13.1.8: Chapter 8

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Be Prepared

8.1

ⓐ 8; ⓑ −1; ⓒ-729

8.2

3.85

8.3

ⓐ $x^{6}$ ; ⓑ $y^{6}$ ; ⓒ $z^{12}$

8.4

$x^{5}$

8.5

$\frac{1}{y^{8}}$

8.6

$n^{12}$

8.7

$\frac{53}{60}$

8.8

$64 x^{6} y^{15}$

8.9

$\frac{1}{125}$

8.10

$2 x^{2} + 11 x - 8$

8.11

$8 + 2 a - a^{2}$

8.12

$81 - 90 y + 25 y^{2}$

8.13

$\frac{5}{8}$

8.14

$x^{6}$

8.15

$49 - 9 x^{2}$

8.16

$y^{2} - 6 y + 9$

8.17

$x = \frac{5}{2}$

8.18

$n = 2 or n = 4$

8.19

$(− \infty, \frac{1}{2}]$

8.20

$f (2) = 2, f (−1) = −7, f (0) = −4$

8.21

$domain: [0, \infty); range: [0, \infty)$

8.22

ⓐ $−4, 0. \bar{5}, \frac{7}{3}, 3, \sqrt{81};$ ⓑ $\sqrt{7};$ ⓒ $−4, - \sqrt{7}, 0. \bar{5}, \frac{7}{3}, 3, \sqrt{81}$

8.23

$2 x^{2} - x - 15$

8.24

$\frac{\sqrt{5} (\sqrt{15})}{2}$

Try It

8.1

ⓐ $−8$ ⓑ 15

8.2

ⓐ 10 ⓑ $−11$

8.3

ⓐ not a real number ⓑ $−9$

8.4

ⓐ $−7$ ⓑ not a real number

8.5

ⓐ 3 ⓑ 4 ⓒ 3

8.6

ⓐ 10 ⓑ 2 ⓒ 4

8.7

ⓐ $−3$ ⓑ not real ⓒ $−2$

8.8

ⓐ $−6$ ⓑ not real ⓒ $−4$

8.9

ⓐ $6 < \sqrt{38} < 7$
ⓑ $4 < \sqrt[3]{93} < 5$

8.10

ⓐ $9 < \sqrt{84} < 10$
ⓑ $5 < \sqrt[3]{152} < 6$

8.11

ⓐ $\approx 3.32$ ⓑ $\approx 4.14$
ⓒ $\approx 3.36$

8.12

ⓐ $\approx 3.61$ ⓑ $\approx 4.38$
ⓒ $\approx 3.15$

8.13

ⓐ $| b |$ ⓑ w ⓒ $| m |$ ⓓ q

8.14

ⓐ $| y |$ ⓑ p ⓒ $| z |$ ⓓ q

8.15

ⓐ $| y^{9} |$ ⓑ $z^{6}$

8.16

ⓐ $m^{2}$ ⓑ $| b^{5} |$

8.17

ⓐ $| u^{3} |$ ⓑ $v^{5}$

8.18

ⓐ $c^{4}$ ⓑ $d^{4}$

8.19

ⓐ $8 | x |$ ⓑ $−10 | p |$

8.20

ⓐ $13 | y |$ ⓑ $−11 | y |$

8.21

ⓐ $3 x^{9}$ ⓑ $3 | q^{7} |$

8.22

ⓐ $5 q^{3}$ ⓑ $3 q^{5}$

8.23

ⓐ $10 | a b |$ ⓑ $12 p^{6} q^{10}$
ⓒ $2 x^{10} y^{4}$

8.24

ⓐ $15 | m n |$ ⓑ $13 | x^{5} y^{7} |$
ⓒ $3 w^{12} z^{5}$

8.25

$4 \sqrt{3}$

8.26

$3 \sqrt{5}$

8.27

ⓐ $12 \sqrt{2}$ ⓑ $3 \sqrt[3]{3}$ ⓒ $2 \sqrt[4]{4}$

8.28

ⓐ $12 \sqrt{3}$ ⓑ $5 \sqrt[3]{5}$ ⓒ $3 \sqrt[4]{9}$

8.29

ⓐ $b^{2} \sqrt{b}$ ⓑ $| y | \sqrt[4]{y^{2}}$ ⓒ $z \sqrt[3]{z^{2}}$

8.30

ⓐ $p^{4} \sqrt{p}$ ⓑ $y \sqrt[5]{y^{3}}$
ⓒ $q^{2} \sqrt[6]{q}$

8.31

ⓐ $4 y^{2} \sqrt{2 y}$ ⓑ $3 p^{3} \sqrt[3]{2 p}$
ⓒ $2 q^{2} \sqrt[4]{4 q^{2}}$

8.32

ⓐ $5 a^{4} \sqrt{3 a}$ ⓑ $4 m^{3} \sqrt[3]{2 m^{2}}$
ⓒ $3 | n | \sqrt[4]{2 n^{3}}$

8.33

ⓐ $7 | a^{3} | b^{2} {\sqrt{2 a b}}^{}$
ⓑ $2 x y \sqrt[3]{7 x^{2} y}$ ⓒ $2 | x | y^{2} \sqrt[4]{2 x}$

8.34

ⓐ $6 m^{4} | n^{5} | \sqrt{5 m n}$
ⓑ $2 x^{2} y \sqrt[3]{9 y^{2}}$ ⓒ $2 | x y | \sqrt[4]{5 x^{3}}$

8.35

ⓐ $−4$ ⓑ $no real number$

8.36

ⓐ $−5 \sqrt[3]{5}$ ⓑ no real number

8.37

ⓐ $5 + 5 \sqrt{3}$ ⓑ $2 - \sqrt{3}$

8.38

ⓐ $2 + 7 \sqrt{2}$ ⓑ $2 - \sqrt{5}$

8.39

ⓐ $\frac{5}{4}$ ⓑ $\frac{3}{5}$ ⓒ $\frac{2}{3}$

8.40

ⓐ $\frac{7}{9}$ ⓑ $\frac{2}{5}$ ⓒ $\frac{1}{3}$

8.41

ⓐ $| a |$ ⓑ $| x |$ ⓒ $y^{3}$

8.42

ⓐ $x^{2}$ ⓑ $m^{2}$ ⓒ $n^{2}$

8.43

$\frac{2 | p | \sqrt{6 p}}{7}$

8.44

$\frac{2 x^{2} \sqrt{3 x}}{5}$

8.45

ⓐ $\frac{4 | m | \sqrt{5 m}}{| n^{3} |}$ ⓑ $\frac{3 c^{3} \sqrt[3]{4 c}}{d^{2}}$
ⓒ $\frac{2 x^{2} \sqrt[4]{5 x^{2}}}{| y |}$

8.46

ⓐ $\frac{3 u^{3} \sqrt{6 u}}{v^{4}}$ ⓑ $\frac{2 r \sqrt[3]{5}}{s^{2}}$
ⓒ $\frac{3 | m^{3} | \sqrt[4]{2 m^{2}}}{| n^{3} |}$

8.47

ⓐ $\frac{5 | y | \sqrt{x}}{6}$ ⓑ $\frac{2 x y \sqrt[3]{y^{2}}}{3}$
ⓒ $\frac{| a b | \sqrt[4]{a}}{2}$

8.48

ⓐ $\frac{2 | m | \sqrt{3}}{5 | n^{3} |}$ ⓑ $\frac{3 x y \sqrt[3]{x^{2}}}{5}$
ⓒ $\frac{2 | a b | \sqrt[4]{a^{2}}}{3}$

8.49

ⓐ $7 z^{2}$ ⓑ $−5 \sqrt[3]{2}$
ⓒ $3 | m | \sqrt[4]{2 m^{2}}$

8.50

ⓐ $8 m^{4}$ ⓑ $−4$ ⓒ $3 | n | \sqrt[4]{2}$

8.51

ⓐ $\sqrt{t}$ ⓑ $\sqrt[3]{m}$ ⓒ $\sqrt[4]{r}$

8.52

ⓐ $\sqrt[6]{b}$ ⓑ $\sqrt[5]{z}$ ⓒ $\sqrt[4]{p}$

8.53

ⓐ ${(10 m)}^{\frac{1}{2}}$ ⓑ ${(3 n)}^{\frac{1}{5}}$
ⓒ $3 {(6 y)}^{\frac{1}{4}}$

8.54

ⓐ ${(3 k)}^{\frac{1}{7}}$ ⓑ ${(5 j)}^{\frac{1}{4}}$
ⓒ $8 {(2 a)}^{\frac{1}{3}}$

8.55

ⓐ 6 ⓑ 2 ⓒ 2

8.56

ⓐ 10 ⓑ 3 ⓒ 3

8.57

ⓐ No real solution ⓑ $−8$
ⓒ $\frac{1}{8}$

8.58

ⓐ No real solution ⓑ $−4$
ⓒ $\frac{1}{4}$

8.59

ⓐ $x^{\frac{5}{2}}$ ⓑ ${(3 y)}^{\frac{3}{4}}$ ⓒ ${(\frac{2 m}{3 n})}^{\frac{5}{2}}$

8.60

ⓐ $a^{\frac{2}{5}}$ ⓑ ${(5 a b)}^{\frac{5}{3}}$
ⓒ ${(\frac{7 x y}{z})}^{\frac{3}{2}}$

8.61

ⓐ 9 ⓑ $\frac{1}{729}$ ⓒ $\frac{1}{8}$

8.62

ⓐ 8 ⓑ $\frac{1}{9}$ ⓒ $\frac{1}{125}$

8.63

ⓐ $−64$ ⓑ $- \frac{1}{64}$ ⓒ not a real number

8.64

ⓐ $−729$ ⓑ $- \frac{1}{729}$ ⓒ not a real number

8.65

ⓐ $x^{\frac{3}{2}}$ ⓑ $x^{8}$ ⓒ $\frac{1}{x}$

8.66

ⓐ $y^{\frac{11}{8}}$ ⓑ $m^{2}$ ⓒ $\frac{1}{d}$

8.67

ⓐ $8 x^{\frac{1}{5}}$ ⓑ $x^{\frac{1}{2}} y^{\frac{1}{3}}$

8.68

ⓐ $729 n^{\frac{3}{5}}$ ⓑ $a^{2} b^{\frac{2}{3}}$

8.69

ⓐ $m^{2}$ ⓑ $\frac{5 n}{m^{\frac{1}{4}}}$

8.70

ⓐ $u^{3}$ ⓑ $3 x^{\frac{1}{5}} y^{\frac{1}{3}}$

8.71

ⓐ $− \sqrt{2}$ ⓑ $11 \sqrt[3]{x}$
ⓒ $3 \sqrt[4]{x} - 5 \sqrt[4]{y}$

8.72

ⓐ $−4 \sqrt{3}$ ⓑ $8 \sqrt[3]{y}$
ⓒ $5 \sqrt[4]{m} - 2 \sqrt[3]{m}$

8.73

ⓐ $−2 \sqrt{7 x}$ ⓑ $− \sqrt[4]{5 x y}$

8.74

ⓐ $− \sqrt{3 y}$ ⓑ $3 \sqrt[3]{7 m n}$

8.75

ⓐ $9 \sqrt{2}$ ⓑ $2 \sqrt[3]{2}$ ⓒ $\sqrt[3]{3}$

8.76

ⓐ $7 \sqrt{3}$ ⓑ $−10 \sqrt[3]{5}$ ⓒ $−3 \sqrt[3]{2}$

8.77

ⓐ $− m^{3} \sqrt{2 m}$ ⓑ $x^{2} \sqrt[3]{5 x}$

8.78

ⓐ $− p \sqrt{3 p}$
ⓑ $4 y \sqrt[3]{4 y^{2}} - 2 n \sqrt[3]{4 n^{2}}$

8.79

ⓐ $12 \sqrt{15}$ ⓑ $−18 \sqrt[3]{4}$

8.80

ⓐ $27 \sqrt{2}$ ⓑ $−36 \sqrt[3]{2}$

8.81

ⓐ $288 x^{3} \sqrt{5}$ ⓑ $8 y \sqrt[4]{6 y^{2}}$

8.82

ⓐ $144 y^{2} \sqrt{5 y}$ ⓑ $−36 a \sqrt[4]{3 a}$

8.83

ⓐ $18 + \sqrt{6}$ ⓑ $−2 \sqrt[3]{4} - 2 \sqrt[3]{3}$

8.84

ⓐ $−40 + 4 \sqrt{2}$ ⓑ $−3 - \sqrt[3]{18}$

8.85

ⓐ $−66 + 15 \sqrt{7}$
ⓑ $\sqrt[3]{x^{2}} - 5 \sqrt[3]{x} + 6$

8.86

ⓐ $41 - 14 \sqrt{11}$
ⓑ $\sqrt[3]{x^{2}} + 4 \sqrt[3]{x} + 3$

8.87

$1 + 9 \sqrt{21}$

8.88

$−12 - 20 \sqrt{3}$

8.89

ⓐ $102 + 20 \sqrt{2}$ ⓑ $55 + 6 \sqrt{6}$

8.90

ⓐ $41 - 12 \sqrt{5}$
ⓑ $121 - 36 \sqrt{10}$

8.91

$−11$

8.92

$−159$

8.93

ⓐ $\frac{5 s}{8}$ ⓑ $\frac{2}{a}$

8.94

ⓐ $\frac{5 q^{2}}{6}$ ⓑ $\frac{2}{b}$

8.95

ⓐ $\frac{9 x^{2}}{y^{2}}$ ⓑ $\frac{−4 x}{y}$

8.96

ⓐ $\frac{10 n^{3}}{m}$ ⓑ $\frac{−3 p}{q^{2}}$

8.97

$4 x y \sqrt{2 x}$

8.98

$4 a b \sqrt{3 b}$

8.99

ⓐ $\frac{5 \sqrt{3}}{3}$ ⓑ $\frac{\sqrt{6}}{8}$ ⓒ $\frac{\sqrt{2 x}}{x}$

8.100

ⓐ $\frac{6 \sqrt{5}}{5}$ ⓑ $\frac{\sqrt{14}}{6}$ ⓒ $\frac{\sqrt{5 x}}{x}$

8.101

ⓐ $\frac{\sqrt[3]{49}}{7}$ ⓑ $\frac{\sqrt[3]{90}}{6}$ ⓒ $\frac{5 \sqrt[3]{3 y^{2}}}{3 y}$

8.102

ⓐ $\frac{\sqrt[3]{4}}{2}$ ⓑ $\frac{\sqrt[3]{150}}{10}$ ⓒ $\frac{2 \sqrt[3]{5 n^{2}}}{5 n}$

8.103

ⓐ $\frac{\sqrt[4]{27}}{3}$ ⓑ $\frac{\sqrt[4]{12}}{4}$ ⓒ $\frac{3 \sqrt[4]{5 x^{3}}}{5 x}$

8.104

ⓐ $\frac{\sqrt[4]{125}}{5}$ ⓑ $\frac{\sqrt[4]{14}}{4}$
ⓒ $\frac{2 \sqrt[4]{4 x^{3}}}{x}$

8.105

$- \frac{3 (1 + \sqrt{5})}{4}$

8.106

$\frac{4 + \sqrt{6}}{5}$

8.107

$\frac{\sqrt{5} (\sqrt{x} - \sqrt{2})}{x - 2}$

8.108

$\frac{\sqrt{10} (\sqrt{y} + \sqrt{3})}{y - 3}$

8.109

${\frac{(\sqrt{p} + \sqrt{2})}{p - 2}}^{2}$

8.110

${\frac{(\sqrt{q} - \sqrt{10})}{q - 10}}^{2}$

8.111

$m = \frac{23}{3}$

8.112

$z = \frac{3}{10}$

8.113

$no solution$

8.114

$no solution$

8.115

$x = 2, x = 3$

8.116

$y = 5, y = 6$

8.117

$x = −6$

8.118

$x = −9$

8.119

$x = 8$

8.120

$x = 6$

8.121

$m = 7$

8.122

$n = 3$

8.123

$a = 63$

8.124

$b = 311$

8.125

$x = 3$

8.126

$x = - \frac{6}{5}$

8.127

$x = 4$

8.128

$x = 9$

8.129

$x = 5$

8.130

$x = 0 x = 4$

8.131

9 seconds

8.132

$3.5$ seconds

8.133

$42.7$ feet

8.134

$54.1$ feet

8.135

ⓐ $f (6) = 4$ ⓑ no value at $x = 0$

8.136

ⓐ $g (4) = 5$ ⓑ no value at $f (−3)$

8.137

ⓐ $g (4) = 2$ ⓑ $g (1) = −1$

8.138

ⓐ $h (2) = 2$
ⓑ $h (−5) = −3$

8.139

ⓐ $f (4) = 2$ ⓑ $f (−1) = 1$

8.140

ⓐ $g (16) = 3$ ⓑ $g (3) = 2$

8.141

$[\frac{5}{6}, \infty)$

8.142

$(− \infty, \frac{4}{5}]$

8.143

$(−3, \infty)$

8.144

$(5, \infty)$

8.145

$(− \infty, \infty)$

8.146

$(− \infty, \infty)$

8.147

ⓐ domain: $[−2, \infty)$
ⓑ

$The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 6. The y-axis runs from 0 to 8. The function has a starting point at (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).$

8.148

ⓐ domain: $[2, \infty)$
ⓑ

$The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 6. The function has a starting point at (2, 0) and goes through the points (3, 1) and (6, 2).$

8.149

ⓐ domain: $(- \infty, \infty)$
ⓑ

$The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 2 to 2. The y-axis runs from negative 2 to 2. The function has a center point at (0, 0) and goes through the points (1, negative 1) and (negative 1, 1).$

8.150

ⓐ domain: $(- \infty, \infty)$
ⓑ

$The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 5. The y-axis runs from negative 3 to 3. The function has a center point at (2, 0) and goes through the points (1, negative 1) and (3, 2).$

8.151

ⓐ $9 i$ ⓑ $\sqrt{5} i$ ⓒ $3 \sqrt{2} i$

8.152

ⓐ $6 i$ ⓑ $\sqrt{3} i$ ⓒ $3 \sqrt{3} i$

8.153

$6 \sqrt{2} i$

8.154

$7 \sqrt{3} i$

8.155

ⓐ $6 + 5 i$ ⓑ $6 - 3 i$

8.156

ⓐ $−2 - 6 i$ ⓑ $2 + 9 i$

8.157

$12 + 20 i$

8.158

$12 - 6 i$

8.159

$−11 - 7 i$

8.160

$−5 - 10 i$

8.161

$−21 - 20 i$

8.162

$9 - 40 i$

8.163

$−14$

8.164

$−54$

8.165

$−12 - 22 \sqrt{3} i$

8.166

$6 + 12 \sqrt{2} i$

8.167

8.168

8.169

109

8.170

8.171

8.172

8.173

$\frac{4}{17} + \frac{16}{17} i$

8.174

$\frac{2}{5} + \frac{4}{5} i$

8.175

$\frac{3}{2} - \frac{3}{2} i$

8.176

$\frac{4}{5} - \frac{2}{5} i$

8.177

$− i$

8.178

$1$

Section 8.1 Exercises

ⓐ 8 ⓑ $−9$

ⓐ 14 ⓑ $−1$

ⓐ $\frac{2}{3}$ ⓑ $−0.1$

ⓐ not real number ⓑ $−17$

ⓐ $−15$ ⓑ not real number

11.

ⓐ 6 ⓑ 4

13.

ⓐ 8 ⓑ 3 ⓒ 1

15.

ⓐ $−2$ ⓑ $not real$ ⓒ $−2$

17.

ⓐ $−5$ ⓑ $not real$ ⓒ $−4$

19.

ⓐ $8 < \sqrt{70} < 9$
ⓑ $4 < \sqrt[3]{71} < 5$

21.

ⓐ $14 < \sqrt{200} < 15$
ⓑ $5 < \sqrt[3]{137} < 6$

23.

ⓐ $\approx 4.36$ ⓑ $\approx 4.46$
ⓒ $\approx 3.14$

25.

ⓐ $\approx 7.28$ ⓑ $\approx 5.28$
ⓒ $\approx 4.61$

27.

ⓐ u ⓑ $| v |$

29.

ⓐ $| y |$ ⓑ $m$

31.

ⓐ $| x^{3} |$ ⓑ $y^{8}$

33.

ⓐ $x^{12}$ ⓑ $| y^{11} |$

35.

ⓐ $x^{3}$ ⓑ $| y^{3} |$

37.

ⓐ $m^{2}$ ⓑ $n^{4}$

39.

ⓐ $7 | x |$ ⓑ $−9 | x^{9} |$

41.

ⓐ $11 m^{10}$ ⓑ $−8 | a |$

43.

ⓐ $2 x^{2}$ ⓑ $2 y^{2}$

45.

ⓐ $6 a^{2}$ ⓑ $2 b^{4}$

47.

ⓐ $12 | x y |$ ⓑ $13 w^{4} | y^{5} |$
ⓒ $2 a^{17} b^{2}$

49.

ⓐ $11 | a b |$ ⓑ $3 c^{4} d^{6}$
ⓒ $4 x^{5} y^{22}$

51.

Answers will vary.

53.

Answers will vary.

Section 8.2 Exercises

55.

$3 \sqrt{3}$

57.

$5 \sqrt{5}$

59.

$7 \sqrt{3}$

61.

$20 \sqrt{2}$

63.

ⓐ $2 \sqrt[4]{2}$ ⓑ $2 \sqrt[5]{2}$

65.

ⓐ $2 \sqrt[4]{4}$ ⓑ $4 \sqrt[3]{4}$

67.

ⓐ $| y^{5} | \sqrt[]{y}$ ⓑ $r \sqrt[3]{r^{2}}$ ⓒ $s^{2} \sqrt[4]{s^{2}}$

69.

ⓐ $n^{10} \sqrt{n}$ ⓑ $q^{2} \sqrt[3]{q^{2}}$
ⓒ $| n | \sqrt[8]{n^{2}}$

71.

ⓐ $5 r^{6} \sqrt[]{5 r}$ ⓑ $3 x \sqrt[3]{4 x^{2}}$
ⓒ $2 | y | \sqrt[4]{3 y^{2}}$

73.

ⓐ $11 | m^{11} | \sqrt[]{2 m}$ ⓑ $3 m^{2} \sqrt[4]{5 m^{2}}$ ⓒ $2 n \sqrt[5]{5 n^{3}}$

75.

ⓐ $7 | m^{3} n^{5} | \sqrt[]{3 m n}$ ⓑ $2 x^{2} y^{2} \sqrt[3]{6 y}$ ⓒ $2 | x y | \sqrt[4]{2 x}$

77.

ⓐ $8 | q r^{3} | \sqrt{3 q r}$ ⓑ $3 m^{3} n^{3} \sqrt[3]{2 n}$ ⓒ $3 a^{2} b^{2} \sqrt[4]{a}$

79.

ⓐ $−6 \sqrt[3]{4}$ ⓑ not real

81.

ⓐ $−2$ ⓑ not real

83.

ⓐ $5 + 2 \sqrt{3}$ ⓑ $5 - \sqrt{6}$

85.

ⓐ $1 + 3 \sqrt{5}$ ⓑ $1 + \sqrt{10}$

87.

ⓐ $\frac{3}{4}$ ⓑ $\frac{2}{3}$ ⓒ $\frac{1}{3}$

89.

ⓐ $\frac{5}{3}$ ⓑ $\frac{3}{5}$ ⓒ $\frac{1}{4}$

91.

ⓐ $x^{2}$ ⓑ $p^{3}$ ⓒ $| q |$

93.

ⓐ $\frac{1}{y^{2}}$ ⓑ $u^{2}$ ⓒ $| v^{3} |$

95.

$\frac{4 | x^{3} | \sqrt{6 x}}{11}$

97.

$\frac{5 m^{2} \sqrt{3 m}}{4}$

99.

$\frac{7 r^{2} \sqrt{2 r}}{10}$

101.

$\frac{2 | q^{3} | \sqrt{7}}{15}$

103.

ⓐ $\frac{5 r^{4} \sqrt{3 r}}{s^{4}}$ ⓑ $\frac{3 a^{2} \sqrt[3]{2 a^{2}}}{b}$
ⓒ $\frac{2 | c | \sqrt[4]{4 c}}{| d |}$

105.

ⓐ $\frac{2 | p^{3} | \sqrt{7 p}}{| q |}$ ⓑ $\frac{3 s^{2} \sqrt[3]{3 s^{2}}}{t}$
ⓒ $\frac{2 | p^{3} | \sqrt[4]{4 p^{3}}}{| q^{3} |}$

107.

ⓐ $\frac{4 | x y |}{3}$ ⓑ $\frac{y^{2} \sqrt[3]{x}}{2}$ ⓒ $\frac{| a b | \sqrt[4]{a}}{2}$

109.

ⓐ $\frac{1}{2 | p q |}$ ⓑ $\frac{2 c d \sqrt[3]{d^{2}}}{5}$
ⓒ $\frac{| m n |}{2}$

111.

ⓐ $\frac{3 p^{4} \sqrt{p}}{| q |}$ ⓑ $2 \sqrt[4]{2}$
ⓒ $2 x \sqrt[5]{2 x}$

113.

ⓐ $5 | m^{3} |$ ⓑ $5 \sqrt[3]{5}$
ⓒ $3 | y | \sqrt[4]{3 y^{2}}$

115.

Answers will vary.

117.

Answers will vary.

Section 8.3 Exercises

119.

ⓐ $\sqrt{x}$ ⓑ $\sqrt[3]{y}$ ⓒ $\sqrt[4]{z}$

121.

ⓐ $\sqrt[5]{u}$ ⓑ $\sqrt[9]{v}$ ⓒ $\sqrt[20]{w}$

123.

ⓐ $x^{\frac{1}{7}}$ ⓑ $y^{\frac{1}{9}}$ ⓒ $f^{\frac{1}{5}}$

125.

ⓐ ${(7 c)}^{\frac{1}{3}}$ ⓑ ${(12 d)}^{\frac{1}{7}}$
ⓒ $2 {(6 b)}^{\frac{1}{4}}$

127.

ⓐ ${(21 p)}^{\frac{1}{2}}$ ⓑ ${(8 q)}^{\frac{1}{4}}$
ⓒ $4 {(36 r)}^{\frac{1}{6}}$

129.

ⓐ 9 ⓑ 5 ⓒ 8

131.

ⓐ 2 ⓑ 4 ⓒ 5

133.

ⓐ $−6$ ⓑ $−6$ ⓒ $\frac{1}{6}$

135.

ⓐ not real ⓑ $−3$ ⓒ $\frac{1}{3}$

137.

ⓐ not real ⓑ $−6$ ⓒ $\frac{1}{6}$

139.

ⓐ not real ⓑ $−10$ ⓒ $\frac{1}{10}$

141.

ⓐ $m^{\frac{5}{2}}$ ⓑ ${(3 y)}^{\frac{7}{3}}$ ⓒ ${(\frac{4 x}{5 y})}^{\frac{3}{5}}$

143.

ⓐ $u^{\frac{2}{5}}$ ⓑ ${(6 x)}^{\frac{5}{3}}$ ⓒ ${(\frac{18 a}{5 b})}^{\frac{7}{4}}$

145.

ⓐ 32,768 ⓑ $\frac{1}{729}$ ⓒ 9

147.

ⓐ 4 ⓑ $\frac{1}{9}$ ⓒ not real

149.

ⓐ $−27$ ⓑ $- \frac{1}{27}$ ⓒ not real

151.

ⓐ $c^{\frac{7}{8}}$ ⓑ $p^{9}$ ⓒ $\frac{1}{r}$

153.

ⓐ $y^{\frac{5}{4}}$ ⓑ $x^{8}$ ⓒ $\frac{1}{m}$

155.

ⓐ $81 q^{2}$ ⓑ $a^{\frac{1}{2}} b$

157.

ⓐ $8 u^{\frac{1}{4}}$ ⓑ $8 p^{\frac{1}{2}} q^{\frac{3}{4}}$

159.

ⓐ $r^{\frac{7}{2}}$ ⓑ $\frac{6 s}{t}$

161.

ⓐ $c^{2}$ ⓑ $\frac{2 x}{3 y}$

163.

Answers will vary.

Section 8.4 Exercises

165.

ⓐ $3 \sqrt{2}$ ⓑ $7 \sqrt[3]{m}$ ⓒ $6 \sqrt[4]{m}$

167.

ⓐ $9 \sqrt{5}$ ⓑ $12 \sqrt[3]{a}$ ⓒ $6 \sqrt[4]{2 z}$

169.

ⓐ $4 \sqrt{2 a}$ ⓑ 0

171.

ⓐ $\sqrt{3 c}$ ⓑ $\sqrt[3]{4 p q}$

173.

ⓐ $−2 \sqrt{3}$ ⓑ $−2 \sqrt[3]{5}$ ⓒ $3 \sqrt[4]{2}$

175.

ⓐ $7 \sqrt{3}$ ⓑ $7 \sqrt[3]{2}$ ⓒ $3 \sqrt[4]{5}$

177.

ⓐ $a^{2} \sqrt{2 a}$ ⓑ 0

179.

ⓐ $2 c^{3} \sqrt{5 c}$ ⓑ $14 r^{2} \sqrt[4]{2 r^{2}}$

181.

$4 y \sqrt{2}$

183.

ⓐ $−18 \sqrt{6}$ ⓑ $−64 \sqrt[3]{9}$

185.

ⓐ $−30 \sqrt{2}$ ⓑ $6 \sqrt[4]{2}$

187.

ⓐ $72 z^{2} \sqrt{3}$ ⓑ $45 x^{2} \sqrt[3]{2}$

189.

ⓐ $−42 z^{5} \sqrt{2 z}$ ⓑ $−8 y \sqrt[4]{6 y}$

191.

ⓐ $14 + 5 \sqrt{7}$ ⓑ $4 \sqrt[3]{6} + 3 \sqrt[3]{4}$

193.

ⓐ $44 - 3 \sqrt{11}$ ⓑ $3 \sqrt[4]{2} + \sqrt[4]{54}$

195.

$60 + 2 \sqrt{3}$

197.

ⓐ $30 + 18 \sqrt{2}$ ⓑ $\sqrt[3]{x^{2}} - 2 \sqrt[3]{x} - 3$

199.

ⓐ $−55 + 13 \sqrt{10}$
ⓑ $2 \sqrt[3]{x^{2}} + 8 \sqrt[3]{x} + 6$

201.

$23 + 3 \sqrt{30}$

203.

$−439 - 2 \sqrt{77}$

205.

ⓐ $14 + 6 \sqrt{5}$ ⓑ $79 - 20 \sqrt{3}$

207.

ⓐ $87 - 18 \sqrt{6}$
ⓑ $163 + 60 \sqrt{7}$

209.

211.

$−227$

213.

$19$

215.

$\sqrt[3]{9 x^{2}} - 4$

217.

$5 \sqrt{3}$

219.

$9 \sqrt{2}$

221.

$− \sqrt[4]{5}$

223.

$10 c^{2} \sqrt{3} - 9 c^{3} \sqrt{3}$

225.

$2 \sqrt{3}$

227.

$17 q^{2}$

229.

$3 \sqrt{7}$

231.

$−42 \sqrt[3]{9}$

233.

235.

$29 - 7 \sqrt{17}$

237.

$54 - 36 \sqrt{2}$

239.

$6 + 3 \sqrt[3]{2}$

241.

Answers will vary.

243.

Answers will vary.

Section 8.5 Exercises

245.

ⓐ $\frac{4}{3}$ ⓑ $\frac{4}{3}$

247.

ⓐ $\frac{10 m^{2}}{7}$ ⓑ $\frac{3}{y}$

249.

ⓐ $\frac{5}{6 r^{2}}$ ⓑ $\frac{2 x}{3}$

251.

ⓐ $\frac{6 p}{q^{2}}$ ⓑ $- \frac{2 a^{2}}{b}$

253.

ⓐ $\frac{8 m^{4}}{3 n^{4}}$ ⓑ $- \frac{2 x^{2}}{3 y^{2}}$

255.

$2 x^{2} \sqrt{7 y}$

257.

$2 a b \sqrt[3]{2 a}$

259.

ⓐ $\frac{5 \sqrt{6}}{3}$ ⓑ $\frac{2 \sqrt{3}}{9}$ ⓒ $\frac{2 \sqrt{5 x}}{x}$

261.

ⓐ $\frac{6 \sqrt{7}}{7}$ ⓑ $\frac{2 \sqrt{10}}{15}$ ⓒ $\frac{4 \sqrt{3 p}}{p}$

263.

ⓐ $\frac{\sqrt[3]{25}}{5}$ ⓑ $\frac{\sqrt[3]{45}}{6}$ ⓒ $\frac{2 \sqrt[3]{6 a^{2}}}{3 a}$

265.

ⓐ $\frac{\sqrt[3]{121}}{11}$ ⓑ $\frac{\sqrt[3]{28}}{6}$ ⓒ $\frac{\sqrt[3]{9 x}}{x}$

267.

ⓐ $\frac{\sqrt[4]{343}}{7}$ ⓑ $\frac{\sqrt[4]{40}}{4}$ ⓒ $\frac{2 \sqrt[4]{4 x^{2}}}{x}$

269.

ⓐ $\frac{\sqrt[4]{9}}{3}$ ⓑ $\frac{\sqrt[4]{50}}{4}$ ⓒ $\frac{2 \sqrt[4]{3 a^{3}}}{a}$

271.

$−2 (1 + \sqrt{5})$

273.

$3 (3 + \sqrt{7})$

275.

$\frac{\sqrt{3} (\sqrt{m} + \sqrt{5})}{m - 5}$

277.

$\frac{\sqrt{2} (\sqrt{x} + \sqrt{6})}{x - 6}$

279.

${\frac{(\sqrt{r} + \sqrt{5})}{r - 5}}^{2}$

281.

${\frac{(\sqrt{x} + 2 \sqrt{2})}{x - 8}}^{2}$

283.

Answers will vary.

285.

Answers will vary.

Section 8.6 Exercises

287.

$x = 14$

289.

no solution

291.

$x = −4$

293.

$m = 14$

295.

$v = 17$

297.

$m = \frac{7}{2}$

299.

no solution

301.

$u = 3, u = 4$

303.

$r = 1, r = 2$

305.

$x = 10$

307.

$x = −8$

309.

$x = 8$

311.

$x = −4$

313.

$x = 7$

315.

$x = 3$

317.

$z = 21$

319.

$x = 42$

321.

$r = 3$

323.

$u = 3$

325.

$r = −2$

327.

$x = 1$

329.

$x = −8, x = 2$

331.

$a = 0$

333.

$u = \frac{9}{4}$

335.

$a = 4$

337.

$x = 0 x = 4$

339.

$x = 1 x = 5$

341.

$x = 9$

343.

$8.7$ feet

345.

$4.7$ seconds

347.

72 feet

349.

Answers will vary.

Section 8.7 Exercises

351.

ⓐ $f (5) = 4$ ⓑ no value at $x = 0$

353.

ⓐ $g (4) = 5$ ⓑ $g (8) = 7$

355.

ⓐ $F (1) = 1$ ⓑ $F (−11) = 5$

357.

ⓐ $G (5) = 2 \sqrt{6}$ ⓑ $G (2) = 3$

359.

ⓐ $g (6) = 2$ ⓑ $g (−2) = −2$

361.

ⓐ $h (−2) = 0$ ⓑ $h (6) = 2 \sqrt[3]{4}$

363.

ⓐ $f (0) = 0$ ⓑ $f (2) = 2$

365.

ⓐ $g (1) = 0$ ⓑ $g (−3) = 2$

367.

$[\frac{1}{3}, \infty)$

369.

$(− \infty, \frac{2}{3}]$

371.

$(2, \infty)$

373.

$(− \infty, −3] \cup (2, \infty)$

375.

$(− \infty, \infty)$

377.

$(− \infty, \infty)$

379.

$[- \frac{3}{8}, \infty)$

381.

$(− \infty, \infty)$

383.

ⓐ domain: $[−1, \infty)$
ⓑ

$The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 1 to 7. The y-axis runs from negative 2 to 10. The function has a starting point at (negative 1, 0) and goes through the points (0, 1) and (3, 2).$

385.

ⓐ domain: $[−4, \infty)$
ⓑ

$The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a starting point at (negative 4, 0) and goes through the points (negative 3, 1) and (0, 2).$

387.

ⓐ domain: $[0, \infty)$
ⓑ

$The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from 0 to 8. The function has a starting point at (0, 2) and goes through the points (1, 3) and (4, 4).$

389.

ⓐ domain: $[0, \infty)$
ⓑ

391.

ⓐ domain: $(− \infty, 3]$
ⓑ

$The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 6 to 4. The y-axis runs from 0 to 8. The function has a starting point at (3, 0) and goes through the points (2, 1), (negative 1, 2), and (negative 6, 3).$

393.

ⓐ domain: $[0, \infty)$
ⓑ

$The figure shows a square root function graph on the x y-coordinate plane. The x-axis of the plane runs from 0 to 8. The y-axis runs from negative 8 to 0. The function has a starting point at (0, 0) and goes through the points (1, negative 1) and (4, negative 2).$

395.

ⓐ domain: $(− \infty, \infty)$
ⓑ

$The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 1, 0) and goes through the points (negative 2, negative 1) and (0, 1).$

397.

ⓐ domain: $(− \infty, \infty)$
ⓑ

$The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 4, 0) and goes through the points (negative 3, negative 1) and (negative 1, 1).$

399.

ⓐ domain: $(− \infty, \infty)$
ⓑ

$The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The function has a center point at (0, 3) and goes through the points (negative 1, 2) and (1, 4).$

401.

ⓐ domain: $(− \infty, \infty)$
ⓑ

$The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (0, 0) and goes through the points (1, 1) and (negative 1, negative 1).$

403.

ⓐ domain: $(− \infty, \infty)$
ⓑ

$The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (0, 0) and goes through the points (1, 2) and (negative 1, negative 2).$

405.

Answers will vary.

407.

Answers will vary.

Section 8.8 Exercises

409.

411.

413.

$9 \sqrt{3} i$

415.

$8 \sqrt{2} i$

417.

$8 + 7 i$

419.

$14 + 2 i$

421.

$−2 + 2 i$

423.

$8 + 5 i$

425.

$7 - 13 i$

427.

$25 - 2 \sqrt{2} i$

429.

$12 + 20 i$

431.

$−12 + 18 i$

433.

$−38 + + 9 i$

435.

$27 + 15 i$

437.

$−7 + 24 i$

439.

$−5 + 12 i$

441.

$−30$

443.

$−30$

445.

$−44 - 4 i \sqrt{3}$

447.

$−20 - 2 \sqrt{2} i$

449.

451.

453.

455.

457.

459.

$\frac{2}{25} + \frac{11}{25} i$

461.

$\frac{6}{13} + \frac{9}{13} i$

463.

$- \frac{12}{13} - \frac{8}{13} i$

465.

$\frac{4}{3} - \frac{1}{3} i$

467.

$- \frac{3}{4} + \frac{1}{2} i$

469.

471.

$−1$

473.

475.

477.

Answers will vary.

479.

Answers will vary.

Review Exercises

481.

ⓐ 15 ⓑ $−4$

483.

ⓐ 2 ⓑ 3 ⓒ 3

485.

ⓐ $8 < \sqrt{68} < 9$
ⓑ $4 < \sqrt[3]{84} < 5$

487.

ⓐ a ⓑ $b$

489.

ⓐ $m^{2}$ ⓑ $n^{4}$

491.

ⓐ $6 a^{2}$ ⓑ $2 b^{4}$

493.

$5 \sqrt{5}$

495.

ⓐ $5 \sqrt[3]{5}$ ⓑ $2 \sqrt[6]{2}$

497.

499.

ⓐ $−2$ ⓑ not real

501.

503.

$\frac{5 m^{2} \sqrt{3 m}}{4}$

505.

507.

509.

ⓐ 5 ⓑ 3 ⓒ 2

511.

513.

515.

517.

519.

521.

$37 y \sqrt{3}$

523.

ⓐ $126 x^{2} \sqrt{x}$ ⓑ $48 a \sqrt[3]{5 a^{2}}$

525.

ⓐ $71 - 22 \sqrt{7}$
ⓑ $\sqrt[3]{x^{2}} - 8 \sqrt[3]{x} + 15$

527.

ⓐ $27 + 8 \sqrt{11}$ ⓑ $29 - 12 \sqrt{5}$

529.

$\sqrt[3]{9 x^{2}} - 4$

531.

ⓐ $\frac{8 m^{4}}{3 n^{4}}$ ⓑ $- \frac{2 x^{2}}{3 y^{2}}$

533.

535.

$- \frac{7 (2 + \sqrt{6})}{2}$

537.

${\frac{(\sqrt{x} + 2 \sqrt{2})}{x - 8}}^{2}$

539.

no solution

541.

$u = 3, u = 4$

543.

$x = −4$

545.

$r = 3$

547.

$x = −8, x = 2$

549.

$x = 3$

551.

$64.8$ feet

553.

ⓐ $G (5) = 2 \sqrt{6}$ ⓑ $G (2) = 3$

555.

ⓐ $g (1) = 0$ ⓑ $g (−3) = 2$

557.

$(2, \infty)$

559.

$(- \infty, \frac{10}{7}]$

561.

ⓐ domain: $[0, \infty)$
ⓑ

563.

ⓐ domain: $(− \infty, \infty)$
ⓑ

565.

$8 \sqrt{2} i$

567.

$8 + 5 i$

569.

$23 + 14 i$

571.

$−8$

573.

$−5 + 12 i$

575.

$\frac{2}{25} + \frac{11}{25} i$

577.

Practice Test

579.

$5 x^{3}$

581.

$2 x^{2} y^{} \sqrt[3]{9 x^{2} y}$

583.

ⓐ $\frac{1}{4}$ ⓑ $−343$

585.

$x^{\frac{7}{4}}$

587.

$− x^{2} \sqrt{3 x}$

589.

$36 x^{4} \sqrt{2}$

591.

$2 - 7 \sqrt{3}$

593.

$\frac{7 x^{2} \sqrt{x} \sqrt{y}}{3 y^{4}}$

595.

$3 (2 - \sqrt{3})$

597.

$−12 + 8 i$

599.

$1$

601.

$x = 4$

603.

ⓐ domain: $[−2, \infty)$
ⓑ

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Section 8.1 Exercises

Section 8.2 Exercises

Section 8.3 Exercises

Section 8.4 Exercises

Section 8.5 Exercises

Section 8.6 Exercises

Section 8.7 Exercises

Section 8.8 Exercises

Review Exercises

Practice Test