13.1.8: Chapter 8

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

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

8.9

$\frac{1}{125}$

8.10

$2x^2+11x-8$

8.11

$8+2a-a^2$

8.12

$81-90y+25y^2$

8.13

$\frac{5}{8}$

8.14

$x^6$

8.15

$49-9x^2$

8.16

$y^2-6y+9$

8.17

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

8.18

$n=2\text{or}n=4$

8.19

$(\text{−}\infty ,\frac{1}{2}]$

8.20

$f(2)=2,f(\mathrm{-1})=\mathrm{-7},f(0)=\mathrm{-4}$

8.21

$\text{domain:}[0,\infty );\text{range:}[0,\infty )$

8.22

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

8.23

$2x^2-x-15$

8.24

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

Try It

8.1

ⓐ $\mathrm{-8}$ ⓑ 15

8.2

ⓐ 10 ⓑ $\mathrm{-11}$

8.3

ⓐ not a real number ⓑ $\mathrm{-9}$

8.4

ⓐ $\mathrm{-7}$ ⓑ not a real number

8.5

ⓐ 3 ⓑ 4 ⓒ 3

8.6

ⓐ 10 ⓑ 2 ⓒ 4

8.7

ⓐ $\mathrm{-3}$ ⓑ not real ⓒ $\mathrm{-2}$

8.8

ⓐ $\mathrm{-6}$ ⓑ not real ⓒ $\mathrm{-4}$

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

ⓐ $\left|b\right|$ ⓑ w ⓒ $\left|m\right|$ ⓓ q

8.14

ⓐ $\left|y\right|$ ⓑ p ⓒ $\left|z\right|$ ⓓ q

8.15

ⓐ $\left|y^9\right|$ ⓑ $z^6$

8.16

ⓐ $m^2$ ⓑ $\left|b^5\right|$

8.17

ⓐ $\left|u^3\right|$ ⓑ $v^5$

8.18

ⓐ $c^4$ ⓑ $d^4$

8.19

ⓐ $8\left|x\right|$ ⓑ $\mathrm{-10}\left|p\right|$

8.20

ⓐ $13\left|y\right|$ ⓑ $\mathrm{-11}\left|y\right|$

8.21

ⓐ $3x^9$ ⓑ $3\left|q^7\right|$

8.22

ⓐ $5q^3$ ⓑ $3q^5$

8.23

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

8.24

ⓐ $15\left|mn\right|$ ⓑ $13\left|x^5{y}^7\right|$
ⓒ $3w^{12}{z}^5$

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

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

8.30

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

8.31

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

8.32

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

8.33

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

8.34

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

8.35

ⓐ $\mathrm{-4}$ ⓑ $\text{no real number}$

8.36

ⓐ $\mathrm{-5}\sqrt[3]{5}$ ⓑ 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

ⓐ $\left|a\right|$ ⓑ $\left|x\right|$ ⓒ $y^3$

8.42

ⓐ $x^2$ ⓑ $m^2$ ⓒ $n^2$

8.43

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

8.44

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

8.45

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

8.46

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

8.47

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

8.48

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

8.49

ⓐ $7z^2$ ⓑ $\mathrm{-5}\sqrt[3]{2}$
ⓒ $3\left|m\right|\sqrt[4]{2m^2}$

8.50

ⓐ $8m^4$ ⓑ $\mathrm{-4}$ ⓒ $3\left|n\right|\sqrt[4]{2}$

8.51

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

8.52

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

8.53

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

8.54

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

8.55

ⓐ 6 ⓑ 2 ⓒ 2

8.56

ⓐ 10 ⓑ 3 ⓒ 3

8.57

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

8.58

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

8.59

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

8.60

ⓐ $a^{\frac{2}{5}}$ ⓑ ${\left(5ab\right)}^{\frac{5}{3}}$
ⓒ ${\left(\frac{7xy}{z}\right)}^{\frac{3}{2}}$

8.61

ⓐ 9 ⓑ $\frac{1}{729}$ ⓒ $\frac{1}{8}$

8.62

ⓐ 8 ⓑ $\frac{1}{9}$ ⓒ $\frac{1}{125}$

8.63

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

8.64

ⓐ $\mathrm{-729}$ ⓑ $-\frac{1}{729}$ ⓒ 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

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

8.68

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

8.69

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

8.70

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

8.71

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

8.72

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

8.73

ⓐ $\mathrm{-2}\sqrt{7x}$ ⓑ $\text{−}\sqrt[4]{5xy}$

8.74

ⓐ $\text{−}\sqrt{3y}$ ⓑ $3\sqrt[3]{7mn}$

8.75

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

8.76

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

8.77

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

8.78

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

8.79

ⓐ $12\sqrt{15}$ ⓑ $\mathrm{-18}\sqrt[3]{4}$

8.80

ⓐ $27\sqrt{2}$ ⓑ $\mathrm{-36}\sqrt[3]{2}$

8.81

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

8.82

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

8.83

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

8.84

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

8.85

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

8.86

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

8.87

$1+9\sqrt{21}$

8.88

$\mathrm{-12}-20\sqrt{3}$

8.89

ⓐ $102+20\sqrt{2}$ ⓑ $55+6\sqrt{6}$

8.90

ⓐ $41-12\sqrt{5}$
ⓑ $121-36\sqrt{10}$

8.91

$\mathrm{-11}$

8.92

$\mathrm{-159}$

8.93

ⓐ $\frac{5s}{8}$ ⓑ $\frac{2}{a}$

8.94

ⓐ $\frac{5q^2}{6}$ ⓑ $\frac{2}{b}$

8.95

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

8.96

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

8.97

$4xy\sqrt{2x}$

8.98

$4ab\sqrt{3b}$

8.99

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

8.100

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

8.101

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

8.102

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

8.103

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

8.104

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

8.105

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

8.106

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

8.107

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

8.108

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

8.109

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

8.110

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

8.111

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

8.112

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

8.113

$\text{no solution}$

8.114

$\text{no solution}$

8.115

$x=2,x=3$

8.116

$y=5,y=6$

8.117

$x=\mathrm{-6}$

8.118

$x=\mathrm{-9}$

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

$x=0x=4$

8.131

9 seconds

8.132

$3.5$ seconds

8.133

$42.7$ feet

8.134

$54.1$ feet

8.135

ⓐ $f\left(6\right)=4$ ⓑ no value at $x=0$

8.136

ⓐ $g\left(4\right)=5$ ⓑ no value at $f(\mathrm{-3})$

8.137

ⓐ $g(4)=2$ ⓑ $g(1)=\mathrm{-1}$

8.138

ⓐ $h(2)=2$
ⓑ $h(\mathrm{-5})=\mathrm{-3}$

8.139

ⓐ $f(4)=2$ ⓑ $f(\mathrm{-1})=1$

8.140

ⓐ $g(16)=3$ ⓑ $g(3)=2$

8.141

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

8.142

$\left(\text{−}\infty ,\frac{4}{5}\right]$

8.143

$\left(\mathrm{-3},\infty \right)$

8.144

$\left(5,\infty \right)$

8.145

$(\text{−}\infty ,\infty )$

8.146

$(\text{−}\infty ,\infty )$

8.147

ⓐ domain: $\left[\mathrm{-2},\infty \right)$
ⓑ

ⓒ range: $\left[0,\infty \right)$

8.148

ⓐ domain: $\left[2,\infty \right)$
ⓑ

ⓒ range: $\left[0,\infty \right)$

8.149

ⓐ domain: $\text{(}-\infty ,\infty )$
ⓑ

ⓒ range: $\text{(}-\infty ,\infty )$

8.150

ⓐ domain: $\text{(}-\infty ,\infty )$
ⓑ

ⓒ range: $\text{(}-\infty ,\infty )$

8.151

ⓐ $9i$ ⓑ $\sqrt{5}i$ ⓒ $3\sqrt{2}i$

8.152

ⓐ $6i$ ⓑ $\sqrt{3}i$ ⓒ $3\sqrt{3}i$

8.153

$6\sqrt{2}i$

8.154

$7\sqrt{3}i$

8.155

ⓐ $6+5i$ ⓑ $6-3i$

8.156

ⓐ $\mathrm{-2}-6i$ ⓑ $2+9i$

8.157

$12+20i$

8.158

$12-6i$

8.159

$\mathrm{-11}-7i$

8.160

$\mathrm{-5}-10i$

8.161

$\mathrm{-21}-20i$

8.162

$9-40i$

8.163

$\mathrm{-14}$

8.164

$\mathrm{-54}$

8.165

$\mathrm{-12}-22\sqrt{3}i$

8.166

$6+12\sqrt{2}i$

8.167

25

8.168

29

8.169

109

8.170

41

8.171

i

8.172

i

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

$\text{−}i$

8.178

$1$

Section 8.1 Exercises

1.

ⓐ 8 ⓑ $\mathrm{-9}$

3.

ⓐ 14 ⓑ $\mathrm{-1}$

5.

ⓐ $\frac{2}{3}$ ⓑ $\mathrm{-0.1}$

7.

ⓐ not real number ⓑ $\mathrm{-17}$

9.

ⓐ $\mathrm{-15}$ ⓑ not real number

11.

ⓐ 6 ⓑ 4

13.

ⓐ 8 ⓑ 3 ⓒ 1

15.

ⓐ $\mathrm{-2}$ ⓑ $\text{not real}$ ⓒ $\mathrm{-2}$

17.

ⓐ $\mathrm{-5}$ ⓑ $\text{not real}$ ⓒ $\mathrm{-4}$

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 ⓑ $\left|v\right|$

29.

ⓐ $\left|y\right|$ ⓑ $m$

31.

ⓐ $\left|x^3\right|$ ⓑ $y^8$

33.

ⓐ $x^{12}$ ⓑ $\left|y^{11}\right|$

35.

ⓐ $x^3$ ⓑ $\left|y^3\right|$

37.

ⓐ $m^2$ ⓑ $n^4$

39.

ⓐ $7\left|x\right|$ ⓑ $\mathrm{-9}\left|x^9\right|$

41.

ⓐ $11m^{10}$ ⓑ $\mathrm{-8}\left|a\right|$

43.

ⓐ $2x^2$ ⓑ $2y^2$

45.

ⓐ $6a^2$ ⓑ $2b^4$

47.

ⓐ $12\left|xy\right|$ ⓑ $13w^4\left|y^5\right|$
ⓒ $2a^{17}{b}^2$

49.

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

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.

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

69.

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

71.

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

73.

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

75.

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

77.

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

79.

ⓐ $\mathrm{-6}\sqrt[3]{4}$ ⓑ not real

81.

ⓐ $\mathrm{-2}$ ⓑ 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.

ⓐ $x^2$ ⓑ $p^3$ ⓒ $\left|q\right|$

93.

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

95.

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

97.

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

99.

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

101.

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

103.

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

105.

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

107.

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

109.

ⓐ $\frac{1}{2\left|pq\right|}$ ⓑ $\frac{2cd\sqrt[3]{d^2}}{5}$
ⓒ $\frac{\left|mn\right|}{2}$

111.

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

113.

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

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.

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

127.

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

129.

ⓐ 9 ⓑ 5 ⓒ 8

131.

ⓐ 2 ⓑ 4 ⓒ 5

133.

ⓐ $\mathrm{-6}$ ⓑ $\mathrm{-6}$ ⓒ $\frac{1}{6}$

135.

ⓐ not real ⓑ $\mathrm{-3}$ ⓒ $\frac{1}{3}$

137.

ⓐ not real ⓑ $\mathrm{-6}$ ⓒ $\frac{1}{6}$

139.

ⓐ not real ⓑ $\mathrm{-10}$ ⓒ $\frac{1}{10}$

141.

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

143.

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

145.

ⓐ 32,768 ⓑ $\frac{1}{729}$ ⓒ 9

147.

ⓐ 4 ⓑ $\frac{1}{9}$ ⓒ not real

149.

ⓐ $\mathrm{-27}$ ⓑ $-\frac{1}{27}$ ⓒ not real

151.

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

153.

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

155.

ⓐ $81q^2$ ⓑ $a^{\frac{1}{2}}b$

157.

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

159.

ⓐ $r^{\frac{7}{2}}$ ⓑ $\frac{6s}{t}$

161.

ⓐ $c^2$ ⓑ $\frac{2x}{3y}$

163.

Answers will vary.

Section 8.4 Exercises

165.

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

167.

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

169.

ⓐ $4\sqrt{2a}$ ⓑ 0

171.

ⓐ $\sqrt{3c}$ ⓑ $\sqrt[3]{4pq}$

173.

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

175.

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

177.

ⓐ $a^2\sqrt{2a}$ ⓑ 0

179.

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

181.

$4y\sqrt{2}$

183.

ⓐ $\mathrm{-18}\sqrt{6}$ ⓑ $\mathrm{-64}\sqrt[3]{9}$

185.

ⓐ $\mathrm{-30}\sqrt{2}$ ⓑ $6\sqrt[4]{2}$

187.

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

189.

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

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.

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

201.

$23+3\sqrt{30}$

203.

$\mathrm{-439}-2\sqrt{77}$

205.

ⓐ $14+6\sqrt{5}$ ⓑ $79-20\sqrt{3}$

207.

ⓐ $87-18\sqrt{6}$
ⓑ $163+60\sqrt{7}$

209.

14

211.

$\mathrm{-227}$

213.

$19$

215.

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

217.

$5\sqrt{3}$

219.

$9\sqrt{2}$

221.

$\text{−}\sqrt[4]{5}$

223.

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

225.

$2\sqrt{3}$

227.

$17q^2$

229.

$3\sqrt{7}$

231.

$\mathrm{-42}\sqrt[3]{9}$

233.

29

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.

ⓐ $\frac{10m^2}{7}$ ⓑ $\frac{3}{y}$

249.

ⓐ $\frac{5}{6r^2}$ ⓑ $\frac{2x}{3}$

251.

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

253.

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

255.

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

257.

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

259.

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

261.

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

263.

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

265.

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

267.

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

269.

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

271.

$\mathrm{-2}\left(1+\sqrt{5}\right)$

273.

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

275.

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

277.

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

279.

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

281.

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

283.

Answers will vary.

285.

Answers will vary.

Section 8.6 Exercises

287.

$x=14$

289.

no solution

291.

$x=\mathrm{-4}$

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.

$x=\mathrm{-8}$

309.

$x=8$

311.

$x=\mathrm{-4}$

313.

$x=7$

315.

$x=3$

317.

$z=21$

319.

$x=42$

321.

$r=3$

323.

$u=3$

325.

$r=\mathrm{-2}$

327.

$x=1$

329.

$x=\mathrm{-8},x=2$

331.

$a=0$

333.

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

335.

$a=4$

337.

$x=0x=4$

339.

$x=1x=5$

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.

ⓐ $F(1)=1$ ⓑ $F(\mathrm{-11})=5$

357.

ⓐ $G(5)=2\sqrt{6}$ ⓑ $G(2)=3$

359.

ⓐ $g(6)=2$ ⓑ $g(\mathrm{-2})=\mathrm{-2}$

361.

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

363.

ⓐ $f(0)=0$ ⓑ $f(2)=2$

365.

ⓐ $g(1)=0$ ⓑ $g(\mathrm{-3})=2$

367.

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

369.

$\left(\text{−}\infty ,\frac{2}{3}\right]$

371.

$\left(2,\infty \right)$

373.

$\left(\text{−}\infty ,\mathrm{-3}\right]\cup \left(2,\infty \right)$

375.

$\left(\text{−}\infty ,\infty \right)$

377.

$\left(\text{−}\infty ,\infty \right)$

379.

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

381.

$\left(\text{−}\infty ,\infty \right)$

383.

ⓐ domain: $\left[\mathrm{-1},\infty \right)$
ⓑ

ⓒ $\left[0,\infty \right)$

385.

ⓐ domain: $\left[\mathrm{-4},\infty \right)$
ⓑ

ⓒ $\left[0,\infty \right)$

387.

ⓐ domain: $\left[0,\infty \right)$
ⓑ

ⓒ $\left[2,\infty \right)$

389.

ⓐ domain: $\left[0,\infty \right)$
ⓑ

ⓒ $\left[0,\infty \right)$

391.

ⓐ domain: $\left(\text{−}\infty ,3\right]$
ⓑ

ⓒ $\left[0,\infty \right)$

393.

ⓐ domain: $\left[0,\infty \right)$
ⓑ

ⓒ $\left(\text{−}\infty ,0\right]$

395.

ⓐ domain: $\left(\text{−}\infty ,\infty \right)$
ⓑ

ⓒ $\left(\text{−}\infty ,\infty \right)$

397.

ⓐ domain: $\left(\text{−}\infty ,\infty \right)$
ⓑ

ⓒ $\left(\text{−}\infty ,\infty \right)$

399.

ⓐ domain: $\left(\text{−}\infty ,\infty \right)$
ⓑ

ⓒ $\left(\text{−}\infty ,\infty \right)$

401.

ⓐ domain: $\left(\text{−}\infty ,\infty \right)$
ⓑ

ⓒ $\left(\text{−}\infty ,\infty \right)$

403.

ⓐ domain: $\left(\text{−}\infty ,\infty \right)$
ⓑ

ⓒ $\left(\text{−}\infty ,\infty \right)$

405.

Answers will vary.

407.

Answers will vary.

Section 8.8 Exercises

409.

ⓐ $4i$ ⓑ $\sqrt{11}i$ ⓒ $2\sqrt{2}i$

411.

ⓐ $10i$ ⓑ $\sqrt{13}i$ ⓒ $3\sqrt{5}i$

413.

$9\sqrt{3}i$

415.

$8\sqrt{2}i$

417.

$8+7i$

419.

$14+2i$

421.

$\mathrm{-2}+2i$

423.

$8+5i$

425.

$7-13i$

427.

$25-2\sqrt{2}i$

429.

$12+20i$

431.

$\mathrm{-12}+18i$

433.

$\mathrm{-38}++9i$

435.

$27+15i$

437.

$\mathrm{-7}+24i$

439.

$\mathrm{-5}+12i$

441.

$\mathrm{-30}$

443.

$\mathrm{-30}$

445.

$\mathrm{-44}-4i\sqrt{3}$

447.

$\mathrm{-20}-2\sqrt{2}i$

449.

5

451.

53

453.

50

455.

85

457.

i

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.

i

471.

$\mathrm{-1}$

473.

1

475.

i

477.

Answers will vary.

479.

Answers will vary.

Review Exercises

481.

ⓐ 15 ⓑ $\mathrm{-4}$

483.

ⓐ 2 ⓑ 3 ⓒ 3

485.

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

487.

ⓐ a ⓑ $b$

489.

ⓐ $m^2$ ⓑ $n^4$

491.

ⓐ $6a^2$ ⓑ $2b^4$

493.

$5\sqrt{5}$

495.

ⓐ $5\sqrt[3]{5}$ ⓑ $2\sqrt[6]{2}$

497.

ⓐ $4\left|s^7\right|\sqrt[]{5s}$ ⓑ $2a\sqrt[5]{3a^2}$
ⓒ $2\left|b\right|\sqrt[6]{2b}$

499.

ⓐ $\mathrm{-2}$ ⓑ not real

501.

ⓐ $\frac{6}{7}$ ⓑ $\frac{2}{3}$ ⓒ $\frac{1}{2}$

503.

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

505.

ⓐ $\frac{1}{2\left|pq\right|}$ ⓑ $\frac{2cd\sqrt[5]{2d^2}}{5}$
ⓒ $\frac{\left|mn\right|}{2}$

507.

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

509.

ⓐ 5 ⓑ 3 ⓒ 2

511.

ⓐ $\mathrm{-2}$ ⓑ $\frac{1}{3}$ ⓒ $\mathrm{-5}$

513.

ⓐ 125 ⓑ $\frac{1}{27}$ ⓒ 16

515.

ⓐ $6^3$ ⓑ $b^9$ ⓒ $\frac{1}{w}$

517.

ⓐ $4\sqrt{2}$ ⓑ $9\sqrt[3]{p}$ ⓒ $2\sqrt[3]{x}$

519.

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

521.

$37y\sqrt{3}$

523.

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

525.

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

527.

ⓐ $27+8\sqrt{11}$ ⓑ $29-12\sqrt{5}$

529.

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

531.

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

533.

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

535.

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

537.

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

539.

no solution

541.

$u=3,u=4$

543.

$x=\mathrm{-4}$

545.

$r=3$

547.

$x=\mathrm{-8},x=2$

549.

$x=3$

551.

$64.8$ feet

553.

ⓐ $G(5)=2\sqrt{6}$ ⓑ $G(2)=3$

555.

ⓐ $g(1)=0$ ⓑ $g(\mathrm{-3})=2$

557.

$\left(2,\infty \right)$

559.

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

561.

ⓐ domain: $\left[0,\infty \right)$
ⓑ

ⓒ range: $\left[0,\infty \right)$

563.

ⓐ domain: $\left(\text{−}\infty ,\infty \right)$
ⓑ

ⓒ range: $\left(\text{−}\infty ,\infty \right)$

565.

$8\sqrt{2}i$

567.

$8+5i$

569.

$23+14i$

571.

$\mathrm{-8}$

573.

$\mathrm{-5}+12i$

575.

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

577.

1

Practice Test

579.

$5x^3$

581.

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

583.

ⓐ $\frac{1}{4}$ ⓑ $\mathrm{-343}$

585.

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

587.

$\text{−}{x}^2\sqrt{3x}$

589.

$36x^4\sqrt{2}$

591.

$2-7\sqrt{3}$

593.

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

595.

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

597.

$\mathrm{-12}+8i$

599.

$1$

601.

$x=4$

603.

ⓐ domain: $\left[\mathrm{-2},\infty \right)$
ⓑ

ⓒ range: $\left[0,\infty \right)$