Elementary and Intermediate Algebra

Marvin L. Bittinger, David J. Ellenbogen,Barbara L. Johnson

Chapter 10

Exponents and Radical Functions

Educators


Problem 1

Select the appropriate word to complete each of the following.
Every positive number has ______ square root(s).

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Problem 2

Select the appropriate word to complete each of the following.
The principal square root is never ______.

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Problem 3

Select the appropriate word to complete each of the following.
For any ____ number $a,$ we have $\sqrt{a^{2}}=a$

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Problem 4

Select the appropriate word to complete each of the following.
For any ______ number $a,$ we have $\sqrt{a^{2}}=-a$

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Problem 5

Select the appropriate word to complete each of the following.
If $a$ is a whole number that is not a perfect square, then $\sqrt{a}$ is a(n) ______ number.

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Problem 6

Select the appropriate word to complete each of the following.
The domain of the function $f$ given by $f(x)=\sqrt[3]{x}$ is the set of all ______ numbers..

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Problem 7

Select the appropriate word to complete each of the following.
If $\sqrt[4]{x}$ is a real number, then $x$ must be ________.

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Problem 8

Select the appropriate word to complete each of the following.
If $\sqrt[3]{x}$ is negative, then $x$ must be _____.

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Problem 9

For each number, find all of its square roots.
$$
49
$$

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Problem 10

For each number, find all of its square roots.
$$
81
$$

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Problem 11

For each number, find all of its square roots.
$$
144
$$

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Problem 12

For each number, find all of its square roots.
$$
9
$$

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Problem 13

For each number, find all of its square roots.
$$
400
$$

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Problem 14

For each number, find all of its square roots.
$$
2500
$$

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Problem 15

For each number, find all of its square roots.
$$
900
$$

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Problem 16

For each number, find all of its square roots.
$$
225
$$

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Problem 17

Simplify.
$$
\sqrt{49}
$$

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Problem 18

Simplify.
$$
\sqrt{144}
$$

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Problem 19

Simplify.
$$
-\sqrt{16}
$$

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Problem 20

Simplify.
$$
-\sqrt{100}
$$

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Problem 21

Simplify.
$$
\sqrt{\frac{36}{49}}
$$

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Problem 22

Simplify.
$$
\sqrt{\frac{4}{9}}
$$

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Problem 23

Simplify.
$$
-\sqrt{\frac{16}{81}}
$$

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Problem 24

Simplify.
$$
-\sqrt{\frac{81}{144}}
$$

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Problem 25

Simplify.
$$
\sqrt{0.04}
$$

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Problem 26

Simplify.
$$
\sqrt{0.36}
$$

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Problem 27

Simplify.
$$
\sqrt{0.0081}
$$

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Problem 28

Simplify.
$$
\sqrt{0.0016}
$$

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Problem 29

Identify the radicand and the index for each expression.
$$
5 \sqrt{p^{2}}+4
$$

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Problem 30

Identify the radicand and the index for each expression.
$$
-7 \sqrt{y^{2}}-8
$$

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Problem 32

Identify the radicand and the index for each expression.
$$
x y \sqrt[5]{\frac{x}{y+4}}
$$

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Problem 32

Identify the radicand and the index for each expression.
$$
\frac{a}{b} \sqrt[6]{a^{2}+1}
$$

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Problem 33

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
f(t)=\sqrt{5 t-10} ; f(3), f(2), f(1), f(-1)
$$

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Problem 34

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
g(x)=\sqrt{x^{2}-25} ; g(-6), g(3), g(6), g(13)
$$

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Problem 35

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
t(x)=-\sqrt{2 x^{2}-1} ; t(5), t(0), t(-1), t\left(-\frac{1}{2}\right)
$$

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Problem 36

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
p(z)=\sqrt{2 z-20} ; p(4), p(10), p(12), p(0)
$$

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Problem 37

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
f(t)=\sqrt{t^{2}+1} ; f(0), f(-1), f(-10)
$$

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Numerade Educator

Problem 38

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
g(x)=-\sqrt{(x+1)^{2}} ; g(-3), g(4), g(-5)
$$

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Problem 39

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{64 x^{2}}
$$

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Problem 40

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{25 t^{2}}
$$

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Problem 41

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{(-4 b)^{2}}
$$

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Problem 42

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{(-7 c)^{2}}
$$

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Problem 43

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{(8-t)^{2}}
$$

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Problem 44

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{(a+3)^{2}}
$$

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Problem 45

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{y^{2}+16 y+64}
$$

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Problem 46

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{x^{2}-4 x+4}
$$

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Problem 47

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{4 x^{2}+28 x+49}
$$

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Problem 48

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{9 x^{2}-30 x+25}
$$

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Problem 49

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
-\sqrt[4]{256}
$$

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Problem 50

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
-\sqrt[4]{625}
$$

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Problem 51

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[5]{-1}
$$

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Problem 52

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
-\sqrt[3]{-1000}
$$

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Problem 53

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
-\sqrt[5]{-\frac{32}{243}}
$$

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Problem 54

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[5]{-\frac{1}{32}}
$$

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Problem 55

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[6]{x^{6}}
$$

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Problem 56

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[8]{y^{8}}
$$

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Problem 57

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[9]{t^{9}}
$$

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Problem 58

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[5]{a^{5}}
$$

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Problem 59

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[4]{(6 a)^{4}}
$$

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Problem 60

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[4]{(8 y)^{4}}
$$

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Problem 61

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[10]{(-6)^{10}}
$$

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Problem 62

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[12]{(-10)^{12}}
$$

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Problem 63

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[414]{(a+b)^{414}}
$$

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Problem 64

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt[1976]{(2 a+b)^{1976}}
$$

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Problem 65

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{a^{22}}
$$

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Problem 66

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{x^{10}}
$$

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Problem 67

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{-25}
$$

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Problem 68

Simplify. Remember to use absolute-value notation when necessary. If a root cannot be simplified, state this.
$$
\sqrt{-16}
$$

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Problem 69

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{16 x^{2}}
$$

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Problem 70

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{25 t^{2}}
$$

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Problem 71

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
-\sqrt{(3 t)^{2}}
$$

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Problem 72

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
-\sqrt{(7 c)^{2}}
$$

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Problem 73

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{(a+1)^{2}}
$$

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Problem 74

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{(5+b)^{2}}
$$

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Problem 75

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{9 t^{2}-12 t+4}
$$

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Problem 76

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{25 t^{2}-20 t+4}
$$

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Problem 77

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt[3]{27 a^{3}}
$$

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Problem 78

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
-\sqrt[3]{64 y^{3}}
$$

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Problem 79

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt[4]{16 x^{4}}
$$

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Problem 80

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt[4]{81 x^{4}}
$$

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Problem 81

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt[5]{(x-1)^{5}}
$$

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Problem 82

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
-\sqrt[7]{(1-x)^{7}}
$$

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Problem 83

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
-\sqrt[3]{-125 y^{3}}
$$

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Problem 84

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
-\sqrt[3]{-64 x^{3}}
$$

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Problem 85

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{t^{18}}
$$

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Problem 86

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{a^{14}}
$$

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Problem 87

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{(x-2)^{8}}
$$

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Problem 88

Simplify. Assume that no radicands were formed by raising negative quantities to even powers.
$$
\sqrt{(x+3)^{10}}
$$

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Problem 89

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
f(x)=\sqrt[3]{x+1} ; f(7), f(26), f(-9), f(-65)
$$

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Problem 90

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
g(x)=-\sqrt[3]{2 x-1} ; g(0), g(-62), g(-13), g(63)
$$

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Problem 91

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
g(t)=\sqrt[4]{t-3} ; g(19), g(-13), g(1), g(84)
$$

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Problem 92

For each function, find the specified function value, if it exists. If it does not exist, state this.
$$
f(t)=\sqrt[4]{t+1} ; f(0), f(15), f(-82), f(80)
$$

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Problem 93

Determine the domain of each function described.
$$
f(x)=\sqrt{x-6}
$$

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Problem 94

Determine the domain of each function described.
$$
g(x)=\sqrt{x+8}
$$

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Problem 95

Determine the domain of each function described.
$$
g(t)=\sqrt[4]{t+8}
$$

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Problem 96

Determine the domain of each function described.
$$
f(x)=\sqrt[4]{x-9}
$$

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Problem 97

Determine the domain of each function described.
$$
g(x)=\sqrt[4]{2 x-10}
$$

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Problem 98

Determine the domain of each function described.
$$
g(t)=\sqrt[3]{2 t-6}
$$

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Problem 99

Determine the domain of each function described.
$$
f(t)=\sqrt[5]{8-3 t}
$$

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Problem 100

Determine the domain of each function described.
$$
f(t)=\sqrt[6]{4-3 t}
$$

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Problem 101

Determine the domain of each function described.
$$
h(z)=-\sqrt[6]{5 z+2}
$$

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Problem 102

Determine the domain of each function described.
$$
d(x)=-\sqrt[4]{7 x-5}
$$

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Problem 103

Determine the domain of each function described.
$$
f(t)=7+\sqrt[8]{t^{8}}
$$

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Problem 104

Determine the domain of each function described.
$$
g(t)=9+\sqrt[6]{t^{6}}
$$

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Problem 105

Determine algebraically the domain of each function described. Then use a graphing calculator to confirm your answer and to estimate the range.
$$
f(x)=\sqrt{5-x}
$$

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Problem 106

Determine algebraically the domain of each function described. Then use a graphing calculator to confirm your answer and to estimate the range.
$$
g(x)=\sqrt{2 x+1}
$$

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Problem 107

Determine algebraically the domain of each function described. Then use a graphing calculator to confirm your answer and to estimate the range.
$$
f(x)=1-\sqrt{x+1}
$$

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Problem 108

Determine algebraically the domain of each function described. Then use a graphing calculator to confirm your answer and to estimate the range.
$$
g(x)=2+\sqrt{3 x-5}
$$

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Problem 109

Determine algebraically the domain of each function described. Then use a graphing calculator to confirm your answer and to estimate the range.
$$
g(x)=3+\sqrt{x^{2}+4}
$$

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Problem 110

Determine algebraically the domain of each function described. Then use a graphing calculator to confirm your answer and to estimate the range.
$$
f(x)=5-\sqrt{3 x^{2}+1}
$$

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Problem 111

Match each function with one of the following graphs without using a calculator.
(GRAPH CAN'T COPY)
$$
f(x)=\sqrt{x-4}
$$

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Problem 112

Match each function with one of the following graphs without using a calculator.
(GRAPH CAN'T COPY)
$$
g(x)=\sqrt{x+4}
$$

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Problem 113

Match each function with one of the following graphs without using a calculator.
(GRAPH CAN'T COPY)
$$
h(x)=\sqrt{x^{2}+4}
$$

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Problem 114

Match each function with one of the following graphs without using a calculator.
(GRAPH CAN'T COPY)
$$
f(x)=-\sqrt{x-4}
$$

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Problem 115

Determine whether a radical function would be a good model of the given situation.
The table below lists the average salary of a major-league baseball player, based on the number of years in his contract.
(TABLE CAN'T COPY)

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Problem 116

Determine whether a radical function would be a good model of the given situation.
The number of gallons per minute discharged from a fire hose depends on the diameter of the hose and the nozzle pressure. The table below lists the amount of water flow for a 2 -in. diameter solid bore nozzle at various nozzle
pressures.
(TABLE CAN'T COPY)

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Problem 117

Determine whether a radical function would be a good model of the given situation.
Koi, a popular fish for backyard pools, grow from $\frac{1}{40} \mathrm{cm}$ when newly hatched to an average length of $80 \mathrm{cm} .$ The table below lists the length of a koi at various ages.
(TABLE CAN'T COPY)

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Problem 118

Determine whether a radical function would be a good model of the given situation.
The table below lists the average size of United States' farms for various years from 1960 to 2007
(TABLE CAN'T COPY)

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Problem 119

Determine whether a radical function would be a good model of the given situation.
The table below lists the amount of federal funds allotted to the National Cancer Institute for cancer research in the United States from 2005 to 2010.
(TABLE CAN'T COPY)

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Problem 120

Determine whether a radical function would be a good model of the given situation.
The table below lists the percent of households with basic cable television service for various years.
(TABLE CAN'T COPY)

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Numerade Educator

Problem 121

Firefighting. The water flow, in number of gallons per minute (GPM), for a 2 -in. diameter solid bore nozzle is given by
$$
f(x)=118.8 \sqrt{x}
$$
where $x$ is the nozzle pressure, in pounds per square inch (psi). (See Exercise $116 .$ ) Use the function to estimate the water flow when the nozzle pressure is 50 psi and when it is 175 psi.

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Problem 122

Koi Growth. The length, in centimeters, of a koi of age $x$ months can be estimated using the function
$$
f(x)=0.27+\sqrt{71.94 x-164.41}
$$
(See Exercise $117 .$ ) Use the function to estimate the length of a koi at 8 months and at 20 months.

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Problem 123

Explain how to write the negative square root of a number using radical notation.

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Problem 124

Does the square root of a number's absolute value always exist? Why or why not?

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Problem 125

Simplify. Do not use negative exponents in your answer.
$$\left(a^{2} b\right)\left(a^{4} b\right)$$

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Problem 126

Simplify. Do not use negative exponents in your answer.
$$
\left(3 x y^{8}\right)\left(5 x^{2} y\right)
$$

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Problem 127

Simplify. Do not use negative exponents in your answer.
$$
\left(5 x^{2} y^{-3}\right)^{3}
$$

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Problem 128

Simplify. Do not use negative exponents in your answer.
$$
\left(2 a^{-1} b^{2} c\right)^{-3}
$$

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Problem 129

Simplify. Do not use negative exponents in your answer.
$$
\left(\frac{10 x^{-1} y^{5}}{5 x^{2} y^{-1}}\right)^{-1}
$$

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Problem 130

Simplify. Do not use negative exponents in your answer.
$$
\left(\frac{8 x^{3} y^{-2}}{2 x z^{4}}\right)^{-2}
$$

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Problem 131

If the domain of $f=[1, \infty)$ and the range of $f=[2, \infty),$ find a possible expression for $f(x)$ and explain how such an expression is formulated.

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Problem 132

Natasha obtains the following graph of
$$
f(x)=\sqrt{x^{2}-4 x-12}
$$
and concludes that the domain of $f$ is $(-\infty,-2]$ Is she correct? If not, what mistake is she making?
(GRAPH CAN'T COPY)

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Numerade Educator

Problem 133

Could the following situation possibly be modeled using a radical function? Why or why not? "For each year, the yield increases. The amount of increase is smaller each year."

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Problem 134

Could the following situation possibly be modeled using a radical function? Why or why not? "For each year, the costs increase. The amount of increase is the same each year."

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Problem 135

A parking lot has attendants to park the cars. The number $N$ of stalls needed for waiting cars before attendants can get to them is given by the formula $N=2.5 \sqrt{A},$ where $A$ is the number of arrivals in peak hours. Find the number of spaces needed for the given number of arrivals in peak hours:
(a) $25 ;$ (b) $36 ;(c) 49 ;$ (d) 64

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Problem 136

Determine the domain of each function described. Then draw the graph of each function.
$$
g(x)=\sqrt{x}+5
$$

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Numerade Educator

Problem 137

Determine the domain of each function described. Then draw the graph of each function.
$$
f(x)=\sqrt{x+5}
$$

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Numerade Educator

Problem 138

Determine the domain of each function described. Then draw the graph of each function.
$$
f(x)=\sqrt{x-2}
$$

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Problem 139

Determine the domain of each function described. Then draw the graph of each function.
$$
g(x)=\sqrt{x}-2
$$

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Numerade Educator

Problem 140

Determine the domain of each function described. Then draw the graph of each function.
Find the domain of $f$ if
$$
f(x)=\frac{\sqrt{x+3}}{\sqrt[4]{2-x}}
$$

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Problem 141

Determine the domain of each function described. Then draw the graph of each function.
Find the domain of $g$ if
$$
g(x)=\frac{\sqrt[4]{5-x}}{\sqrt[6]{x+4}}
$$

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Numerade Educator

Problem 142

Determine the domain of each function described. Then draw the graph of each function.
Find the domain of $F$ if $F(x)=\frac{x}{\sqrt{x^{2}-5 x-6}}$

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Problem 143

Determine the domain of each function described. Then draw the graph of each function.
Examine the graph of the data in Exercise $119 .$ What type of function could be used to model the data?

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Problem 144

Determine the domain of each function described. Then draw the graph of each function.
In Exercise $120,$ a radical function could be used to model the data through $2000 .$ What might have caused the number of basic cable subscribers to change its pattern of growth?

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Numerade Educator