chapter 1 the real number system = 9 - (12 - [8]) = 9 - (4 ... · pdf file4 • 9 + 1...

Chapter 1 The Real Number System

Section 1.1 Introduction to Algebra

Practice 1.1.11. 25 + 4 is a sum whose terms are 25

and 4.

2. 5w is a product whose factors are 5 and w.

3. 6x + 2 is a sum whose terms are 6x and 2. 6x is a product whose factors are 6 and x.

4. 2m + 3n is a sum whose terms are2m and 3n. 2m is a product whose

factors are 2 and m. 3n is a product Whose terms are 3 and n.

5. 5abc is a product whose factors are 5,a,b,and c.

Practice 1.1.21. 8 + 12 ÷ 3 • 4 = 8 + 4 • 4 = 8 + 16 =

24

2. 5 2 − 2 2 = 25 - 4 = 21

3. (5 - 2)2 = (3)2 = 9

4. (4 -1)(12 - 7)= (3)(5) = 15

5. 24 ÷ (4)(16 - 10) = 24 ÷ (4)(6) = 6(6) = 36

6. 8 − 2 + 5•4

15 − 2 • 2 =

8 − 2 + 2015 − 4

= 8 − 2211

=

8 - 2 = 6

Practice 1.1.31. 14 - [(5 - 3)2 + 6] = 14 - [(2)2 + 6]= 14 - [4 + 6] = 14 - [10] = 4

2. 9 - (12 - [22 + 4]) = 9 - (12 - [4 + 4])

= 9 - (12 - [8]) = 9 - (4) = 5

3. 60 - 3(2 + 3[2 + 3]) = 60 - 3(2 + 3[5]) = 60 - 3(2 + 15) = 60 - 3(17) = 60 - 51 = 9

4. 15 - [(2 + 4) • 2 - (12 - 8)] =15 - [6 •2 - 4] = 15 - [12 - 4] = 15 - 8 = 7

Practice 1.1.41. For t = 7

4t + 3 = 4(7) + 3 = 28 + 3 = 312. For r = 3

π r2 = 3.14 • (3)2 = 3.14 •9 = 28.26

3. For x = 95x + 25(x + 30) = 5(9) + 25(9 + 30) = 5(9) + 25(39) = 45 + 975 = 1020

4. For l = 8, w = 2 and h = 3lwh = (8)(2)(3) = 48

5. For a = 9 and b= 4a2 + b2 = (9)2 + (4)2 = 81 + 16 = 97

6. For a = 9 and b= 4(a + b)2 = (9 + 4)2 = (13)2 = 169

Practice 1.1.51. Variable - Def (pg. 103): A

symbol, usually a letter, that is usedto represent

something, usually a quantity or aquantity that can take on manydifferent values. Ex. x, y

2. Simplify - Def (pg. 106): toperform some or all of the allowableoperations in an expression.Ex: The simplified version of 2+3 is 5.

3. Factor - Def (pg. 103): Numbersbeing multiplied.Ex: In 2x, both 2 and x are factors

4. Exponent - Def (pg. 106): In

1CHAPTER 1 THE REAL NUMBER SYSTEM

exponential notation, the number ofbases being multiplied.

Ex: In 32, 2 is the exponent. 32 means 3 • 3

Exercise Set 1.1

1. 35 + 21 is a sum . The terms are 35and 21.

3. (3.5) • 2 is a product . The factorsare 3.5 and 2.

5. 4u is a product. The factors are 4 andu.

7. 5v + 3w is a sum. The terms are vand 3w. 5v is a product

. The factors are 5 andv. 3w is a product. The

factors are 3 and w.

9. 31 - 8 •3 = 31 - 24 = 7

11. 18 - 22/11 •(5) = 18 - 2 •(5) = 18 - 10 = 8

13. (2 •3)2 = (6)2 = 36

15. 23 • 33 = 8 • 27 = 216

17. 2 + 3(2 + 1) = 2 + 3(3) = 2 + 9 = 11

19. 15 - 3- 6 + 4 • 2 = 15 - 3 - 6 + 8 = 12 - 6 + 8 = 6 + 8 = 14

21. [12 - (12 - 6)] + 8 = [12 - 6] + 8 = 6 + 8 = 14

23. [3 + (10-7)2]2 = [3 + (3)2]2 =[3 + 9]2 = 144

25. 12 + 12 − 2 •3

10 − 3 2 =

12 + 12 − 6 10 − 9

=

12 + 6 1

= 12 + 6 = 18

27. 15 - [(8 - 5)2 - 2] = 15 - [(3)2 - 2] = 15 - [9 - 2] = 15 - 7 = 8

29. 5 + 4(5 +4[5 - 2]) =5 + 4(5 + 4[3]) = 5 + 4(5 + 12) = 5 + 4(17) = 5 + 68 =73

31. 23 - [(8 -2) •3 - (9-5)] =23 - [6 •3 - 4] = 23 - [18 - 4] = 23 - 14 = 9

33. For a = 8 and b = 6a2 - b2 = (8)2 - (6)2 = 64 - 16 = 48

35. For a = 3, b= 9 and c = 2b2 - 4ac = (9)2 - 4(3)(2) = 81 - 24 =57

37. For x = 6 and y = 5x2 + 2xy + y2 = (6)2 + 2(6)(5) + (5)2 = 36 + 60 + 25 = 121

39. For x = 150.05(x) + 0.10(30 - x) = 0.05(15) + 0.10(30 - 15) =0.75 + 0.10 (15) = 0.75 + 1.5 = 2.25

2 CHAPTER 1 THE REAL NUMBER SYSTEM

Section 1.2 Fractions

Practice 1.2.11. 18 = 2 • 3 • 3=2•32

1

3 3

3 9

2 18

2. 54 = 2 • 3 • 3 •3 =2•33

3. 385 = 5 • 7 •11

4. 119 = 7 •17

5. 105 = 3 • 5 •7

6. 60 = 2 •2 •3 •5=22•3•5

Practice 1.2.2

1.1820

= 2•3•32•2•5

= 3•3 2•5

= 9 10

2.2464

= 2 •2 •2 •3

2 •2 •2 •2 •2 •2=

3 2 •2 •2

= 3 8

3.7240

= 2 •2 •2 •3 •32 •2 •2 •5

= 3 •35

= 9 5

4.104156

= 2 •2 •2 •132 •2 •3 •13

= 2 3

5.95152

= 5 •19

2 •2 •2 •19=

5 2 •2 •2

= 5 8

6.84120

= 2 •2 •3 •7

2 •2 •2 •3 •5=

7 2 •5

= 7 10

1

5 5

3 15

2 30

2 60

1

7 7

3 21

5 105

1

17 17

7 119

1

11 11

7 77

5 385

1

3 3

3 9

3 27

2 54

Practice 1.2.3

1.1225

• 1027

= 2•2•35•5

• 2•5

3•3•3=

2•25

• 2

3•3=

2•2• 2 5•3•3

= 8 45

2.203

• 1520

= 203

• 3 • 5 20

= 1 1

• 5 1

= 5

3.4 42

• 146

= 2 • 2

2 • 3 • 7 •

2 • 7 2 • 3

=


1 3

• 2 3

= 1 • 2 3 • 3

= 2 9

4.1825

• 3512

= 2•3•35•5

• 5•7

2•2•3=

3 5

• 7 2

= 3•75•2

= 2110

5.9257

• 190115

= 2•2•233•19

• 2•5•195•23

=

2•23

• 2 1

= 2•2•23•1

= 8 3

6.4298

• 175165

= 2•3•72•7•7

• 5•5•73•5•11

=

1 1

• 5 11

= 5 11

Practice 1.2.4

1.3332

2.1 6

3.x 1

, x ≠ 0

4.1 y

, y≠ 0

Practice 1.2.5

1.5 3

= 1 3

• 5

2.x 3

= 1 3

x

3. 1w

4.5 p 4

= 5 4

p

5.3 c 5

= 3 5

c

Practice 1.2.6

1.3 4

÷ 9 16

= 3 4

• 169

= 1 1

• 4 3

= 4 3

2.

5 9 3527

= 5 9

÷ 3527

= 5 9

• 2735

= 1 1

• 3 7

= 3 7

3.108

÷ 5 = 108

• 1 5

= 2 8

• 1 1

= 1 4

4.7 1420

= 7 1

÷ 1420

= 7 1

• 2014

= 202

= 10

5.5176

÷ 10295

= 5176

• 95102

=

512 • 2 • 19

• 5 • 192 • 51

= 5 8

6.4856

÷ 12 = 4856

• 1 12

= 4 56

= 1 14

7.

152114

= 1521

÷ 14 = 1521

• 1 14

=

5 7

• 1 14

= 5 98

8.

703

17515

= 703

÷ 17515

= 703

• 15175

=

703

• 3 35

= 2

Practice 1.2.7

1.5 8

+ 7 8

= 128

= 3 2


2.1 12

+ 5 12

= 6 12

= 1 2

3.4 15

+ 1 15

− 2 15

= 3 15

= 1 5

4.7 20

− 3 20

+ 1 20

= 5 20

= 1 4

Practice 1.2.81. 9 = 32

12 = 22•3LCD = 22•32 = 36

2. 18 = 2•32

24 = 23•3LCD = 23•32 = 72

3. 8 = 23

9 = 32

10 = 2•5LCD = 23•32•5=360

4. 34 = 2•1751 = 3•1785 = 5•17LCD = 2•3•5•17 = 510

Practice 1.2.9

1.4 5

= 4 • 8 5 • 8

= 3240

2.5 16

= 5 • 3 16• 3

= 1548

3. 4 = 4 • 181 • 18

= 7218

4. 3 = 3 • 541 • 54

= 16254

Practice 1.2.101. 12 = 22•3 18 = 2•32

LCD = 22•32 = 365

12 +

5 18

=

5 • 3 12• 3

+ 5 • 2 18• 2

= 1536

+ 1036

= 2536

2. 8 = 22 24 = 22•3 LCD = 22•3=247 8

− 1724

= 7 • 3 8 • 3

− 1724

=

2124

− 1724

= 4 24

= 1 6

3. 10 = 2•5 9 = 32 15 = 3•5LCD = 2•32•5 = 909 10

− 1 9

+ 1415

=

9 • 9 10• 9

− 1 • 109 • 10

+ 14• 6 15• 6

=

8190

− 1090

+ 8490

= 15590

= 5 • 315 • 18

= 3118

4. 9 =32 14 = 2•7 7 = 1•7LCD = 2•32•7 = 1264 9

+ 4 7

− 1 14

=

4 • 149 • 14

+ 4 • 187 • 18

− 1 • 9 14• 9

=

56126

+ 72126

− 9

126=

119126

=

7 • 177 • 18

= 1718

5. 15 =3•5 12 = 22•3 10=2•5LCD = 22•3•5=607 15

+ 5 12

+ 7 10

=

7 • 4 15• 4

+ 5 • 5 12• 5

+ 7 • 6 10• 6

=

2860

+ 2560

+ 4260

= 9560

= 5 • 195 • 12

= 1912

6. 4 = 22 9 =32 36=22•32

LCD = 22•32=363 4

+ 1 9

− 1336

= 3 • 9 4 • 9

+ 1 • 4 9 • 4

− 1336

=

273 6

+ 4 36

− 1336

= 1836

= 1 2


Practice 1.2.11

1. 10 − 4 ( 3 ) 2 + 4

=

10 − 126

= 10 − 2 = 8

2. 5 − 2 • 6 4

= 5 − 124

= 5 − 3 = 2

3.1 2

+ 1 3

( 5 4

− 1 6

) =

1 2

+ 1 3

( 1512

− 2 12

) = 1 2

+ 1 3

( 1312

)

1 2

+ 1336

= 1836

+ 1336

= 3136

4.2 3

+ 2 7

( 1 4

+ 1 3

) = 2 3

+ 2 7

( 3 12

+ 4 12

)

2 3

+ 2 7

( 7 12

) = 2 3

+ 1 6

= 4 6

+ 1 6

= 5 6

5. ( 4 5

+ 2 10

) 2 − 3 4

+ 1 4

=

( 8 10

+ 2 10

) 2 − 3 4

+ 1 4

=

( 1010

) 2 − 3 4

+ 1 4

= ( 1 ) 2 − 3 4

+ 1 4

=

1 − 3 4

+ 1 4

= 4 4

− 3 4

+ 1 4

= 2 4

= 1 2

6.5 + 2 3

6 − 2 ÷

24 + 2

3 2 + 5 =

5 + 8 4

÷ 26

9 + 5 =

134

÷ 2614

= 134

• 1426

= 1 2

• 7 2

= 7 4

Exercise Set 1.2

1. 96 =25•3

3. 378 = 2•33•7

5. 825=3•52•11

7. 484 = 22•112

9.6 20

= 3 • 2

2 • 2 • 5 =

3 10

11.3690

= 2 • 2 • 3 • 3 2 • 3 • 3 • 5

= 2 5

13.135126

= 3 • 3 • 3 • 5 2 • 3 • 3 • 7

= 3 • 5 2 • 7

= 1514

15.8 15

• 2012

= 2 • 4 3 • 5

• 4 • 5 3 • 4

= 2 3

• 4 3

= 8 9

17.3042

• 8 60

= 1 • 302 • 21

• 2 • 4 2 • 30

= 1 21

• 4 2

=

1 21

• 2 = 2 21

19.11030

• 2133

= 10• 113 • 10

• 3 • 7 3 • 11

=

1 1

• 7 3

= 7 3

21.9850

• 60315

= 2 • 7 • 7 2 • 5 • 5

• 2 • 5 • 2 • 3 3 • 3 • 5 • 7

=

7 5

• 2 • 2 3 • 5

= 2875

23.1 8

25.6 5

27.a 7

, a≠0

29.1 t

, t≠0

31.1 15

• 13

33.1 2

x

35.3 8

•t


37.4 9

÷ 1227

= 4 9

• 2712

= 1 1

• 3 3

= 1

39.9 185

= 9 ÷ 185

= 9 • 5 18

= 1 • 5 2

= 5 2

41.135210

÷ 18 = 135210

• 1 18

=

5 • 3 • 3 • 3 2 • 3 • 5 • 7

• 1

2 • 3 • 3 =

1 2 • 7

• 1 2

= 1 28

43.1415

÷ 6090

= 1415

• 9060

=

2 • 7 3 • 5

• 2 • 3 • 3 • 5 2 • 2 • 3 • 5

= 7 1

• 1 5

= 7 5

45.

3 4 1 8

÷ 6 = 3 4

÷ 1 8

÷ 6 =

3 4

• 8 1

• 1 6

= 2424

= 1

47.1 15

+ 1115

= 1215

= 4 5

49.139

− 2 9

+ 109

= 219

= 7 3

51.1954

− 5 54

− 5 54

= 9 54

= 1 6

53. 8 = 23 14 = 2•7LCD = 23•7=56

55. 9 = 32 15 = 3•5 35 = 5•7LCD = 32•5•7= 315

57. 27 = 33 36 = 22•32 48 = 24•3

LCD = 24•33=432

59.1 6

= 1 • 3 6 • 3

= 3 18

61.1 36

= 1 • 5 36• 5

= 5

180

63. 3 = 3 1

= 3 • 911 • 91

= 27391

65. 9 = 32 24 = 23•3 LCD = 23•32 = 72

2 9

+ 5 24

= 2 • 8 9 • 8

+ 5 • 3 24• 3

=

1672

+ 1572

= 3172

67. 8 = 23 20 = 22•5 LCD = 23•5=40

6 8

− 3 20

= 6 • 5 8 • 5

− 3 • 2 20• 2

=

3040

− 6 40

= 2440

= 3 5

69. 6 = 2•3 2 = 2•1 3 = 3•1 LCD = 6

5 6

+ 1 2

+ 8 3

= 5 6

+ 1 • 3 2 • 3

+ 8 • 2 3 • 2

=

5 6

+ 3 6

+ 166

= 246

= 4

71. 28 = 22•7 21 = 3•7 9 = 32

LCD = 22•32•7 =252

9 28

+ 1021

− 7 9

=

9 • 9 28• 9

+ 10• 1221• 12

− 7 • 289 • 28

=

81252

+ 120252

− 196252

= 5

252

73. ( 2 3

) 2 + ( 1 4

) 2 = 4 9

+ 1 16


9 = 32 16 =24 LCD = 24•32 = 144

4 9

+ 1 16

= 4 • 169 • 16

+ 1 • 9 16• 9

=

64144

+ 9

144=

73144

75. 10 = 2•5 6 = 2•3 LCD=2•3•5=30

( 3 7

) ( 1 10

+ 5 6

) =

( 3 7

) ( 1 • 3 10• 3

+ 5 • 5 6 • 5

) =

( 3 7

) ( 3 30

+ 2530

) = ( 3 7

) ( 2830

) = 4 10

=

2 5

77. 2 + 3 + 5 3 ( 2 )

= 2 + 8 6

= 2 + 4 3

=

2 • 3 1 • 3

+ 4 3

= 6 3

+ 4 3

= 103

79. ( 4 15

+ 1 5

) ( 7 6

+ 4 3

) =

15 = 3•5 LCD=15 6=2•3 LCD = 6

( 4 15

+ 1 • 3 5 • 3

) ( 7 6

+ 4 • 2 3 • 2

) =

( 4 15

+ 3 15

) ( 7 6

+ 8 6

) = ( 7 15

) ( 156

) =

7 6

81.3 8

÷ 9 4

+ 1 9

÷ 1 3

= 3 8

• 4 9

+ 1 9

• 3 1

=

1 6

+ 1 3

= 1 6

+ 2 6

= 3 6

= 1 2

83.1 3

+ 2 3

( 1 2

+ 1 7

) =

1 3

+ 2 3

( 7 14

+ 2 14

) = 1 3

+ 2 3

( 9 14

)

1 3

+ 3 7

= 7 21

+ 9 21

= 1621

85. Sum is -2r + 5s with the terms being -2r and 5s. The products are-2r and 5s. The factors of -2r are -2 and r and the factors of 5s are 5and s.

86. 18/3•2+32= 18/3•2+9=6•2+912 + 9=21

87. 3•22-32= 3•4 - 9 = 12 - 9= 3

88. 8+24÷2•3=8 +12•3=8+36=44

89. 5 + 8 − 2 • 2 2

4 + 3 2 = 5 +

8 − 2 • 4 4 + 9

=

5 + 8 − 8 13

= 5 + 0 13

= 5 + 0 = 5

90. (32-2•4)(20-12/2•3)=(9 - 2•4)(20-12/2•3) = (9 - 8)(20-

6•3)(1)(20-18) = 1(2) = 2

91. 4+2(109-5(2 + 62÷2)) = 4+2(109-5(2+36÷2))=4+2(109-5(2+18)=4+2(109-5(20))=4+2(109-100)=4+2(9)=4+18=22

92. 18-3(8-3(22-3)) = 18-3(8-3(4 -3))=18-3(8-3(1))= 18-3(8-3)=18-3(5)=18-15 = 3

93. 6÷2• 9 = 6÷2•3=3•3=9

94. For c = 38c - c2 = 8(3) - 32=8(3) -9=24-9=15

95. For a = 62a(a2÷2•3-50) = 2(6)((6)2÷2•3-50)=


12(36÷2•3-50)= 12(18•3-50)=12(54 - 50)= 12(4) = 48

Section 1.3 Real Numbers

Practice 1.3.11.<--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

2.<--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Practice 1.3.21. {-9.9,-4.7,-0.6, 3.1, 5.8,10}

2. {-7.4, -4, -2.1,0.5,3,5.7}

Practice 1.3.31.<--|-----|-----|-----|-----|-----|-----|-----|-----|----->

-300 -250 -200 -150 -100- -50 0 50 100

2.<--|-----|-----|-----|-----|-----|-----|-----|-----|----|->

-200 -100 0 100 200- 300 400 500 600 700

3.<--|-----|-----|-----|-----|-----|-----|---->

20 30 40 50 60 70 80

4.<--|-----|-----|-----|-----|-----|-----|-----|-----|----->

5 10 15 20 25 30 35 40 45

5.<--|---|---|---|---|---|--|--|---|---|---|--|---|---|--|--|---|--|->

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

6. <--|-----|-----|-----|-----|-----|-----|---->

-2.5 -2.0 -1.5 -1.0 -0.5 0 0.5

Practice 1.3.41. 7 is to the left of 12 on the number

line so 7<12.

2.1 2

is to the left of 3 4

on the number

line so 1 2

< 3 4

.

3. -12 is to the right of -22 on thenumber line so -12>-22.

4. 5 is to the right of -12 on thenumber line so 5> -12.

Practice 1.3.51. -36

2. 45 or +453. 2w

4. -7t

5. -(-(-2)) = -2


Practice 1.3.61. 12 = 12 since the distance from 12

to 0 on the numberline is 12.

2. − 57 = 57 since the distance from-57 to 0 on the number line is 57.

3. − 4445

= 4445

since the distance from

− 4445

to 0 is 4445

4. − 179 = -179 since this is theopposite of 179 . The absolutevalue of 179 is 179 and

the opposite of 179 is -179 so − 179 = − 179

5. − − 59 = − 59 since this is theopposite of the absolute value of

-59. The absolute value of -59 is 59and the opposite of 59 is -59.

Practice 1.3.71. 15 + − 12 = 15 + 12 = 27

2. 25 − 7 = 25 − 7 = 18

3. − 17 + 2 • 21 = 17 + 2 • 21 = 17+42 =59

4. 5 + − 45 ÷ 5 + − 7 = 5 + 45÷ 5 + 7 =5+9+7=21

5. 22 − − 10 − 3 = 22 − 10 − 3 = 9

6. − 32 + − 15 • 3 = 32 + 15• 3 = 32+45=77

Exercise Set 1.31.<--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

3. {-8,-4.5,-1,3,6,9.9}

5.<--|-------|-------|-------|------•-------|-------|->

1940 1950 `960 1970 1980 1990 2000

7.<--|------|------|------|------|------|----•-|------|------|-->

0 5 10 15 20 25 30 3540

9.<--|--•--|-----|-----|-----|-----|-----|-----|-----|----->

-2 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0

11. -3>-4 since -3 is to the right of -4 onthe number line.

13. − 1 2

< − 1 3

since − 1 2

is to the left of

− 1 3

on the number line.

15. 5>-12 since 5 is to the right of -12 onthe number line.

17. 0 >− 7 9

since 0 is to the right of− 7 9

on the number line.

19.5 12

> − 6 7

since 5 12

is to the right of

− 6 7

on the number line.

21. -12

23. 5x

25. -(-(-7))=-(7) = -7

27. − 577 = 577 since the distance onthe number line from -

577 to 0 is 577.

29. − − 7 . 5 = − 7 . 5 since the absolutevalue of -7.5 is 7.5 and the oppositeof 7.5 is -7.5.

31. 8 − − 5 = 8 − 5 = 3

33. 15 − 3 ÷ 5 − 3 = 12 ÷ 2 = 12÷ 2 = 6

35. − ( 8 + − 6 ) = − ( 8 + 6 ) = − 14

37. 8 − 2 ÷ ( 6 − 6 ) = 6 ÷ ( 6 − 6 ) 6 ÷ 0 is not possible since we cannotdivide by zero.


39. The sum is 8 + 2 and the terms are8 and 2.

40.9 8 343

= 2 • 7 • 7 7 • 7 • 7

= 2 7

41.2 x

, x≠0

42.1 3

• 7

43. 16/2•(13-2(5-1)-2)=16/2•(13-2(4)-2)=16/2•(13-8-2)=16/2•(3)=8•3=24

44. 8 + 5•2 = 8+10=18

45. 8 − 1

2 • 14

1

3 + 2

= 8 − 7 1

3 + 6

3

=

8 − 7 7

3

= 8 − 7 ÷ 7 3

=

8 − 7 • 3 7

= 8 - 3 = 5

46.32

25

24

35

= 3225

÷ 2435

= 3225

• 3524

=2815

47.84

15

4

25

= 8415

÷ 4

25= 84

15•

254

=

3 • 4 • 7 3 • 5

• 5 • 5 4

= 7 • 5 = 35

48.4

15+ 7

25= 4 • 5

15• 5 + 7 • 3

25• 3 =

2075

+ 2175

= 4175

49.3 4

+ 7 10

+ 1 5

= 3 • 5 4 • 5

+ 7 • 2 10• 2

+ 1 • 4 5 • 4

=

1520

+ 1420

+ 4 20

= 3320

50.6 2 − 2 • 3 15 − 5 • 2

= 36 − 6 15 − 10

= 305

= 6

51.9 24

• 6072

= 9

2 • 12•

5 • 128 • 9

= 5 16

Section 1.4 Multiplying and Dividing Real Numbers

Practice 1.4.11. 6(-5) = -30

2. (-6)(-9) = 54

3. (-12)(5) = -60

4. ( 8 ) ( − 3 4

) = 2 ( − 3 ) = − 6

5. ( − 12) ( − 2 3

) = 8

6. (-2)(5)(4) = -40

7. (-2)(-7)(-4) = -56

Practice 1.4.21. For m = -5

-7m = -7(-5)=35

2. For m = -55m = 5(-5) = -25

3. For m = -5 and n = -4n2m = (-4)2(-5) = 16(-5) = -80

4. For m = -5 and n = -4m3n = (-5)3(-4) = (-125)(-4) = 500

5. For n = -4

− 3 16

n = − 3 16

( − 4 ) = 3 4

6. For m = -5 and n = -4(mn)3 = ((-5)(-4))3= (20)3 = 8000

Practice 1.4.31. For m= -8

-m = -(-8) = 8


2. For n = 3-n2 = -(3)2= -9

3. For n= 3(-n)2 = (-3)2 = 9

4. For m = -8 and n = 3-(-m)(-n) = -(-(-8)(-3)) = -(-24) =

24

Practice 1.4.41. The cube of -2 = (-2)3 = -82. The product of 5 and -1 =

(5)(-1) = -5

3. Twice -10 = 2(-10) = -20

4. Two thirds of nine = 2 3

• 9 = 6

5. Five percent of 8 = 0.05(8) = 0.4

6. 15 fifteen multiplied by -4 = -60

Practice 1.4.51. Product of -2 and 4, or -2 multiplied

by 4 or -2 times 4.

2. Product of 7 and 8, or 7 multiplied by8 or 7 times 8.

3. Product of 1/2 and 5, or 1/2multiplied by 5 or 1/2 of 5.

4. Six cubed or 6 to the third power

Practice 1.4.61. -20/4 = -5

2.49− 7

= − 7

3. -54÷0 is undefined. We can’t divideby zero.

4. 0/-9 = 0

5. (-1.5)/(-0.3) = 5

Practice 1.4.7

1.

5 6

− 109

= 5 6

÷ − 10

9

5 6

• 9

− 10=

3 2 • ( − 2 )

=

3 − 4

= − 3 4

2. ( 24) ÷ ( 1

− 3 ) = 24• (

− 3 1

) = − 72

3.−

4 5

− 6 = −

4 5

÷ − 6 = − 4 5

• 1

− 6 =

− 2 5

• 1

− 3 =

− 2 − 15

= 2 15

Practice 1.4.81. For x = -5 and y = -3

y x

= − 3 − 5

= 3 5

2. For y = -3 and z = 1/4

z y

=

1 4

− 3 =

1 4

÷ − 3 =

1 4

• 1

− 3 =

1 − 12

= − 1 12

3. For x = -5 , y = -3 and z = 1/4

− xy2

z =

− ( − 5 ) ( − 3 ) 2

1 4

= 5 ( 9 )

1 4

=

45÷ 1 4

= 45• 4 = 180

4. For x = -5 and y = -3


x 2

y

y 2

x

= x 2

y ÷

y 2

x =

( − 5 ) 2

− 3 ÷

( − 3 ) 2

− 5 =

25− 3

÷ 9

− 5 =

25− 3

• − 5 9

=

− 125− 27

= 12527

Practice 1.4.91. The quotient of -8 and 4 = -8/4 = -2

2. The quotient of -24 and 4 3

=

− 24÷ 4 3

= − 24• 3 4

= − 18

3.8 9

divided by -3 = 8 9

÷ − 3

8 9

• 1

− 3 =

8 − 27

= − 8 27

4. -36 divided by -4 = -36/-4 = 9

5. The reciprocal of 2 is 1/2

6. The reciprocal of 4 less than six is1

6 − 4 =

1 2

Practice 1.4.101. 18 divided by 4 or the quotient of 18

and 4.

2. 39 divided by -5 or the quotient of 39and -5.

3. The reciprocal of three more than 5.

4. The reciprocal of twice -4.

Exercise Set 1.41. (-15)(-5) = 75

3. (-2)(-5)(-3) = -30

5. ( − 6 5

) ( − 5 ) = ( − 6 ) ( − 1 ) = 6

7. ( 1518

) ( − 6 5

) = ( 5 6

) ( − 6 5

) = − 1

9. For x = -22x = 2(-2) = -4

11. For y = -3-6y = -6(-3) = 18

13. For x = -2 and y = -3-xy = -(-2)(-3) = -6

15. For x = -2-x2 = -(-2)2= -4

17. For y = -3 and z = 4(yz)2 = ((-3)(4))2 = (-12)2 = 144

19. For x = -2 and y = -3xy2 = (-2)(-3)2 = (-2)(9) = -18

21. For x = -2

− 5 4

x 2 = − 5 4

( − 2 ) 2 = − 5 4

( 4 ) = − 5

23. The square of -5 = (-5)2 = 25

25. Double -8 = 2(-8) = -16

27. Six-sevenths of -14 =6 7

• ( − 141

) =

6 • ( − 2 ) = − 12

29. 12% of 80 = 0.12(80) = 9.6

31. Product of 6 and 3 or 6 multiplied by3 or 6 times 3.

33. Product of 3/4 and 12 or 3/4multiplied by 12 or 3/4 of 12.

35. -45/9 = -5

37.− 27− 3

= 9

39. 54÷(-9) = -6


41.15− 3 4

= 15÷ − 3 4

= 15• 4

− 3 =

− 5 • 4 = − 20

43.− 5 8

÷ ( − 1516

) = − 5 8

• ( 16

− 15) =

2 3

45.2 3

÷ ( − 6 ) = 2 3

• ( 1

− 6 ) =

1 − 9

= − 1 9

47.18

− 258

− 15

= 18

− 25÷

8 − 15

=

18− 25

• − 15

8 =

− 27− 20

= 2720

49.

24− 5 − 8

= 24− 5

÷ ( − 8 ) =

24− 5

• 1

− 8 =

3 5

51.− 2 − 3 − 4

= − 2 ÷ − 3 − 4

= − 2 • − 4 − 3

=

8 − 3

= − 8 3

53. For p = -4 and r = -1/3p r

= − 4

− 1 3

= − 4 ÷ ( − 1 3

) =

− 4 • ( − 3 ) = 12

55. For p = -4 and q = -2

q p•q

= − 2

− 4 • ( − 2 ) =

1 − 4

= − 1 4

57. For p = -4 and q = -2

( q

p 2 ) 2 = (

− 2

( − 4 ) 2 ) 2 =

( − 2 16

) 2 = ( − 1 8

) 2 = 1 64

59. For p = -4, q = -2 and r = -1/3

r(p÷q) = (− 1 3

) ( − 4 ÷ ( − 2 ) ) =

( − 1 3

) ( 2 ) = − 2 3

61. 50÷(-150) = 1/(-3) = -1/3

63.1

12 − 5 =

1 7

65.4 5

÷ ( − 2 ) = 4 5

• 1

− 2 = − 2

5

67.6 15

÷ − 2 5

= 6 15

• 5

− 2 =

− 3 3

= − 1

69. The quotient of 64 and -8 or 64divided by -8

71. The quotient of 2 and 1/3 or 2divided by 1/3.

73. The quotient of 1/3 and 2/9 or 1/3divided by 2/9.

75. 12 + 5 is a sum whose terms are

76. 300=22•3•52

1

5 5

5 25

3 75

2 150

2 300


77.8 3

78.1 9

x

79.1620

= 4 • 4 4 • 5

= 4 5

80. -8

81. <--|---|---|---|---|---|---|---|---|---|---|---|•--|---|-->

-10 -5 0 5 10 15 20 25 30 35 40 45 50 55

82. 6•2+3 = 12+3 = 15

83. 8 - [(2+3)2 - 18] =8 -[(5)2 - 18] =8 - [25-18]= 8 -7=1

84.1224

• 6 30

= 1 2

• 1 5

= 1 10

85.1812

÷ 6 15

= 1812

• 156

=

3 • 5 4

= 154

86.1 2

+ 5 6

= 1 • 3 2 • 3

+ 5 6

= 3 6

+ 5 6

=

8 6

= 4 3

87.24 − 2 • 3 2

3 • 2 2 − 5 • 2 =

24 − 2 • 9 3 • 4 − 5 • 2

=

24 − 1812 − 10

= 6 2

= 3

88. − 2 + 5 = 2 + 5 = 7

89. − 3 • 18÷ ( − 9 • 2 ) = 3 • 18÷ ( − 18) = 54÷ ( − 18) = − 3

Section 1.5 Adding and SubtractingReal Numbers

Practice 1.5.11. (-8) +(-10)

Add absolute values |-8| +|-10| = 18Attach the common sign -18

2. (-25)+(-16) Add absolute values |-25| +|-16| = 41Attach the common sign -41

3. 9 + (-36) Subtract absolute values |-36| -|9| =27Attach sign of larger absolute value

-27

4. (-15) + 18Subtract absolute values |18| -|15| =


3Attach sign of larger absolute value +3 or 3

5. (-25) +(-10) + 14 = (-35) + 14 = -21

6. 8 + (-31) + 34 = -23 + 34 = 11

7. 21 + (-27) + (-14) = -6 + (-14) = -20

8. (-12) + (-26) + (-13) = -38 + (-13) = -51

9. ( − 1 2

) + ( − 3 4

) =

( − 2 4

) + ( − 3 4

) = − 5 4

= -1 1 4

10. ( − 5 6

) + ( 3 4

) =

( − 1012

) + ( 9 12

) = − 1 12

11. − 3 8

+ ( − 3 1 2

) =

− 3 8

+ ( − 3 4 8

) = − 3 7 8

12. − 2 1 4

+ 3 2

= − 9 4

+ 3 2

=

− 9 4

+ 6 4

= − 3 4

Practice 1.5.21. The box is -11 since 11 + (-11) =02. The box is 22 since 22 + (-22) = 0

3. The box is 5 6

5 6

+ ( − 5 6

) = 0

4. The box is -0.002 since -0.002+0.002 = 0

Practice 1.5.31. -18 increased by 10

‘Increased by’ tells us to add-18 + 10 <---- Translated-8

2. -8.6 increased by 15‘Increased by’ tells us to add-8.6 + 15 <---- Translated6.4 <----Simplified

3.3 7

more than − 5 7

‘More than’ tells us to add3 7

+ − 5 7

<---- Translated

− 2 7

<----Simplified

4. 1 more than − 7 8

‘More than’ tells us to add

1 + − 7 8

<---- Translated

8 8

+ − 7 8

= 1 8

<----Simplified

5. -10 added to -16‘Added to’ tells us to add-10 + -16 <----Translated-26 <---- Simplified

6. The sum of 25 and -36‘The sum of’ tells us to add25 + -36 <---Translated-11 <----Simplified

Practice 1.5.41. The sum of -12 and 7.

-12 increased by 77 more than -127 added to -12

2. The sum of 15.9 and 25.15.9 increased by 2525 more than 15.925 added to 15.9

Practice 1.5.51. The current temperature is -40

degrees and it warms up 3 degrees.What is the new temperature?-40 + 3 <----Translated-37 degrees or 37 degrees below 0

2. You begin on a hill 650 feet abovesea level and descend


780 feet from the top of the hill.What is your new elevation?

650 feet above sea level is +650Descending 780 feet would berepresented by -780.650 + -780 <-----Translated-130 <---- Simplified-130 feet or 130 feet below sea level

3. Checking account overdrawn by $36is represented by -

$36. When $15 is depositedthen you are adding $15 to the -$36.

-36 + 15 <---- Translated-21 <--- Simplified-$21 is the balance. The account isstill overdrawn by $21.

4. A balance of $45 would berepresented by +45. A $60 checkwould be represented by -60.45 + (-60) <---- Translated-15 <---- Simplified-$15 is the balance. The account isstill overdrawn by $15.

5. The new share price is represented

by the sum 136 + ( − 2 1 2

) <--

Translated

136 + ( − 2 1 2

) = 136 + ( − 5 2

) =

1361

• 2 2

+ ( − 5 2

) =

2722

+ ( − 5 2

) = 2672

= 1331 2

The new price is 133 1/2 or $133.50.

6. The new share price is represented

by 255 + ( − 3 1 4

) <----

Translated

255 + ( − 3 1 4

) = 255 + ( − 134

) =

2551

• 4 4

+ ( − 134

) =

10204

+ ( − 134

) = 1007

4 =

2513 4

The new price is 2513 4

or $251.75.

Practice 1.5.61. -8 -10 = -8 + (-10) = -18

2. -9 -12 = -9 + (-12)= -21

3. 42 - (-27)= 42 + (+27) = 69

4. 4 - (-36) = 4 + (+36) = 40

5. -10 - (-4) = -10 + (4) = -6

6. -7 - (-3) = -7 + (+3) = -4

7. 18 - 27 = 18 + (-27) = -9

8. 11 - 32 = 11 + (-32) =-21

9. -25 - (-10) -14 = -25+(+10)+ (-14)=-15 + (-14) = -29

10. -8 -(-31) - 34= -8 + (+31) +(-34)=23 + (-34) = -11

11. ( − 1 2

) − ( 3 4

) =

( − 2 4

) + ( − 3 4

) = − 5 4

= -1 1 4

12. ( − 3 8

) − ( 3 5

) =

( − 1540

) + ( − 2440

) = − 3940

Practice 1.5.71. 7 - (6-9) = 7 - (6 + (-9))=

7 - (-3) = 7 +(+3)= 10

2. -7 -(4+2) = -7 -(6) = -7 + (-6) = -13

3. 25 - (-10 + 13) = 25 - (3) = 22

4. 19 - (25 - 36) = 19 - (25 + (-36)) = 19 - (-11) = 19 + (+11) = 30


5. 25 - (-10 -13) = 25 - (-10 + (-13)) = 25 - (-23) = 25 + (+23) = 48

6. (19 - 25) - 36 = (19 + (-25)) - 36 = -6 - 36 = -6 + (-36) = -42

7. (-8 -2) - (-5 + 4) = (-8 + (-2)) - (-1)=(-10) - (-1) = (-10) + (+1) = -9

8. (-5 -1) - (-3 + 8) = (-5 + (-1)) - (5) = (-6) - (5) = -6 + (- 5) = -11

Practice 1.5.81. For x = -8 and y = 7

x - y = (-8) - 7 = (-8) + (-7) = -15

2. For p = 10 and q = -4-3p - q = -3(10) - (-4) = -30 - (-4) =-30 + (+4) = -26

3. For w = -32 and v = -25w - (v - w) = -32 -(-25 -(-32)) = -32 - (-25 + (+32)) = -32 - 7 = -32 + (-7) = -39

4. For p=-5 and q = 3p-2(q - 3p) = -5 -2(3 - 3(-5)) = -5 - 2(3 - (-15)) = -5 -2(3 + (+15)) = -5 -2(18) - -5 -36 = -5 + (-36) = -41

Practice 1.5.91. -18 decreased by 10

‘Decreased by’ tells us to subtract-18 -10 <----Translated-18 + (-10) = -28 <--- Simplified

2. 15 decreased by 9‘Decreased by’ tells us to subtract15 -9 <----Translated6 <----Simplified

3. 1 less than − 7 8

− 7 8

− 1 <---Translated with

rearranged order tomatch Englishmeaning

− 7 8

− 1 = − 1 7 8

4. -10 subtracted from -16

-16 - (-10) <---Translated with rearranged order tomatch Englishmeaning

-16 + (+ 10) = -6

5. -13 subtracted from 2828 - (-13) <---Translated with

rearranged order tomatch Englishmeaning

28 + (+13) = 41

6. The difference of 5 and − 3 4

5 − ( − 3 4

) = 5 + ( + 3 4

) = 5 3 4

Practice 1.5.101. The difference of -12 and 7

-12 decreased by 77 less than -12 7 subtracted from -12

2. The difference of -1 and -3-1 decreased by -3-3 less than -1-3 subtracted from -1

Exercise Set 1.51. -5 + (-9) = -14

3. 24 + (-26) = -2

5. -39 + 54 + (-7) = 15 + (-7) = 8

7. 36 + (-45) + (-22) + 73 =-9 + (-22) + 73 = -31 + 73 = 42

9. − 5 12

+ 3 4

= − 5 12

+ 9 12

= 4 12

= 1 3

11. − 1 + 1 8

+ ( − 5 6

) =

− 2124

+ ( − 2024

) = − 4124

= − 1 1724

13. The box is -29 since 29 + (-29) = 0

15. The box is 1315


since 1315

+ ( − 1315

) = 0

17. The box is 24.9 since -24.9+24.9 =0

19. -32 increased by 24‘Increased by’ tells us to add-32 + 24 <----Translated-8 <----Simplified

21. -22 added to 13‘Added to’ tells us to add-22 + 13 <-----Translated-9 <---- Simplified

23. The sum of 5 8

and − 1 6

‘The sum of’ tells is to add5 8

+ − 1 6

<---- Translated

5 8

+ − 1 6

= 1524

+ − 4 24

= 1124

25. 5 more than − 2 3

‘More than‘tells us to add

5 + − 2 3

<---Translated

5 + − 2 3

= 153

+ − 2 3

=

133

= 4 1 3

<----Simplified

27. The sum of -36 and 1111 more than -36-36 increased by 1111 added to -36

29. The sum of -5 and (-2)(-2) added to -5

31. Checking account is overdrawn by$25 and $40 is

deposited. What is thenew balance?Overdrawn is represented by -25

and the deposit is representedby 40.

-25 + 40 <---Translated15 <-----Simplified$15 is in the account

33. Price was $25 and the value changed

by - 3 4

. What is the current value?

25 + ( − 3 4

) <----Translated

25 + ( − 3 4

) = 251

• 4 4

+ ( − 3 4

) =

1004

+ ( − 3 4

) =

974

= 241 4

<--Simplified

Current price is $241 4

or $24.25

35. -5 -9 = -5 + (-9) = -14

37. -24 - (-26) = -24 + (+26) = 2

39. -36 - (-22) - 18 = -36 + (+22) -18=-14 -18 = -14 + (-18) = -32

41. 11 - 18 - (-30) = 11 + (-18) -(-30) = -7 -(-30) = -7 +(+30) = 23

43. 36 - (-45) - (-22) - 73= 36 + (+45) + (+22) + (-73)=81 + 22 + (-73) = 103 + (-73) = 30

45.− 3 14

− 6 7

= − 3 14

− 1214

=

− 3 14

+ ( − 1214

) = − 1514

= − 1 1 14

47. 12 - (4 - 8)= 12 - (4 + (-8)) = 12 - (-4) = 12 + (+4) = 16

49. (12 - 4) - 8 = 8 - 8 = 0

51. (-10 - 7) - 2 = (-10 + (-7)) - 2=-17 - 2 = -17 + (-2) = -19

53. (36 - 45) - (6 -8) = (36 + (-45)) - (6 + (-8)) = -9 - (-2) = -9 + (+2) = -7

55. For x = -28 and y = 15x - y = -28 - 15 = -28 + (-15) =-43

57. For m= -2 and n= -5


m - (m -n) = -2 - (-2 - (-5)) = -2 -(-2 +(+5)) = -2 -3 = -2 +(-3) =-5

59. For x = 4 and y = 93x - 5y = 3(4) - 5(9) = 12 - 45= 12 + (-45) = -33

61. -32 decreased by 24-32 - 24 <-----Translated-32 + (-24) = -56 <----Simplified

63. -22 subtracted from 1313 - (-22) <---Translated withrearranged order to match Englishmeaning13 -(-22) = 13 + (+22) = 35

65. the difference of -1 3

and 3

− 1 3

− 3 <---Translated

− 1 3

− 3 = − 1 3

− 3 1

• 3 3

=

− 1 3

− 9 3

= − 1 3

+ − 9 3

= − 10

3 =

− 3 1 3

<-----Simplified

67. 5 less than -24-24 - 5 <---Translated withrearranged order to match Englishmeaning-24 - 5 = -24 + (-5) = -29

69. The difference of -36 and 11-36 decreased by 1111 less than -3611 subtracted from -36

71. The difference of 4/9 and 44/9 decreased by 44 less than 4/94 subtracted from 4/9

73. 150 = 2•3•5•5 = 2•3•52

74.1 x

, x≠0

75. 4•2 is a product whose factors are 4and 2.

76.2 t 3

= 2 3

t

77. -0.23

78.2436

= 2 • 123 • 12

= 2 3

79. 8 + 2 • 3 + 6 12 − 2 • 3

= 8 + 6 + 6 12 − 6

=

8 + 126

= 8 + 2 = 10

80.1835

• 2124

= 3 • 6 5 • 7

• 3 • 7 4 • 6

=

3 5

• 3 4

= 9 20

81.5 8

+ 2 3

= 1524

+ 1624

= 3124

= 1 7 24

82.2 4 − 5 • 2

5 2 − 2 • 3 2 − 1 =

16 − 1025 − 2 • 9 − 1

=

6 25 − 18 − 1

= 6 6

= 1

83. 8 - 2(8 + 2(8 - 2) -19) =8 - 2(8 + 2(6) - 19) =8 - 2(8 + 12 - 19) = 8 - 2(1) = 8 - 2= 6

84. 2 • |-3| -4 = 2 • 3 - 4 = 6 - 4 = 2

85. 6 • 6 • ( − 1 2

) = 6 1

• ( − 1 2

) = − 3

86. -0.5 •2 ÷ (-0.4) =-1.0 ÷(-0.4) = 2.5

87. -5 •12

88. x is the number

x÷7 or x 7


Section 1.6 Properties of Real Numbers

Practice 1.6.11. The box is 0 since 0 + -36 = -36

2. The box is 1 since -12 • 1 = -12

3. The box is 7 since 7 + -7 + 2z = 0 + 2z = 2z

4. The box is -33 since -33 + 33 + 8n = 0 + 8n = 8n

5. The box is 3 2

since ( 3 2

) ( 2 3

) ( 9 ) = 1 ( 9 ) = 9

6. The box is 1 23

since ( 1 23

) ( 23) ( 11) = 1 ( 11) = 11

Practice 1.6.21. 22 +5

2. 8 - 23 = 8 + (-23) = -23 + 8

3. 5y2 +3y+2

4. 5w3 -2w2 + w

5. -8x -12

6. p - 5 - p2 = p -5 + (-p2)= -p2 + p - 5

7. a2 - 2a - 3

Practice 1.6.31. 5•4

2. 5w

3. 12(y - 8)

4. (d - 3)8

5. (x +2 )(x - 12)

6. a2 bc since this is alphabetical order

Practice 1.6.4

1. ( 5 6

+ 1 6

) + 9

2. − 10 + ( 1 3

+ 2 3

)

3. 6 + (7x + 3x)

4. (3x + 4x) + 12

Practice 1.6.51. 7 • (3 • 5)

2. -4 • (2 • (-8))

3. (4 • 5) • 3

4. ( 2 3

• 3 ) x

5. ( 12• 1 12

) y

6. ( 8 • 1 8

) z

Practice 1.6.61. 2(y + 7) = 2•y + 2•7=2y +14


2. -4(x + 2) = -4•x + (-4)•2 = -4x - 8

3. 6(r + 3s + 2) = 6•r + 6•3s + 6•2 = 6r + 18s + 12

4. -2(5x + y + 4) =-2•5x + (-2)•y + (-2)•4= -10x -2y -8

5. -7(y - 4) = -7•y - (-7)•4 = -7y -(-28)=-7y + 28

6. -4(-2x - 3)= -4(-2x) -(-4)•3 = 8x - (-12) = 8x + 12

Practice 1.6.71. 2y + 12y = y(2 +12)

2. 5y + 8y + 11y = y(5 + 8 + 11)

3. 3x - 5x = x(3 - 5)

4. xy + wy = y(x + w)

5. nm + mp = m(n + p)Practice 1.6.81. 63

• base is 6• exponent is 3• 63 = 6•6•6 = 216

2. 45

• base is 4• exponent is 5• 45 =4•4•4•4•4 =1024

3. 5•42 • base is 5• exponent is 1• base is 4• exponent is 2• 42 =4•4=165•42 =80

4. (3•2)3

• base is (3•2)• exponent is 3• (3•2)3 =(3•2)•(3•2)•(3•2)(3•2)3=63=6•6•6 = 216

5. (-7)2

• base is -7• exponent is 2• (-7)2 =-7•-7=49

6. -(8)2

• base is 8• exponent is 2• -(8)2 =-8•8=-64

7. -72

• base is 7• exponent is 2• -72 =-7•7= -49

8. -(-8)2

• base is -8• exponent is 2• -(-8)2 =-(-8)•(-8)= - 64

Practice 1.6.91. y 7

•y is the base and its exponent is 7•the expanded form is y•y•y•y•y•y•y

2. 8z5

•8 is a base and its exponent is 1•z is a base and its exponent is 5•the expanded form is 8•z•z•z•z•z

3. (4y)3

•4y is the base and its exponent is 3•the expanded form is 4y•4y•4y

4. 53 b4

•5 is the base of the first factor andits exponent is3

•b is the base of the second factorand its exponentis 4

•the expanded form is 5•5•5•b•b•b•b

Practice 1.6.101. 3•3•3•3

• the base is 3• there are 4 factors so the exponent should be 4• written in exponential notation we have 34

2. xxyyyyy• one base is x, the other base is y• there are 2 factors of x so its exponent should be 2• there are 5 factors of y so its exponent should be 5


• written in exponential notation we have x2y5

3. 8•8•8•x•x•x•x• one base is 8, the other base is x• there are 3 factors of 8 so its exponent should be 3• there are 4 factors of x so its exponent should be 4• written in exponential notation we have 83x4

4. 2•2•2•2•2•2•y• one base is 2, the other base is y• there are 6 factors of 2 so its exponent should be 6• there is 1 factor of y so its exponent should be 1• written in exponential notation we have 26y

5. (-3)(-3)(-3)(-3)mmmmmnnn• one base is (-3), another is m and the other base is n• there are 4 factors of (-3) so its exponent should be 4• there are 5 factors of m so its exponent should be 5• there are 3 factors of n so its exponent should be 3• written in exponential notation we have(-3)4m5n3

Exercise Set 1.61. The box is 0 since 0 + -18 = -18

3. The box is 1 since 8•1 = 8

5. The box is 7.7 since 7.7 + (-7.7) + 6z = 0 + 6z=6z

7. The box is − 4 5

since − 4 5

+ 4 5

+ 12n =

0 + 12n = 12n

9. The box is 6 5

since

6 5

• ( 5 6

) • ( 33) = 1 • ( 33) = 33

11. 7 - 13 = 7 + (-13) = -13 + 7

13. 3x - 2x2 + 5 = 3x + (-2x2) + 5=-2x2 + 3x + 5

15. 12•26 = 26•12

17. r3 = 3r

19. (x - 5)9 = 9(x - 5)

21. (2x + 3)(x + 9) = (x + 9)(2x + 3)

23.1 16

+ ( 1516

+ 10) = ( 1 16

+ 1516

) + 10

25. (-2 + 6y) + 8y = -2 + (6y + 8y)

27. (6•2)•3 = 6•(2•3)

29.7 9

( 9 y ) = ( 7 9

• 9 ) y

31. 12(y + 3) = 12•y + 12•3 = 12y + 36

33. 8(x - 2) = 8•x - 8•2 = 8x - 16

35. -3(y - 5x) = -3•y - (-3)(5x) = -3y - (-15x) = -3y + 15x

37. -2(x + y - 6) = -2•x + -2•y -(-2)(6)=-2x -2y -(-12)= -2x - 2y +12

39. 3(y - 2x + 6) = 3•y -3•2x + 3•6 = 3y - 6x + 18

41. 3h + 6h = h(3 + 6)

43. 8x - 2x = x(8-2)

45. 2y - 3y + 5y = y(2 -3 +5)

47. rs - rt = r(s - t)

49. 73

• 7 is the base• 3 is the exponent• 73=7•7•7 = 343

51. 2 •43

• 2 is a base whose exponent is 1•4 is a base whose exponent is 3• 2 •43 =2•4•4•4 = 128


53. (2•4)3

• 2•4 is the base whose exponent is3

• (2 •4)3 =(2 •4)(2 •4)(2 •4) = • 83=512

55. -62

• 6 is the base whose exponent is 2• -62 =-6•6 = -36

57. 9p4

• 9 is a base whose exponent is 1•p is a base whose exponent is 4• The expanded form is 9•p•p•p•p

59. (-8r)3

• -8 is a base whose exponent is 3• The expanded form is (-8r)(-8r)(-8r)

61. 42q4

• 4 is a base whose exponent is 2•q is a base whose exponent is 4• The expanded form is 4•4•q•q•q•q

63. -r3

• r is a base whose exponent is 3• The expanded form is -r•r•r

65. -(-t)4

• -t is a base whose exponent is 4• The expanded form is -(-t)(-t)(-t)(-

t)

67. 11•11•11• 11 is the base with three factors so the

exponent should be 3• exponential notation is 113

69. pppppprrr• p is a base with six factors so the exponent should be 6• r is a base with three factors so the exponent should be 3• exponential notation is p6r3

71. (zzz)(zzzz)• In the first parenthesis z is the

base with 3 factors so the exponent should be 3.

• In the second parenthesis z is the base with 4 factors so the exponent should be 4.• exponential notation is (z3)(z4)

73. -5•5•5•5• 5 is a base with 4 factors so the exponent should be 4• exponential form -54

75. (-3)•(-3)•(-3)•(-3)• (-3) is a base with 4 factors so the exponent should be

4• exponential form (-3)4

77. − 5 4

78.5 x 2

= 5 2

x

79. 12x

80. Let x = the number

x÷6 or x 6

81. Let x = the number x - 4

82. 2( 8 9

)

83. 4[12 - (6 -4)2] = 4[12 - (2)2] = 4[12 - 4] = 4(8) = 32

84.6480

• 3015

= 4 • 165 • 16

• 2 • 151 • 15

=

4 5

• 2 1

= 8 5

= 1 3 5

85.

15214549

= 1521

÷ 4549

= 1521

• 4945

=

153 • 7

• 7 • 7 3 • 15

= 1 3

• 7 3

= 7 9


86.5 6

+ 7 8

= 2024

+ 2124

= 4124

= 1 1724

87. 12 - |-5| + 2 = 12 - 5 + 2 = 7 + 2 = 9

88. For x = -2 and y = -312 - y - xy = 12 - (-3) - (-2)(-3) = 12 + 3 - 6 = 9

89.4812

= 4 • 1212

= 4

90. <--|----|-----|----|-----|----|----|----•----|-----|-----|->

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Section 1.7 Simplifying Algebraic Expressions

Practice 1.7.11. 2x2, 3x2 are like terms

2. 5y, 3 2

y are like terms

3. 2x2y4,x 2 y 4

π , 7x2y4 are like terms

0.07y3x2, -x2y3 are like terms

-2x3y2 is not like any other terms

4. 5x5y2, 1.25y2x5 are like terms-9x2y5, 1.1x2y5 are like terms10x3y2 is not like any other term

5. None of these are like any otherterm.

6. 7r2s, 8sr2 are like terms-4rs2, 15r2s 2, 6r2, 2s2 are not likeany other terms

Practice 1.7.21. 6m + 3m = 9m

2. -6r + 8r = 2r

3. 7w - 15w = 7w + (-15w) = -8w

4. 4s - s = 4s - 1s = 3s

5. -8x3 + 5x3 - 12x3 = -3x3 + (-12x3) = -15x3

6. -15a + 7a - 14a = -8a + (-14a) = -22a

7. 8x - 2y + 3y - x = 8x - 1x - 2y + 3y =7x + y

8. 9xy2 + 3x2y - 7xy2 + 2x2y = 2xy2 + 5x2y

Practice 1.7.31. 4x + (2x - 12x) = 4x + (-10x) =-6x

2. -8y + (3y - 6y) = -8y + (-3y) = -11y

3. 5w - (4w - 6w) = 5w - (-2w) =5w + 2w = 7w

4. -7m - (4m - 3m) = -7m - 1m = -8m


5. (6x + 4x) - (3x + 7x) = 10x - 10x = 0

6. (5m + 7m) - (m - 8m) = 12m - (-7m) =12m + 7m = 19m

7. (5r - r) - (9r - 6r) = 4r - 3r = r

8. (5n + 2n) - (6n - 15n) = 7n - (-9n) =7n + 9n = 16n

Practice 1.7.41. 5x + 3(x + 3) = 5x + 3x + 9 = 8x + 9

2. 3m + 6(m+6) = 3m + 6m + 36 = 9m + 36

3. 4y + 3(2y - 7) = 4y + 3(2y) - 3(7) = 4y + 6y - 21 = 10y - 21

4. 2(3x) = 6x

5. -12(3x + 8) = -12(3x) + (-12)(8) = -36x - 96

6. -5(4x + 4) = -5(4x) + (-5)(4) = -20x -20

7. -3(2x - 5) = -3(2x) -(-3)(5) = -6x - (-15) = -6x + 15

8. 6x - 3(5x) = 6x - 15x = -9x

9. 1.5(6x + 10) = 1.5(6x) + 1.5(10) = 9x + 15

10.1 3

( 6 x − 9 ) = 1 3

( 6 x ) − 1 3

( 9 ) =

2x - 3

Practice 1.7.51. -(-4x) = 4x

2. -(6m3 + 3m2 - 4m) = -6m3 - 3m2 + 4m

3. -(-3n2 - 2n + 5) = 3n2+ 2n - 5

4. -(-2x2 - 3x - 5) = 2x2 + 3x + 5

Practice 1.7.6

1. 5x - 6(3x + 7) = 5x - 18x - 42 = -13x - 42

2. 2n - 7(4n + 8) = 2n - 28n - 56 = -26n - 56

3. 3r - 12(5 - 2r) = 3r - 60 + 24r = 27r - 60

4. 5t - 3(7 - 4t) = 5t - 21 + 12t = 17t - 21

5. 4x -(2x2 + 3x - 6) = 4x - 2x2 - 3x + 6= -2x2 + x + 6

6. 4n -(-3n2 - 2n + 5) = 4n + 3n2 + 2n - 5 = 3n2 + 6n - 5

7. -(4x + 5) - (3x - 3) =-4x - 5- 3x + 3= -7x - 2

8. -(2y + 9) - (4y - 2) = -2y - 9 - 4y + 2-6y - 7

Practice 1.7.71. x - 6[3x + (7x - 1) + 2] =

x - 6[3x + 7x - 1 + 2] =x - 6[10x + 1]=x - 60x - 6 = -59x - 6

2. 5x - 3[2 + 2(x + 1) + 4x] = 5x - 3[2 + 2x + 2 + 4x] = 5x - 3[6x + 4] = 5x - 18x -12 = -13x - 12

3. 2x - [2x - (2x + 3) - 4] = 2x - [2x - 2x - 3 - 4] = 2x - [-7]=2x + 7

4. 5x - [5x + 2(x + 3) + 3]=5x - [5x + 2x + 6 + 3] = 5x - [7x + 9] = 5x - 7x - 9 = -2x -9

5. 5x + [3x - (2x + 4x)] = 5x + [3x - 6x] = 5x + [-3x] = 2x

6. 8w + [4w - (5w - 7w)] = 8w + [4w-(-2w)]=8w + [4w + 2w] = 8w + 6w = 14w

7. -3[2x + 4(x - 1)] = -3[2x + 4x - 4]=-3[6x - 4] = -18x + 12

8. -2[5x + 3(x - 4)] = -2[5x + 3x - 12]=


-2[8x - 12] = -16x + 24

9. 5[3x - 4(2x + 1)] = 5[3x - 8x - 4]=5[-5x - 4] = -25x - 20

10. 2[5x - 3(3x + 1)] = 2[5x - 9x - 3] = 2[-4x - 3] = -8x - 6

Exercise Set 1.7

1. -4m + 7m = 3m

3. -5mn + 2mn - 13mn = -3mn - 13mn=- 3mn +(-13mn ) = -16mn

5. 1.2xy + 2.4xy - 5.7xy = 3.6xy + (-5.7xy) = -2.1xy

7. x − 1 2

x + 3 4

x =

4 4

x − 2 4

x + 3 4

x = 5 4

x = 1 1 4

x

9. 8x + (5x - 11x) = 8x +(5x + (-11x))=8x + (-6x ) = 2x

11. -2w+(4w - 5w)=-2w+ (4w +(-5w))= -2w + (-1w) = -3w

13. (-5x + 3x) - (8x + 12x) =- 2x- 20x = -2x + (-20x) = -22x

15. 5x+[3x-(2x +4x)]= 5x+[3x-6x]=5x +[3x + (-6x)] = 5x + (-3x) = 2x

17. -12m +7(2m - 6) = -12m + 14m -42=2m - 42

19. -8(2y -5) = -8•2y - (-8)(5)=-16y - (-40) = -16y + 40

21. 2.5(4x + 6) = 2.5(4x) +2.5(6) = 10x + 15

23. − 1 2

( 6 x − 2 ) =

− 1 2

( 6 x ) − ( − 1 2

) ( 2 ) =

− 3 x − ( − 1 ) = − 3 x + 1

25. -(-12y) = 12y

27. -(x2 - 3x + 5) = -x2 + 3x - 5 29. 2w -3(2w + 2) = 2w - 6w - 6 =

-4w - 6

31. 3r -(2r2 + 5r - 3)=3r -2r2 - 5r + 3=-2r2 - 2r + 3

33. (2x + 5) -3(x - 7) = 2x + 5-3x +21=-x + 26

35. (4t2 + 3t - 8)-2(3t2 + 5t -5)= 4t2 + 3t - 8 - 6t2 - 10t + 10 = -2t2 -7t+ 2

37. x + 4(2x + 6(x + 1) - 8)=x + 4(2x + 6x + 6 - 8) =x + 4(8x - 2) = x + 32x - 8 = 33x - 8

39. x -2(5x + 3(x - 8) + 7) = x - 2(5x + 3x - 24 + 7) = x - 2(8x - 17) = x - 16x + 34=-15x + 34

41. 3x - 2(4x - 3(x - 5) + 6) =3x - 2(4x - 3x + 15 + 6) = 3x - 2(x + 21) = 3x - 2x - 42= x - 42

43. -2[-4x -(1-4x)] = -2[-4x - 1 +4x]=-2[-1] = 2

45. − 1 6

46.1 6

w

47. 0 has no opposite

48. 7x + 2x = x(7 + 2)

49. 23 • 8÷4 = 8 • 8÷4 = 64 ÷4 = 16

50. 39 - 2[3 + 22(5 - 1)]=39 - 2[3 + 22(4)]= 39 - 2[3 + 4(4)]=39 - 2[3 + 16] = 39 - 2[19] = 39 - 38 = 1


51.4228

• 6 24

= 2 • 3 • 7 2 • 2 • 7

• 6

4 • 6 =

3 2

• 1 4

= 3 8

52. 8 - |-6|÷2 + 3 •4 = 8 - 6÷2 + 3•4 = 8 - 3 + 12 = 5 + 12 = 17

53. Three percent of 9 = 0.03(9)

54. Let n = the number

n÷3 or n 3

55. Let n= the numbern - 5

56. For x = -2 and y = -33x - y = 3(-2) - (-3) = -6 + 3= -3

57. The commutative property ofaddition

58. 3w5

• One base is 3 and its exponent is 1• The other base is w and its exponent is 5

Chapter 1 Review Exercises

1. -8rs is a product whose factors are--8,r and s

2. 3w + 5p is a sum whose terms are3w and 5p. 3w is a

product whose factors are 3and w. 5p is a product

whose factors are 5 and p.

3. 392 = 2•2•2•7•7 = 23•72

4. 572 = 2•2•11•13 = 22•11•13

5.128312

= 8 • 168 • 39

= 1639

6.11848

= 2 • 592 • 24

= 5924

7.7 5

8. − 1 2

9.5

3 m

10.5 9

b

11. (18 - 5•2)2 = (18 - 10)2 = 82 = 64


12. 72/32 •5 = 72/9 •5 = 8•5 = 40

13. 10 - [(8 - 5)2 - 12/(3•4)]10 - [(3)2 - 12/12]= 10 - [9 - 1]=10 - 8 = 2

14. 8 + 2[35 - 2(24/6)2 - 3] = 8 + 2[35 - 2(4)2 - 3]=8 + 2[35 - 2(16) - 3]=8 + 2[35 - 32 - 3] = 8 + 2[0] = 8 + 0 = 8

15.4062

• 2430

= 2 • 2 • 102 • 31

• 3 • 8 3 • 10

=

2 31

• 8 1

= 1631

16.1870

• 6030

= 2 • 9 2 • 35

• 2 • 3030

=

9 35

• 2 1

= 1835

17.9012

÷ 1015

= 9012

• 1510

=

3 • 3 • 103 • 4

• 3 • 5 10

= 3 4

• 3 • 5 1

=

454

= 111 4

18.4556

÷ 2735

= 4556

• 3527

=

5 • 9 7 • 8

• 5 • 7 3 • 9

= 5 8

• 5 3

= 2524

= 1 1 24

19.9 16

+ 5 12

= 9 • 3 16• 3

+ 5 • 4 12• 4

=

2748

+ 2048

= 4748

20.6 25

+ 7 30

+ 8 75

=

6 • 6 25• 6

+ 7 • 5 30• 5

+ 8 • 2 75• 2

=

36150

+ 35150

+ 16150

= 87150

= 2950

21.4 • 3 2 − 2 3

2 • 6 − 5 =

4 • 9 − 8 12 − 5

=

36 − 8 7

= 287

= 4

22. 12 − 30 −

1 3

• 12

3 − 2 5

=

12 − 30 − 4 155

− 2 5

= 12 − 26135

=

12 − 26÷ 135

= 12 − 261

• 5 13

=

12 - 2•5 = 12- 10 = 2

23.3 2

+ 185

( 5 6

) 2 =

3 2

+ 185

( 5 6

) ( 5 6

) =

3 2

+ 3 • 6 5

( 5

2 • 3 ) (

5 6

) =

3 2

+ 1 1

( 1 2

) ( 5 1

) = 3 2

+ 5 2

=

8 2

= 4

24.30 + 5 2 • 2 3 • 7 − 5

= 30 + 25• 2

21 − 5 =

30 + 5016

= 8016

= 5

25. 12 - |-8| ÷4 + 2 •5 = 12 - 8÷4 + 10=12 - 2+ 10 = 10 + 10 = 20


26. |2 - 11| • 2 - |12| ÷3= |-9| • 2 -12÷3=9 • 2- 4 = 18 - 4 = 14

27. -34 = - 3•3•3•3 = -81

28. -(x - 5x) + 3x = -(-4x) + 3x = 4x + 3x = 7x

29. 8x - 12x2 - 13x + 19x2 = 7x2 - 5x

30. -(7x - x) + 3x = -(6x) + 3x = -6x + 3x = -3x

31. 4x - (3x + 2) = 4x - 3x - 2= x - 2

32. -(3x - 2) - (5 - x) = -3x + 2 -5 + x=-2x - 3

33. z + 3[4z - 3(2z + 1)] = z+3[4z - 6z -3] = z + 3[-2z - 3]=z + (-6z) - 9= -5z -9

34. m + 4[5m -2(5m + 1)]=m + 4[5m - 10m -2] = m +4[-5m -2]= m + (-20m) - 8 = -19m - 8

35. <--|--|--|--|--•--|--|--|•-|--|--|•|--|--|--|•-|--|--|--|--|->

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

9

36. 12.6

37. − 5 8

38. -3.5, -2, 0.5, 6

39.<--|---|---|---|--•|---|---|---|---|---|---|---|---|---|---|---|---|>

-800 -700 -600 -500 -400 -300 -300 -100 0 100 200 300 400 500 600 700 800

40. <--|----|----|----|----|----|----|----|----|---•---|-->

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0

41. 0.08 •12 = 0.96

42. Let x = the number 2x

43. Let x= the number

x÷(-8) or x

− 8

44. Let x = the number

2x÷3 or 2 x 3

45. Let x = the numberx + 5

46. For x = -2 and y = -38 - (x - y) = 8 - (-2 -(-3)) = 8 - (-2 + 3) = 8 - 1 = 7

47. For x = -2 and y = -3y - xy2 - 3x = -3 -(-2)(-3)2 - 3(-2)=-3 - (-2)(9) -(-6) = -3 - (-18) + 6=-3 + 18 + 6 = 15 + 6 = 21

48. Let x = the numberx - (-2) = x + 2

49. Let x = the numberx - 4

50. Let x = the number1 2

x + 3

51. The Commutative Property ofAddition5 - x = 5 + (-x) = -x + 5

52. Multiplication Property of one

53. 2x + 7x = x(2 + 7)

54. 3a - a = a(3 - 1)


Chapter 1 Test

1. 8 + x is a sum whose terms are 8and x

2. <--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

3. -5.5, -2, 0, 3.5

4. <--|-----|-----|----•-----|-----|-----|-----|-----|-->

-100 -25 50 125 200 275 350 425 500

5. - 1 3


6. − 3 2

7.1 2

w

8. 5b + b = b(5 + 1)

9. 459 = 3•3•3•17 = 33 •17

10. Let x = the number1 3

x + 2

11.56112

= 56

2 • 56 =

1 2

12. For x = -2 and y = -32x3 - xy2 = 2(-2)3 - (-2)(-3)2 = 2(-8) - (-2)(9) = -16 - (-18) = -16 + 18 = 2

13. The Commutative Property ofAddition

14. Associative Property of Addition

15. 36/22 • (5 - 2) = 36/4 • 3 = 9 •3 = 27

16.2039

• 5125

= 4 • 5 3 • 13

• 3 • 175 • 5

=

4 13

• 175

= 6865

= 1 3 65

17.2130

÷ 4220

= 2130

• 2042

=

213 • 10

• 2 • 102 • 21

= 1 3

• 1 1

= 1 3

18. -26 = -2•2•2•2•2•2= -64

19.8 15

+ 3 40

= 8 • 8 15• 8

+ 3 • 3 40• 3

=

64120

+ 9

120=

73120

20. 20 −

7 5

• 20

2 − 1 4

=

20 − 7 • 4

8 4

− 1 4

= 20 − 287 4

=

20 − 28÷ 7 4

= 20 − 28• 4 7

=

20 − 4 • 4 = 20 − 16 = 4

21. 9 − − 16 ÷ 8 + 12 / 3 = 9 -16÷8 + 4=9 - 2 + 4 = 7 + 4 = 11

22. 12x3 - 3x2 + 5x3 - 4x2 = 17x3 - 7x2

23. 6x - (3x - 8x) + 2x = 6x - 3x + 8x + 2x = 13x

24. 9x - 3(3x - 2) = 9x - 9x + 6 = 6

25. 6r -2[r +4(r - 3) + 1] = 6r - 2[r + 4r - 12 +1] = 6r -2[5r - 11]=6r - 10r + 22 = -4r + 22

26. 26 - 3(12 - (5 - 2)2 + 36/12) = 26 - 3(12 - (3)2 + 3)= 26 - 3(12 - 9 + 3) = 26 - 3(6) = 26 - 18 = 8


chapter 1 the real number system = 9 - (12 - [8]) = 9 - (4 ... · pdf file4 • 9 + 1...

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