQBDBO n.

%; > EJ@@

LO KV KLKWBOL OBI KRJBO a KA KQROI KRJBO n, QEB KBDQFSB ORIB LC BUMLKBKQP PQQBP QEQ

(1.5)a−n = 1an

E


46/1147

1.18

. θ 3

θ 10= θ 3 − 1 0 = θ −7 = 1

θ 7

. z2 ⋅ z z4

= z2 + 1

z4= z

3

z4= z3 − 4 = z−1 = 1 z

@.

⎛⎝−5t

3⎞⎠4

⎛⎝−5t

3⎞⎠

8 = ⎛⎝−5t

3⎞⎠

4 − 8= ⎛⎝−5t

3⎞⎠

−4= 1

⎛⎝−5t

3⎞⎠

4

OFQB B@E LC QEB CLIILTFKD NRLQFBKQP TFQE PFKDIB PB. L KLQ PFJMIFCV CROQEBO. OFQB KPTBOP TFQE

MLPFQFSB BUMLKBKQP.

.(−3t )2

(−3t )8

. f 47

f 49 ⋅ f

@. 2k 4

5k 7

E


47/1147

1.19 OFQB B@E LC QEB CLIILTFKD MOLAR@QP TFQE PFKDIB PB. L KLQ PFJMIFCV CROQEBO. OFQB KPTBOP TFQE

MLPFQFSB BUMLKBKQP.

. t −11 ⋅ t 6

. 2512

2513

F;@;@ I

L PFJMIFCV QEB MLTBO LC MOLAR@Q LC QTL BUMLKBKQFI BUMOBPPFLKP, TB @K RPB QEB ; ;/@? @ ;99?,

TEF@E OBHP RM QEB MLTBO LC MOLAR@Q LC C@QLOP FKQL QEB MOLAR@Q LC QEB MLTBOP LC QEB C@QLOP. LO FKPQK@B, @LKPFABO

( pq)3. B BDFK V RPFKD QEB PPL@FQFSB KA @LJJRQQFSB MOLMBOQFBP LC JRIQFMIF@QFLK QL OBDOLRM QEB C@QLOP.

( pq)3 = ( pq) ⋅ ( pq) ⋅ ( pq)3 factors

= p ⋅ q ⋅ p ⋅ q ⋅ p ⋅ q

= p ⋅ p ⋅ p3 factors

⋅ q ⋅ q ⋅ q3 factors

= p3 ⋅ q3

K LQEBO TLOAP, ( pq)3 = p 3 ⋅ q3.

I > EJ@@

LO KV OBI KRJBOP a KA b KA KV FKQBDBO n, QEB MLTBO LC MOLAR@Q ORIB LC BUMLKBKQP PQQBP QEQ

(1.6)(ab)n = an bn

E


48/1147

1.20

. ⎛⎝ab

2⎞⎠

3= (a)3 ⋅ ⎛⎝b

2⎞⎠

3= a1 ⋅ 3 ⋅ b2 ⋅ 3 = a3 b6

. 2t 15 = (2)15 ⋅ (t )15 = 215 t 15 = 32, 768t 15

@. ⎛

⎝−2w3⎞

⎠

3= (−2)3 ⋅ ⎛

⎝w3⎞

⎠

3= − 8 ⋅ w3 ⋅ 3 = −8w9

A. 1(−7 z)4

= 1(−7)4 ⋅ ( z)4

= 12, 401 z4

B. ⎛⎝e

−2 f 2⎞⎠

7= ⎛⎝e

−2⎞⎠

7⋅ ⎛⎝ f

2⎞⎠

7= e− 2 ⋅ 7 ⋅ f 2 ⋅ 7 = e−14 f 14 = f

14

e14

FJMIFCV B@E LC QEB CLIILTFKD MOLAR@QP P JR@E P MLPPFIB RPFKD QEB MLTBO LC MOLAR@Q ORIB. OFQB

KPTBOP TFQE MLPFQFSB BUMLKBKQP.

. ⎛⎝g

2 h3⎞⎠5

. (5t )3

@. ⎛⎝−3 y

5⎞⎠

3

A. 1⎛⎝a

6 b7⎞⎠

3

B. ⎛⎝r

3 s−2⎞⎠4

F;@;@ I ;@

L PFJMIFCV QEB MLTBO LC NRLQFBKQ LC QTL BUMOBPPFLKP, TB @K RPB QEB ; == ;; > ?://cnx.org/>==/>;11759/1.3


49/1147

⎛⎝e

−2 f 2⎞⎠

7=

⎛

⎝⎜ f 2

e2

⎞

⎠⎟

7

= ( f 2)7

(e2)7

= f 2 ⋅ 7

e2 ⋅ 7

= f 14

e14

I ;@ > EJ@@

LO KV OBI KRJBOP a KA b KA KV FKQBDBO n, QEB MLTBO LC NRLQFBKQ ORIB LC BUMLKBKQP PQQBP QEQ

(1.7)⎛⎝ab⎞⎠

n

= an

bn

E


50/1147

1.21

@. ⎛⎝

−1t 2⎞⎠

27

= (−1)27

⎛⎝t

2⎞⎠

27 =−1

t 2 ⋅ 2 7= −1

t 54= − 1

t 54

A.

⎛

⎝ j

3

k

−2⎞⎠

4

=

⎛

⎝⎜

j3

k 2

⎞

⎠⎟

4

=

⎛⎝ j

3⎞⎠

4

⎛⎝k

2⎞⎠4 =

j3 ⋅ 4

k 2 ⋅ 4 =

j12

k 8

B. ⎛⎝m

−2 n−2⎞⎠

3= ⎛⎝

1m2 n2

⎞⎠

3

= (1)3

⎛⎝m

2 n2⎞⎠3 =

1⎛⎝m

2⎞⎠

3⎛⎝n

2⎞⎠

3 =1

m2 ⋅ 3 ⋅ n2 ⋅ 3= 1

m6 n6

FJMIFCV B@E LC QEB CLIILTFKD NRLQFBKQP P JR@E P MLPPFIB RPFKD QEB MLTBO LC NRLQFBKQ ORIB. OFQB

KPTBOP TFQE MLPFQFSB BUMLKBKQP.

. ⎛⎝b5c ⎞⎠

3

. ⎛⎝

5u8⎞⎠

4

@. ⎛⎝

−1w3

⎞⎠

35

A. ⎛⎝ p

−4 q3⎞⎠8

B. ⎛

⎝c−5 d −3

⎞

⎠

4

;?>;K;@ EJ@@;> EJ;@

B@II QEQ QL PFJMIFCV K BUMOBPPFLK JBKP QL OBTOFQB FQ V @LJFKD QBOJP LO BUMLKBKQP; FK LQEBO TLOAP, QL TOFQB QEB

BUMOBPPFLK JLOB PFJMIV TFQE CBTBO QBOJP. EB ORIBP CLO BUMLKBKQP JV B @LJFKBA QL PFJMIFCV BUMOBPPFLKP.

E


51/1147

A. ⎛⎝−2a

3 b−1⎞⎠⎛⎝5a

−2 b2⎞⎠

B. ⎛⎝ x

2 2⎞⎠4⎛⎝ x

2 2⎞⎠−4

C.

⎛⎝3w

2⎞⎠

5

⎛⎝6w−2⎞⎠2

>;@

.

⎛⎝6m

2 n−1⎞⎠

3= (6)3 ⎛⎝m

2⎞⎠

3⎛⎝n

−1⎞⎠

3The power of a product rule

= 63 m2 ⋅ 3 n− 1 ⋅ 3 The power rule

= 216m6 n−3 Simplify.

= 216m6

n3The negative exponent rule

.175 ⋅ 17−4 ⋅ 17−3 = 175 − 4 − 3 The product rule

= 17−2 Simplify.

= 1172

or 1289 The negative exponent rule

@.

⎛⎝

u−1 v

v−1⎞⎠

2

= (u−1 v)2

(v−1)2The power of a quotient rule

= u−2 v2

v−2The power of a product rule

= u−2

v

2−(−2)

The quotient rule= u−2 v4 Simplify.

= v4

u2The negative exponent rule

A.⎛⎝−2a

3 b−1⎞⎠⎛⎝5a

−2 b2⎞⎠ = −2 ⋅ 5 ⋅ a3 ⋅ a−2 ⋅ b−1 ⋅ b2 Commutative and associative laws of multiplication

= −10 ⋅ a3 − 2 ⋅ b− 1 + 2 The product rule= −10ab Simplify.

B.

⎛⎝ x

2 2⎞⎠4⎛⎝ x

2 2⎞⎠−4

= ⎛⎝ x2 2⎞⎠

4 − 4The product rule

= ⎛⎝ x2 2⎞⎠

0Simplify.

= 1 The zero exponent rule

C? 1 @ 45


52/1147

1.22

C.

(3w2)5

(6w−2)2= (3)

5 ⋅ (w2)5

(6)2 ⋅ (w−2)2The power of a product rule

= 35 w2 ⋅ 5

62 w− 2 ⋅ 2The power rule

= 243w1036w−4

Simplify.

= 27w10−(−4)

4The quotient rule and reduce fraction

= 27w14

4Simplify.

FJMIFCV B@E BUMOBPPFLK KA TOFQB QEB KPTBO TFQE MLPFQFSB BUMLKBKQP LKIV.

.

⎛

⎝2uv−2⎞

⎠

−3

. x8 ⋅ x−12 ⋅ x

@.⎛

⎝⎜e2 f −3

f −1

⎞

⎠⎟

2

A. ⎛⎝9r

−5 s3⎞⎠⎛⎝3r

6 s−4⎞⎠

B. ⎛⎝49

tw−2⎞⎠

−3⎛⎝49

tw−2⎞⎠

3

C.

⎛

⎝2h2

k

⎞

⎠

4

⎛⎝7h

−1 k 2⎞⎠2

;@ ;@;; %;@

B@II Q QEB BDFKKFKD LC QEB PB@QFLK QEQ TB CLRKA QEB KRJBO 1.3 × 1013 TEBK ABP@OFFKD FQP LC FKCLOJQFLK FK AFDFQIFJDBP. QEBO BUQOBJB KRJBOP FK@IRAB QEB TFAQE LC ERJK EFO, TEF@E FP LRQ 0.00005 J, KA QEB OAFRP LC K BIB@QOLK,

TEF@E FP LRQ 0.00000000000047 J. LT @K TB BCCB@QFSBIV TLOH OBA, @LJMOB, KA @I@RIQB TFQE KRJBOP PR@E P

QEBPB?

PELOQEKA JBQELA LC TOFQFKD SBOV PJII KA SBOV IODB KRJBOP FP @IIBA scientific notation, FK TEF@E TB BUMOBPP

KRJBOP FK QBOJP LC BUMLKBKQP LC 10. L TOFQB KRJBO FK P@FBKQFCF@ KLQQFLK, JLSB QEB AB@FJI MLFKQ QL QEB OFDEQ LC QEB

CFOPQ AFDFQ FK QEB KRJBO. OFQB QEB AFDFQP P AB@FJI KRJBO BQTBBK 1 KA 10. LRKQ QEB KRJBO LC MI@BP 9 QEQ VLRJLSBA QEB AB@FJI MLFKQ. RIQFMIV QEB AB@FJI KRJBO V 10 OFPBA QL MLTBO LC 9. C VLR JLSBA QEB AB@FJI IBCQ P FK

SBOV IODB KRJBO, n FP MLPFQFSB. C VLR JLSBA QEB AB@FJI OFDEQ P FK PJII IODB KRJBO, n FP KBDQFSB.

LO BUJMIB, @LKPFABO QEB KRJBO 2,780,418. LSB QEB AB@FJI IBCQ RKQFI FQ FP QL QEB OFDEQ LC QEB CFOPQ KLKWBOL AFDFQ, TEF@E

FP 2.

46 C? 1 @

+ >== ;; > ?://cnx.org/>==/>;11759/1.3


53/1147

B LQFK 2.780418 V JLSFKD QEB AB@FJI MLFKQ 6 MI@BP QL QEB IBCQ. EBOBCLOB, QEB BUMLKBKQ LC 10 FP 6, KA FQ FP MLPFQFSB

B@RPB TB JLSBA QEB AB@FJI MLFKQ QL QEB IBCQ. EFP FP TEQ TB PELRIA BUMB@Q CLO IODB KRJBO.

2.780418 × 106

LOHFKD TFQE PJII KRJBOP FP PFJFIO. HB, CLO BUJMIB, QEB OAFRP LC K BIB@QOLK, 0.00000000000047 J. BOCLOJ QEB

PJB PBOFBP LC PQBMP P LSB, BU@BMQ JLSB QEB AB@FJI MLFKQ QL QEB OFDEQ.

B @OBCRI KLQ QL FK@IRAB QEB IBAFKD 0 FK VLRO @LRKQ. B JLSB QEB AB@FJI MLFKQ 13 MI@BP QL QEB OFDEQ, PL QEB BUMLKBKQ LC

10 FP 13. EB BUMLKBKQ FP KBDQFSB B@RPB TB JLSBA QEB AB@FJI MLFKQ QL QEB OFDEQ. EFP FP TEQ TB PELRIA BUMB@Q CLO

PJII KRJBO.

4.7 × 10−13

;@;; %;@

KRJBO FP TOFQQBK FK scientific notation FC FQ FPTOFQQBKFK QEB CLOJ a × 10n, TEBOB 1 ≤ |a| < 10 KA n FP K FKQBDBO.

E


54/1147

1.23

A.

0.00000000000094 m0.00000000000094 m

→ 6 places

9.4 × 10−13 m

B.0.000001430.00000143

→ 6 places

1.43 × 10−6

A@>K;

PBOSB QEQ, FC QEB DFSBK KRJBO FP DOBQBO QEK 1, P FK BUJMIBP b@, QEB BUMLKBKQ LC 10 FP MLPFQFSB; KA FC QEB

KRJBO FP IBPP QEK 1, P FK BUJMIBP AbB, QEB BUMLKBKQ FP KBDQFSB.

OFQB B@E KRJBO FK P@FBKQFCF@ KLQQFLK.

. .. KQFLKI ABQ MBO QUMVBO (MOFI 2014): $152,000

. LOIA MLMRIQFLK (MOFI 2014): 7,158,000,000

@. LOIA DOLPP KQFLKI FK@LJB (MOFI 2014): $85,500,000,000,000

A. FJB CLO IFDEQ QL QOSBI 1 J: 0.00000000334 P

B. OLFIFQV LC TFKKFKD ILQQBOV (JQ@E 6 LC 49 MLPPFIB KRJBOP): 0.0000000715

C@;@ ? ;@;; @ %;@

L @LKSBOQ KRJBO FK scientific notation QL PQKAOA KLQQFLK, PFJMIV OBSBOPB QEB MOL@BPP. LSB QEB AB@FJI n MI@BP QL

QEB OFDEQ FC n FP MLPFQFSB LO n MI@BP QL QEB IBCQ FC n FP KBDQFSB KA AA WBOLP P KBBABA. BJBJBO, FC n FP MLPFQFSB, QEB

SIRB LC QEB KRJBO FP DOBQBO QEK 1, KA FC

n

FP KBDQFSB, QEB SIRB LC QEB KRJBO FP IBPP QEK LKB.

E


55/1147

1.24

.

3.547 × 1014

3.54700000000000→ 14 places

354,700,000,000,000

.

−2 × 106

−2.000000→ 6 places

−2,000,000

@.

7.91 × 10−7

0000007.91→ 7 places

0.000000791

A.

−8.05

×

10

−12

−000000000008.05→ 12 places

−0.00000000000805

LKSBOQ B@E KRJBO FK P@FBKQFCF@ KLQQFLK QL PQKAOA KLQQFLK.

. 7.03 × 105

. −8.16 × 1011

@. −3.9

×

10−13

A. 8 × 10−6

;@ ;@;; %;@ ;@ A>;;@

@FBKQFCF@ KLQQFLK, RPBA TFQE QEB ORIBP LC BUMLKBKQP, JHBP @I@RIQFKD TFQE IODB LO PJII KRJBOP JR@E BPFBO QEK ALFKD

PL RPFKD PQKAOA KLQQFLK. LO BUJMIB, PRMMLPB TB OB PHBA QL @I@RIQB QEB KRJBO LC QLJP FK 1 LC TQBO. @E

TQBO JLIB@RIB @LKQFKP 3 QLJP (2 EVAOLDBK KA 1 LUVDBK). EB SBODB AOLM LC TQBO @LKQFKP OLRKA 1.32 × 1021

JLIB@RIBP LC TQBO KA 1 LC TQBO ELIAP LRQ 1.22 × 104 SBODB AOLMP. EBOBCLOB, QEBOB OB MMOLUFJQBIV

3 ⋅ ⎛⎝1.32 × 1021⎞

⎠ ⋅ ⎛⎝1.22 × 10

4⎞⎠ ≈ 4.83 × 10

25 QLJP FK 1 LC TQBO. B PFJMIV JRIQFMIV QEB AB@FJI QBOJP KA AA QEB

BUMLKBKQP. JDFKB ESFKD QL MBOCLOJ QEB @I@RIQFLK TFQELRQ RPFKD P@FBKQFCF@ KLQQFLK!

EBK MBOCLOJFKD @I@RIQFLKP TFQE P@FBKQFCF@ KLQQFLK, B PROB QL TOFQB QEB KPTBO FK MOLMBO P@FBKQFCF@ KLQQFLK. LO

BUJMIB, @LKPFABO QEB MOLAR@Q ⎛⎝7 × 10

4⎞⎠ ⋅

⎛⎝5 × 10

6⎞⎠ = 35 × 10

10. EB KPTBO FP KLQ FK MOLMBO P@FBKQFCF@ KLQQFLK B@RPB

35 FP DOBQBO QEK 10. LKPFABO 35 P 3.5 × 10. EQ AAP QBK QL QEB BUMLKBKQ LC QEB KPTBO.

(35) × 1010 = (3.5 × 10) × 1010 = 3.5 × ⎛⎝10 × 1010⎞

⎠ = 3.5 × 1011

C? 1 @ 49


56/1147

E


57/1147

1.25

1.26

BOCLOJ QEB LMBOQFLKP KA TOFQB QEB KPTBO FK P@FBKQFCF@ KLQQFLK.

. ⎛⎝−7.5 × 10

8⎞⎠⎛⎝1.13 × 10

−2⎞⎠

. ⎛⎝1.24 × 10

11⎞⎠ ÷

⎛⎝1.55 × 10

18⎞⎠

@. ⎛⎝3.72 × 10

9⎞⎠⎛⎝8 × 10

3⎞⎠

A. ⎛⎝9.933 × 10

23⎞⎠ ÷

⎛⎝−2.31 × 10

17⎞⎠

B. ⎛⎝−6.04 × 10

9⎞⎠⎛⎝7.3 × 10

2⎞⎠⎛⎝−2.81 × 10

2⎞⎠

E


58/1147

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

79.

80.

81.

82.

83.

84.

85.

86.

87.

88.

89.

90.

91.

92.

93.

94.

95.

96.

97.

98.

99.

100.

101.

102.

103.

104.

1.2 EECE

>

P 23 QEB PJB P 32 ? UMIFK.

EBK @K VLR AA QTL BUMLKBKQP?

EQ FP QEB MROMLPB LC P@FBKQFCF@ KLQQFLK?

UMIFK TEQ KBDQFSB BUMLKBKQ ALBP.

%?;

LO QEB CLIILTFKD BUBO@FPBP, PFJMIFCV QEB DFSBK BUMOBPPFLK.

OFQB KPTBOP TFQE MLPFQFSB BUMLKBKQP.

92

15−2

32

×

33

44 ÷ 4

⎛⎝2

2⎞⎠

−2

(5−8)0

113 ÷ 114

65 × 6−7

⎛⎝8

0⎞⎠

2

5−2 ÷ 52

LO QEB CLIILTFKD BUBO@FPBP, TOFQB B@E BUMOBPPFLK TFQE

PFKDIB PB. L KLQ PFJMIFCV CROQEBO. OFQB KPTBOP TFQE

MLPFQFSB BUMLKBKQP.

42 × 43 ÷ 4−4

612

69

⎛⎝12

3 × 12⎞⎠

10

106 ÷ ⎛⎝1010⎞

⎠

−2

7−6 × 7−3

⎛⎝3

3 ÷ 34⎞⎠5

LO QEB CLIILTFKD BUBO@FPBP, BUMOBPP QEB AB@FJI FK

P@FBKQFCF@ KLQQFLK.

0.0000314

148,000,000

LO QEB CLIILTFKD BUBO@FPBP, @LKSBOQ B@E KRJBO FK

P@FBKQFCF@ KLQQFLK QL PQKAOA KLQQFLK.

1.6

×

1010

9.8 × 10−9

A>;

LO QEB CLIILTFKD BUBO@FPBP, PFJMIFCV QEB DFSBK BUMOBPPFLK.OFQB KPTBOP TFQE MLPFQFSB BUMLKBKQP.

a3 a2a

mn2

m−2

⎛⎝b

3 c4⎞⎠2

⎛

⎝

⎜ x−3

y2

⎞

⎠

⎟

−5

ab2 ÷ d −3

⎛⎝w

0 x5⎞⎠

−1

m4

n0

y−4 ⎛⎝ y2⎞⎠

2

p−4 q2

p2 q−3

(l × w)2

⎛⎝ y

7⎞⎠

3÷ x14

52 C? 1 @

+ >== ;; > ?://cnx.org/>==/>;11759/1.3


59/1147

105.

106.

107.

108.

109.

110.

111.

112.

113.

114.

115.

116.

117.

118.

119.

120.

121.

122.

123.

124.

125.

126.

127.

⎛⎝

a

23⎞⎠

2

52 m ÷ 50 m

(16 x)2

y

−1

23

(3a)−2

⎛⎝ma

6⎞⎠

2 1m3 a2

⎛⎝b

−3 c⎞⎠3

⎛⎝ x

2 y13 ÷ y0⎞⎠2

⎛⎝9 z

3⎞⎠

−2 y

>-> A>;;@

L OB@E BP@MB SBIL@FQV, OL@HBQ JRPQ QOSBI Q QEB

OQB LC 2.2 × 106 CQ/JFK. BTOFQB QEB OQB FK PQKAOAKLQQFLK.

AFJB FP QEB QEFKKBPQ @LFK FK .. @ROOBK@V. AFJBdP

QEF@HKBPP JBPROBP 2.2 × 106 J. BTOFQB QEB KRJBO FKPQKAOA KLQQFLK.

EB SBODB AFPQK@B BQTBBK OQE KA QEB RK FP

92,960,000 JF. BTOFQB QEB AFPQK@B RPFKD P@FBKQFCF@

KLQQFLK.

QBOVQB FP JAB LC MMOLUFJQBIV

1,099,500,000,000 VQBP. BTOFQB FK P@FBKQFCF@ KLQQFLK.

EB OLPP LJBPQF@ OLAR@Q () CLO QEB KFQBA

QQBP FK QEB CFOPQ NROQBO LC 2014 TP $1.71496 × 1013. BTOFQB QEB FK PQKAOA KLQQFLK.

KB MF@LJBQBO FP MMOLUFJQBIV 3.397 × 10−11 FK.BTOFQB QEFP IBKDQE RPFKD PQKAOA KLQQFLK.

EB SIRB LC QEB PBOSF@BP PB@QLO LC QEB .. B@LKLJV

FK QEB CFOPQ NROQBO LC 2012 TP $10,633.6 FIIFLK. BTOFQB

QEFP JLRKQ FK P@FBKQFCF@ KLQQFLK.

@>K

LO QEB CLIILTFKD BUBO@FPBP, RPB DOMEFKD @I@RIQLO QL

PFJMIFCV. LRKA QEB KPTBOP QL QEB KBOBPQ ERKAOBAQE.

⎛⎝

123 m33

4−3⎞⎠

2

173 ÷ 152 x3

EJ@;@

LO QEB CLIILTFKD BUBO@FPBP, PFJMIFCV QEB DFSBK BUMOBPPFLK.OFQB KPTBOP TFQE MLPFQFSB BUMLKBKQP.

⎛⎝

32

a3⎞⎠

−2⎛⎝

a4

22⎞⎠

2

⎛⎝6

2 −24⎞⎠2

÷ ⎛⎝ x y⎞⎠

−5

m2 n3

a2 c−3⋅ a

−7 n−2

m2 c4

⎛⎝⎜ x6 y3 x3 y−3

⋅ y−7 x−3

⎞⎠⎟

10

⎛

⎝⎜⎜⎛⎝ab

2 c⎞⎠−3

b−3

⎞

⎠⎟⎟

2

SLDAOLdP @LKPQKQ FP RPBA QL @I@RIQB QEB KRJBO

LC MOQF@IBP FK JLIB. JLIB FP PF@ RKFQ FK @EBJFPQOV QL

JBPROB QEB JLRKQ LC PRPQK@B. EB @LKPQKQ FP

6.0221413 × 1023. OFQB SLDAOLdP @LKPQKQ FKPQKAOA KLQQFLK.

IK@HdP @LKPQKQ FP K FJMLOQKQ RKFQ LC JBPROB FK

NRKQRJ MEVPF@P. Q ABP@OFBP QEB OBIQFLKPEFM BQTBBK

BKBODV KA COBNRBK@V. EB @LKPQKQ FP TOFQQBK P

6.62606957 × 10−34. OFQB IK@HdP @LKPQKQ FK PQKAOAKLQQFLK.

C? 1 @ 53


60/1147

1.3 M ;> @ ;@> EJ;@

@;@ &= = ;;:

1.3.1 E; @ >>.1.3.2 ?> ; >


61/1147

1.27

;@;>

EB principal square root LC a FP QEB KLKKBDQFSB KRJBO QEQ, TEBK JRIQFMIFBA V FQPBIC, BNRIP a. Q FP TOFQQBK P radical expression, TFQE PVJLI @IIBA radical LSBO QEB QBOJ @IIBA QEB radicand: a.

Does 25 = ± 5?

. A?@ ?

52

9/

(−5)2

25, ? / 8 8; 9 999? ?, ? ;9;


62/1147

1.28

;@ > ;?>;K

L PFJMIFCV PNROB OLLQ, TB OBTOFQB FQ PR@E QEQ QEBOB OB KL MBOCB@Q PNROBP FK QEB OAF@KA. EBOB OB PBSBOI MOLMBOQFBP LC

PNROB OLLQP QEQ IILT RP QL PFJMIFCV @LJMIF@QBA OAF@I BUMOBPPFLKP. EB CFOPQ ORIB TB TFII ILLH Q FP QEB ;/@? @

8;9 ;?>;K;@

C a KA b OB KLKKBDQFSB, QEB PNROB OLLQ LC QEB MOLAR@Q ab FP BNRI QL QEB MOLAR@Q LC QEB PNROB OLLQP LC a KA b.

ab = a ⋅ b

Given a square root radical expression, use the product rule to simplify it.

1. @QLO KV MBOCB@Q PNROBP COLJ QEB OAF@KA.

2. OFQB QEB OAF@I BUMOBPPFLK P MOLAR@Q LC OAF@I BUMOBPPFLKP.

3. FJMIFCV.

E


63/1147

1.29

E


64/1147

1.30

1.31

536

Write as quotient of two radical expressions.

56

Simplify denominator.

FJMIFCV 2 x2

9 y4.

E


65/1147

1.32

1.33

A;@

AA 5 12 + 2 3.

>;@

B @K OBTOFQB 5 12

P 5 4 · 3.

@@LOAFKD QEB MOLAR@Q ORIB, QEFP B@LJBP 5 4 3.

EB PNROB OLLQ LC 4

FP 2,

PL QEB BUMOBPPFLK B@LJBP 5(2) 3, TEF@E FP 10 3. LT TB @K QEB QBOJP ESB QEB PJB OAF@KA PL TB @K

AA.

1

collegealgebra lr

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