Contents

Preface
1Foundations
- Introduction
- 1.1Use the Language of Algebra
- 1.2Integers
- 1.3Fractions
- 1.4Decimals
- 1.5Properties of Real Numbers
- Chapter Review
  Key Terms
  Key Concepts
- Exercises
  Review Exercises
  Practice Test
2Solving Linear Equations
3Graphs and Functions
4Systems of Linear Equations
5Polynomials and Polynomial Functions
6Factoring
7Rational Expressions and Functions
8Roots and Radicals
9Quadratic Equations and Functions
10Exponential and Logarithmic Functions
11Conics
12Sequences, Series and Binomial Theorem
- Introduction
- 12.1Sequences
- 12.2Arithmetic Sequences
- 12.3Geometric Sequences and Series
- 12.4Binomial Theorem
- Chapter Review
  Key Terms
Answer Key
Index

Chapter 8

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8.1

ⓐ $−8 −8$ ⓑ 15

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8.2

ⓐ 10 ⓑ $−11 −11$

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8.3

ⓐ not a real number ⓑ $−9 −9$

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8.4

ⓐ $−7 −7$ ⓑ not a real number

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8.5

ⓐ 3 ⓑ 4 ⓒ 3

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8.6

ⓐ 10 ⓑ 2 ⓒ 3

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8.7

ⓐ $−3 −3$ ⓑ not real ⓒ $−2 −2$

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8.8

ⓐ $−6 −6$ ⓑ not real ⓒ $−4 −4$

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8.9

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

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8.10

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

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8.11

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

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8.12

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

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8.13

ⓐ $| b | | b |$ ⓑ w ⓒ $| m | | m |$ ⓓ q

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8.14

ⓐ $| y | | y |$ ⓑ p ⓒ $| z | | z |$ ⓓ q

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8.15

ⓐ $| y^{9} | | y^{9} |$ ⓑ $z^{6} z^{6}$

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8.16

ⓐ $m^{2} m^{2}$ ⓑ $| b^{5} | | b^{5} |$

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8.17

ⓐ $| u^{3} | | u^{3} |$ ⓑ $v^{5} v^{5}$

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8.18

ⓐ $c^{4} c^{4}$ ⓑ $d^{4} d^{4}$

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8.19

ⓐ $8 | x | 8 | x |$ ⓑ $−10 | p | −10 | p |$

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8.20

ⓐ $13 | y | 13 | y |$ ⓑ $−11 | y | −11 | y |$

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8.21

ⓐ $3 x^{9} 3 x^{9}$ ⓑ $3 | q^{7} | 3 | q^{7} |$

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8.22

ⓐ $5 p^{3} 5 p^{3}$ ⓑ $3 q^{5} 3 q^{5}$

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8.23

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

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8.24

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

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8.25

$4 \sqrt{3} 4 \sqrt{3}$

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8.26

$3 \sqrt{5} 3 \sqrt{5}$

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8.27

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

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8.28

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

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8.29

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

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8.30

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

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8.31

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

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8.32

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

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8.33

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

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8.34

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

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8.35

ⓐ $−4 −4$ ⓑ $no real number no real number$

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8.36

ⓐ $−5 \sqrt[3]{5} −5 \sqrt[3]{5}$ ⓑ no real number

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8.37

ⓐ $5 + 5 \sqrt{3} 5 + 5 \sqrt{3}$ ⓑ $2 - \sqrt{3} 2 - \sqrt{3}$

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8.38

ⓐ $2 + 7 \sqrt{2} 2 + 7 \sqrt{2}$ ⓑ $2 - \sqrt{5} 2 - \sqrt{5}$

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8.39

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

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8.40

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

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8.41

ⓐ $| a | | a |$ ⓑ $| x | | x |$ ⓒ $y^{3} y^{3}$

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8.42

ⓐ $x^{2} x^{2}$ ⓑ $m^{2} m^{2}$ ⓒ $n^{2} n^{2}$

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8.43

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

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8.44

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

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8.45

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

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8.46

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

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8.47

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

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8.48

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

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8.49

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

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8.50

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

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8.51

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

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8.52

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

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8.53

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

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8.54

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

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8.55

ⓐ 6 ⓑ 2 ⓒ 2

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8.56

ⓐ 10 ⓑ 3 ⓒ 3

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8.57

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

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8.58

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

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8.59

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

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8.60

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

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8.61

ⓐ 9 ⓑ $\frac{1}{729} \frac{1}{729}$ ⓒ $\frac{1}{8} \frac{1}{8}$

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8.62

ⓐ 8 ⓑ $\frac{1}{9} \frac{1}{9}$ ⓒ $\frac{1}{125} \frac{1}{125}$

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8.63

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

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8.64

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

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8.65

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

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8.66

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

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8.67

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

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8.68

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

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8.69

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

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8.70

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

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8.71

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

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8.72

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

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8.73

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

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8.74

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

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8.75

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

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8.76

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

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8.77

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

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8.78

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

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8.79

ⓐ $12 \sqrt{15} 12 \sqrt{15}$ ⓑ $−18 \sqrt[3]{2} −18 \sqrt[3]{2}$

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8.80

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

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8.81

ⓐ $36 x^{3} \sqrt{5} 36 x^{3} \sqrt{5}$ ⓑ $8 y \sqrt[4]{3 y^{2}} 8 y \sqrt[4]{3 y^{2}}$

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8.82

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

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8.83

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

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8.84

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

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8.85

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

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8.86

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

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8.87

$1 + 9 \sqrt{21} 1 + 9 \sqrt{21}$

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8.88

$−12 - 20 \sqrt{3} −12 - 20 \sqrt{3}$

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8.89

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

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8.90

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

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8.91

$−11 −11$

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8.92

$−159 −159$

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8.93

ⓐ $\frac{5 s}{8} \frac{5 s}{8}$ ⓑ $\frac{2}{a} \frac{2}{a}$

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8.94

ⓐ $\frac{5 q^{2}}{6} \frac{5 q^{2}}{6}$ ⓑ $\frac{2}{b} \frac{2}{b}$

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8.95

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

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8.96

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

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8.97

$4 x y \sqrt{2 x} 4 x y \sqrt{2 x}$

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8.98

$4 a b \sqrt{3 b} 4 a b \sqrt{3 b}$

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8.99

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

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8.100

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

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8.101

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

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8.102

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

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8.103

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

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8.104

ⓐ $\frac{\sqrt[4]{125}}{5} \frac{\sqrt[4]{125}}{5}$ ⓑ $\frac{\sqrt[4]{224}}{8} \frac{\sqrt[4]{224}}{8}$
ⓒ $\frac{\sqrt[4]{64 x^{3}}}{x} \frac{\sqrt[4]{64 x^{3}}}{x}$

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8.105

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

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8.106

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

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8.107

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

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8.108

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

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8.109

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

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8.110

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

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8.111

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

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8.112

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

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8.113

$no solution no solution$

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8.114

$no solution no solution$

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8.115

$x = 2, x = 3 x = 2, x = 3$

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8.116

$y = 5, y = 6 y = 5, y = 6$

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8.117

$x = −6 x = −6$

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8.118

$x = −9 x = −9$

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8.119

$x = 8 x = 8$

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8.120

$x = 6 x = 6$

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8.121

$m = 7 m = 7$

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8.122

$n = 3 n = 3$

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8.123

$a = 63 a = 63$

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8.124

$b = 311 b = 311$

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8.125

$x = 3 x = 3$

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8.126

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

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8.127

$x = 4 x = 4$

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8.128

$x = 9 x = 9$

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8.129

$x = 5 x = 5$

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8.130

$x = 0 x = 4 x = 0$

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8.131

9 seconds

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8.132

$3.5 3.5$ seconds

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8.133

$42.7 42.7$ feet

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8.134

$54.1 54.1$ feet

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8.135

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

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8.136

ⓐ $g (4) = 5 g (4) = 5$ ⓑ no value at $f (−3) f (−3)$

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8.137

ⓐ $g (4) = 2 g (4) = 2$ ⓑ $g (1) = −1 g (1) = −1$

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8.138

ⓐ $h (2) = 2 h (2) = 2$
ⓑ $h (−5) = −3 h (−5) = −3$

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8.139

ⓐ $f (4) = 2 f (4) = 2$ ⓑ $f (−1) = 1 f (−1) = 1$

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8.140

ⓐ $g (16) = 3 g (16) = 3$ ⓑ $g (3) = 2 g (3) = 2$

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8.141

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

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8.142

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

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8.143

$(−3, \infty) (−3, \infty)$

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8.144

$(5, \infty) (5, \infty)$

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8.145

$(− \infty, \infty) (− \infty, \infty)$

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8.146

$(− \infty, \infty) (− \infty, \infty)$

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8.147

ⓐ domain: $[−2, \infty) [−2, \infty)$
ⓑ

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).

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8.148

ⓐ domain: $[2, \infty) [2, \infty)$
ⓑ

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).

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8.149

ⓐ domain: $(- \infty, \infty) (- \infty, \infty)$
ⓑ

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).

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8.150

ⓐ domain: $(- \infty, \infty) (- \infty, \infty)$
ⓑ

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).

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8.151

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

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8.152

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

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8.153

$6 \sqrt{2} i 6 \sqrt{2} i$

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8.154

$7 \sqrt{3} i 7 \sqrt{3} i$

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8.155

ⓐ $6 + 5 i 6 + 5 i$ ⓑ $6 - 3 i 6 - 3 i$

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8.156

ⓐ $−2 - 6 i −2 - 6 i$ ⓑ $2 + 9 i 2 + 9 i$

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8.157

$12 + 20 i 12 + 20 i$

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8.158

$12 + 6 i 12 + 6 i$

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8.159

$−11 - 7 i −11 - 7 i$

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8.160

$−5 - 10 i −5 - 10 i$

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8.161

$−21 - 20 i −21 - 20 i$

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8.162

$9 - 40 i 9 - 40 i$

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8.163

$−14 −14$

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8.164

$−54 −54$

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8.165

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

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8.166

$6 + 12 \sqrt{2} i 6 + 12 \sqrt{2} i$

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8.167

Show solution

8.168

Show solution

8.169

109

Show solution

8.170

Show solution

8.171

Show solution

8.172

Show solution

8.173

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

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8.174

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

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8.175

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

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8.176

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

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8.177

$− i − i$

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8.178

$11$

Section 8.1 Exercises

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ⓐ 8 ⓑ $−9 −9$

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ⓐ 14 ⓑ $−1 −1$

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ⓐ $\frac{2}{3} \frac{2}{3}$ ⓑ $−0.1 −0.1$

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ⓐ not real number ⓑ $−17 −17$

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ⓐ $−15 −15$ ⓑ not real number

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11.

ⓐ 6 ⓑ 4

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13.

ⓐ 8 ⓑ 3 ⓒ 1

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15.

ⓐ $−2 −2$ ⓑ $not real not real$ ⓒ $−2 −2$

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17.

ⓐ $−5 −5$ ⓑ $not real not real$ ⓒ $−4 −4$

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19.

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

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21.

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

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23.

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

Show solution

25.

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

Show solution

27.

ⓐ u ⓑ $| v | | v |$

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29.

ⓐ $| y | | y |$ ⓑ $m m$

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31.

ⓐ $| x^{3} | | x^{3} |$ ⓑ $y^{8} y^{8}$

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33.

ⓐ $x^{12} x^{12}$ ⓑ $| y^{11} | | y^{11} |$

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35.

ⓐ $x^{3} x^{3}$ ⓑ $| y^{3} | | y^{3} |$

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37.

ⓐ $m^{2} m^{2}$ ⓑ $n^{4} n^{4}$

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39.

ⓐ $7 | x | 7 | x |$ ⓑ $−9 | x^{9} | −9 | x^{9} |$

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41.

ⓐ $11 m^{10} 11 m^{10}$ ⓑ $−8 | a | −8 | a |$

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43.

ⓐ $2 x^{2} 2 x^{2}$ ⓑ $2 y^{2} 2 y^{2}$

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45.

ⓐ $6 a^{2} 6 a^{2}$ ⓑ $2 b^{4} 2 b^{4}$

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47.

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

Show solution

49.

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

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51.

Answers will vary.

Show solution

53.

Answers will vary.

Section 8.2 Exercises

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55.

$3 \sqrt{3} 3 \sqrt{3}$

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57.

$5 \sqrt{5} 5 \sqrt{5}$

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59.

$7 \sqrt{3} 7 \sqrt{3}$

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61.

$20 \sqrt{2} 20 \sqrt{2}$

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63.

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

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65.

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

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67.

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

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69.

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

Show solution

71.

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

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73.

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

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75.

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

Show solution

77.

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

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79.

ⓐ $−6 \sqrt[3]{4} −6 \sqrt[3]{4}$ ⓑ not real

Show solution

81.

ⓐ $−2 −2$ ⓑ not real

Show solution

83.

ⓐ $5 + 2 \sqrt{3} 5 + 2 \sqrt{3}$ ⓑ $5 - \sqrt{6} 5 - \sqrt{6}$

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85.

ⓐ $1 + 3 \sqrt{5} 1 + 3 \sqrt{5}$ ⓑ $1 + \sqrt{10} 1 + \sqrt{10}$

Show solution

87.

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

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89.

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

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91.

ⓐ $x^{2} x^{2}$ ⓑ $p^{3} p^{3}$ ⓒ $| q | | q |$

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93.

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

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95.

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

Show solution

97.

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

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99.

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

Show solution

101.

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

Show solution

103.

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

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105.

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

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107.

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

Show solution

109.

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

Show solution

111.

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

Show solution

113.

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

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115.

Answers will vary.

Show solution

117.

Answers will vary.

Section 8.3 Exercises

Show solution

119.

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

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121.

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

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123.

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

Show solution

125.

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

Show solution

127.

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

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129.

ⓐ 9 ⓑ 5 ⓒ 8

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131.

ⓐ 2 ⓑ 4 ⓒ 5

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133.

ⓐ $−6 −6$ ⓑ $−6 −6$ ⓒ $\frac{1}{6} \frac{1}{6}$

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135.

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

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137.

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

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139.

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

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141.

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

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143.

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

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145.

ⓐ 32,768 ⓑ $\frac{1}{729} \frac{1}{729}$ ⓒ 9

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147.

ⓐ 4 ⓑ $\frac{1}{9} \frac{1}{9}$ ⓒ not real

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149.

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

Show solution

151.

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

Show solution

153.

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

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155.

ⓐ $81 q^{2} 81 q^{2}$ ⓑ $a^{\frac{1}{2}} b a^{\frac{1}{2}} b$

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157.

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

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159.

ⓐ $r^{\frac{7}{2}} r^{\frac{7}{2}}$ ⓑ $\frac{6 s}{t} \frac{6 s}{t}$

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161.

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

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163.

Answers will vary.

Section 8.4 Exercises

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165.

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

Show solution

167.

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

Show solution

169.

ⓐ $4 \sqrt{2 a} 4 \sqrt{2 a}$ ⓑ 0

Show solution

171.

ⓐ $−2 \sqrt{3} −2 \sqrt{3}$ ⓑ $\sqrt[3]{4 p q} \sqrt[3]{4 p q}$

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173.

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

Show solution

175.

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

Show solution

177.

ⓐ $a^{2} \sqrt{2 a} a^{2} \sqrt{2 a}$ ⓑ 0

Show solution

179.

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

Show solution

181.

$4 y \sqrt{2} 4 y \sqrt{2}$

Show solution

183.

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

Show solution

185.

ⓐ $−30 \sqrt{2} −30 \sqrt{2}$ ) ⓑ $6 \sqrt[4]{2} 6 \sqrt[4]{2}$

Show solution

187.

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

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189.

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

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191.

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

Show solution

193.

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

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195.

$60 + 2 \sqrt{3} 60 + 2 \sqrt{3}$

Show solution

197.

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

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199.

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

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201.

$23 + 3 \sqrt{30} 23 + 3 \sqrt{30}$

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203.

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

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205.

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

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207.

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

Show solution

209.

Show solution

211.

$−227 −227$

Show solution

213.

$1919$

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215.

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

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217.

$5 \sqrt{3} 5 \sqrt{3}$

Show solution

219.

$9 \sqrt{2} 9 \sqrt{2}$

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221.

$− \sqrt[4]{5} − \sqrt[4]{5}$

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223.

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

Show solution

225.

$2 \sqrt{3} 2 \sqrt{3}$

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227.

$17 q^{2} 17 q^{2}$

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229.

$3 \sqrt{7} 3 \sqrt{7}$

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231.

$−42 \sqrt[3]{9} −42 \sqrt[3]{9}$

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233.

Show solution

235.

$29 - 7 \sqrt{17} 29 - 7 \sqrt{17}$

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237.

$54 - 36 \sqrt{2} 54 - 36 \sqrt{2}$

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239.

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

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241.

Answers will vary.

Show solution

243.

Answers will vary.

Section 8.5 Exercises

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245.

ⓐ $\frac{4}{3} \frac{4}{3}$ ⓑ $\frac{4}{3} \frac{4}{3}$

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247.

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

Show solution

249.

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

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251.

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

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253.

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

Show solution

255.

$x^{2} \sqrt{28 y} x^{2} \sqrt{28 y}$

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257.

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

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259.

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

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261.

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

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263.

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

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265.

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

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267.

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

Show solution

269.

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

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271.

$−2 (1 + \sqrt{5}) −2 (1 + \sqrt{5})$

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273.

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

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275.

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

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277.

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

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279.

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

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281.

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

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283.

Answers will vary.

Show solution

285.

Answers will vary.

Section 8.6 Exercises

Show solution

287.

$x = 14 x = 14$

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289.

no solution

Show solution

291.

$x = −4 x = −4$

Show solution

293.

$m = 14 m = 14$

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295.

$v = 17 v = 17$

Show solution

297.

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

Show solution

299.

no solution

Show solution

301.

$u = 3, u = 4 u = 3, u = 4$

Show solution

303.

$r = 1, r = 2 r = 1, r = 2$

Show solution

305.

$x = 10 x = 10$

Show solution

307.

$x = −8 x = −8$

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309.

$x = 8 x = 8$

Show solution

311.

$x = −4 x = −4$

Show solution

313.

$x = 7 x = 7$

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315.

$x = 3 x = 3$

Show solution

317.

$z = 21 z = 21$

Show solution

319.

$x = 42 x = 42$

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321.

$r = 3 r = 3$

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323.

$u = 3 u = 3$

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325.

$r = −2 r = −2$

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327.

$x = 1 x = 1$

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329.

$x = −8, x = 2 x = −8, x = 2$

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331.

$a = 0 a = 0$

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333.

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

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335.

$a = 4 a = 4$

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337.

$x = 0 x = 4 x = 0$

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339.

$x = 1 x = 5 x = 1$

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341.

$x = 9 x = 9$

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343.

$8.7 8.7$ feet

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345.

$4.7 4.7$ seconds

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347.

72 feet

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349.

Answers will vary.

Section 8.7 Exercises

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351.

ⓐ $f (5) = 4 f (5) = 4$ ⓑ no value at $x = 0 x = 0$

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353.

ⓐ $g (4) = 5 g (4) = 5$ ⓑ $g (8) = 7 g (8) = 7$

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355.

ⓐ $F (1) = 1 F (1) = 1$ ⓑ $F (−11) = 5 F (−11) = 5$

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357.

ⓐ $G (5) = 2 \sqrt{6} G (5) = 2 \sqrt{6}$ ⓑ $G (2) = 3 G (2) = 3$

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359.

ⓐ $g (6) = 2 g (6) = 2$ ⓑ $g (−2) = −2 g (−2) = −2$

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361.

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

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363.

ⓐ $f (0) = 0 f (0) = 0$ ⓑ $f (2) = 2 f (2) = 2$

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365.

ⓐ $g (1) = 0 g (1) = 0$ ⓑ $g (−3) = 2 g (−3) = 2$

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367.

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

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369.

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

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371.

$(2, \infty) (2, \infty)$

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373.

$(− \infty, −3] \cup (2, \infty) (− \infty, −3] \cup (2, \infty)$

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375.

$(− \infty, \infty) (− \infty, \infty)$

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377.

$(− \infty, \infty) (− \infty, \infty)$

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379.

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

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381.

$(− \infty, \infty) (− \infty, \infty)$

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383.

ⓐ domain: $[−1, \infty) [−1, \infty)$
ⓑ

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).

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385.

ⓐ domain: $[−4, \infty) [−4, \infty)$
ⓑ

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).

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387.

ⓐ domain: $[0, \infty) [0, \infty)$
ⓑ

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).

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389.

ⓐ domain: $[0, \infty) [0, \infty)$
ⓑ

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391.

ⓐ domain: $(− \infty, 3] (− \infty, 3]$
ⓑ

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).

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393.

ⓐ domain: $[0, \infty) [0, \infty)$
ⓑ