Introductory &amp; Intermediate Algebra for College Students

Problem 1

evaluate each expression, or state that the expression is not a real number.
$\sqrt{36}$

Amy Jiang

Numerade Educator

Problem 2

evaluate each expression, or state that the expression is not a real number.
$\sqrt{16}$

Amy Jiang

Numerade Educator

Problem 3

evaluate each expression, or state that the expression is not a real number.
$-\sqrt{36}$

Amy Jiang

Numerade Educator

Problem 4

evaluate each expression, or state that the expression is not a real number.
$-\sqrt{16}$

Amy Jiang

Numerade Educator

00:30

Problem 5

evaluate each expression, or state that the expression is not a real number.
$\sqrt{-36}$

Amy Jiang

Numerade Educator

00:21

Problem 6

evaluate each expression, or state that the expression is not a real number.
$\sqrt{-16}$

Amy Jiang

Numerade Educator

00:15

Problem 7

evaluate each expression, or state that the expression is not a real number.
$\sqrt{\frac{1}{25}}$

Amy Jiang

Numerade Educator

00:19

Problem 8

evaluate each expression, or state that the expression is not a real number.
$\sqrt{\frac{1}{49}}$

Amy Jiang

Numerade Educator

Problem 9

evaluate each expression, or state that the expression is not a real number.
$-\sqrt{\frac{9}{16}}$

Amy Jiang

Numerade Educator

Problem 10

evaluate each expression, or state that the expression is not a real number.
$-\sqrt{\frac{4}{25}}$

Amy Jiang

Numerade Educator

Problem 11

evaluate each expression, or state that the expression is not a real number.
$\sqrt{0.81}$

Amy Jiang

Numerade Educator

00:24

Problem 12

evaluate each expression, or state that the expression is not a real number.
$\sqrt{0.49}$

Amy Jiang

Numerade Educator

00:27

Problem 13

evaluate each expression, or state that the expression is not a real number.
$-\sqrt{0.04}$

Amy Jiang

Numerade Educator

00:34

Problem 14

evaluate each expression, or state that the expression is not a real number.
$-\sqrt{0.64}$

Amy Jiang

Numerade Educator

00:18

Problem 15

evaluate each expression, or state that the expression is not a real number.
$\sqrt{25-16}$

Amy Jiang

Numerade Educator

00:24

Problem 16

evaluate each expression, or state that the expression is not a real number.
$\sqrt{144+25}$

Amy Jiang

Numerade Educator

Problem 17

evaluate each expression, or state that the expression is not a real number.
$\sqrt{25}-\sqrt{16}$

Amy Jiang

Numerade Educator

Problem 18

evaluate each expression, or state that the expression is not a real number.
$\sqrt{144}+\sqrt{25}$

Amy Jiang

Numerade Educator

00:18

Problem 19

evaluate each expression, or state that the expression is not a real number.
$\sqrt{16-25}$

Amy Jiang

Numerade Educator

00:22

Problem 20

evaluate each expression, or state that the expression is not a real number.
$\sqrt{25-144}$

Amy Jiang

Numerade Educator

01:01

find the indicated function values for each function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, so state.
$f(x)=\sqrt{x-2} ; \quad f(18), f(3), f(2), f(-2)$

Amy Jiang

Numerade Educator

00:52

Problem 22

Amy Jiang

Numerade Educator

00:49

Problem 23

find the indicated function values for each function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, so state.
$g(x)=-\sqrt{2 x+3} ; \quad g(11), g(1), g(-1), g(-2)$

Amy Jiang

Numerade Educator

00:39

Problem 24

Amy Jiang

Numerade Educator

01:03

Problem 25

find the indicated function values for each function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, so state.
$h(x)=\sqrt{(x-1)^2} ; \quad h(5), h(3), h(0), h(-5)$

Amy Jiang

Numerade Educator

00:37

Problem 26

Amy Jiang

Numerade Educator

00:17

Problem 27

find the domain of each square root function. Then use the domain to match the radical function with its graph. [The graphs are labeled (a) through (f) and are shown in $[-10,10,1]$ by $[-10,10,1]$ viewing rectangles below and on the next page.]
$f(x)=\sqrt{x-3}$

Amy Jiang

Numerade Educator

00:41

Problem 28

Amy Jiang

Numerade Educator

00:19

Problem 29

Amy Jiang

Numerade Educator

Problem 30

Amy Jiang

Numerade Educator

Problem 31

Amy Jiang

Numerade Educator

00:29

Problem 32

Amy Jiang

Numerade Educator

Problem 33

simplify each expression.
$\sqrt{5^2}$

Matt Gibson

Numerade Educator

Problem 34

simplify each expression.
$\sqrt{7^2}$

Matt Gibson

Numerade Educator

Problem 35

simplify each expression.
$\sqrt{(-4)^2}$

Amy Jiang

Numerade Educator

Problem 36

simplify each expression.
$\sqrt{(-10)^2}$

Amy Jiang

Numerade Educator

01:50

Problem 37

simplify each expression.
$\sqrt{(x-1)^2}$

Noah Musser

Numerade Educator

Problem 38

simplify each expression.
$\sqrt{(x-2)^2}$

Amy Jiang

Numerade Educator

Problem 39

simplify each expression.
$\sqrt{36 x^4}$

Amy Jiang

Numerade Educator

00:19

Problem 40

simplify each expression.
$\sqrt{81 x^4}$

Amy Jiang

Numerade Educator

Problem 41

simplify each expression.
$-\sqrt{100 x^6}$

Amy Jiang

Numerade Educator

00:21

Problem 42

simplify each expression.
$-\sqrt{49 x^6}$

Amy Jiang

Numerade Educator

Problem 43

simplify each expression.
$\sqrt{x^2+12 x+36}$

Amy Jiang

Numerade Educator

Problem 44

simplify each expression.
$\sqrt{x^2+14 x+49}$

Amy Jiang

Numerade Educator

Problem 45

simplify each expression.
$-\sqrt{x^2-8 x+16}$

Amy Jiang

Numerade Educator

00:16

Problem 46

simplify each expression.
$-\sqrt{x^2-10 x+25}$

Amy Jiang

Numerade Educator

Problem 47

find each cube root.
$\sqrt[3]{27}$

Amy Jiang

Numerade Educator

Problem 48

find each cube root.
$\sqrt[3]{64}$

Amy Jiang

Numerade Educator

Problem 49

find each cube root.
$\sqrt[3]{-27}$

Amy Jiang

Numerade Educator

Problem 50

find each cube root.
$\sqrt[3]{-64}$

Amy Jiang

Numerade Educator

00:13

Problem 51

find each cube root.
$\sqrt[3]{\frac{1}{125}}$

Amy Jiang

Numerade Educator

Problem 52

find each cube root.
$\sqrt[3]{\frac{1}{1000}}$

Amy Jiang

Numerade Educator

00:17

Problem 53

find each cube root.
$\sqrt[3]{\frac{-27}{1000}}$

Amy Jiang

Numerade Educator

Problem 54

find each cube root.
$\sqrt[3]{\frac{-8}{125}}$

Amy Jiang

Numerade Educator

00:47

Problem 55

find the indicated function values for each function.
$f(x)=\sqrt[3]{x-1} ; f(28), f(9), f(0), f(-63)$

Amy Jiang

Numerade Educator

00:36

Problem 56

find the indicated function values for each function.
$f(x)=\sqrt[3]{x-3} ; f(30), f(11), f(2), f(-122)$

Amy Jiang

Numerade Educator

Problem 57

find the indicated function values for each function.
$g(x)=-\sqrt[3]{8 x-8} ; g(2), g(1), g(0)$

Amy Jiang

Numerade Educator

00:40

Problem 58

find the indicated function values for each function.
$g(x)=-\sqrt[3]{2 x+1} ; g(13), g(0), g(-63)$

Amy Jiang

Numerade Educator

00:05

Problem 59

find the indicated root, or state that the expression is not a real number.
$\sqrt[4]{1}$

Amy Jiang

Numerade Educator

00:04

Problem 60

find the indicated root, or state that the expression is not a real number.
$\sqrt[5]{1}$

Amy Jiang

Numerade Educator

Problem 61

find the indicated root, or state that the expression is not a real number.
$\sqrt[4]{16}$

Amy Jiang

Numerade Educator

Problem 62

find the indicated root, or state that the expression is not a real number.
$\sqrt[4]{81}$

Amy Jiang

Numerade Educator

Problem 63

find the indicated root, or state that the expression is not a real number.
$-\sqrt[4]{16}$

Amy Jiang

Numerade Educator

Problem 64

find the indicated root, or state that the expression is not a real number.
$-\sqrt[4]{81}$

Amy Jiang

Numerade Educator

Problem 65

find the indicated root, or state that the expression is not a real number.
$\sqrt[4]{-16}$

Amy Jiang

Numerade Educator

Problem 66

find the indicated root, or state that the expression is not a real number.
$\sqrt[4]{-81}$

Amy Jiang

Numerade Educator

Problem 67

find the indicated root, or state that the expression is not a real number.
$\sqrt[5]{-1}$

Amy Jiang

Numerade Educator

Problem 68

find the indicated root, or state that the expression is not a real number.
$\sqrt[7]{-1}$

Amy Jiang

Numerade Educator

00:15

Problem 69

find the indicated root, or state that the expression is not a real number.
$\sqrt[6]{-1}$

Amy Jiang

Numerade Educator

Problem 70

find the indicated root, or state that the expression is not a real number.
$\sqrt[8]{-1}$

Amy Jiang

Numerade Educator

Problem 71

find the indicated root, or state that the expression is not a real number.
$-\sqrt[4]{256}$

Amy Jiang

Numerade Educator

Problem 72

find the indicated root, or state that the expression is not a real number.
$-\sqrt[4]{10,000}$

Amy Jiang

Numerade Educator

Problem 73

find the indicated root, or state that the expression is not a real number.
$\sqrt[6]{64}$

Amy Jiang

Numerade Educator

Problem 74

find the indicated root, or state that the expression is not a real number.
$\sqrt[5]{32}$

Amy Jiang

Numerade Educator

Problem 75

find the indicated root, or state that the expression is not a real number.
$-\sqrt[5]{32}$

Amy Jiang

Numerade Educator

01:02

Problem 76

find the indicated root, or state that the expression is not a real number.
$-\sqrt[6]{64}$

Ernest Castorena

Numerade Educator

Problem 77

simplify each expression. Include absolute value bars where necessary.
$\sqrt[3]{x^3}$

Amy Jiang

Numerade Educator

Problem 78

simplify each expression. Include absolute value bars where necessary.
$\sqrt[5]{x^5}$

Amy Jiang

Numerade Educator

Problem 79

simplify each expression. Include absolute value bars where necessary.
$\sqrt[4]{y^4}$

Amy Jiang

Numerade Educator

Problem 80

simplify each expression. Include absolute value bars where necessary.
$\sqrt[6]{y^6}$

Amy Jiang

Numerade Educator

Problem 81

simplify each expression. Include absolute value bars where necessary.
$\sqrt[3]{-8 x^3}$

Amy Jiang

Numerade Educator

00:29

Problem 82

simplify each expression. Include absolute value bars where necessary.
$\sqrt[3]{-125 x^3}$

Amy Jiang

Numerade Educator

Problem 83

simplify each expression. Include absolute value bars where necessary.
$\sqrt[3]{(-5)^3}$

Amy Jiang

Numerade Educator

Problem 84

simplify each expression. Include absolute value bars where necessary.
$\sqrt[3]{(-6)^3}$

Amy Jiang

Numerade Educator

Problem 85

simplify each expression. Include absolute value bars where necessary.
$\sqrt[4]{(-5)^4}$

Amy Jiang

Numerade Educator

Problem 86

simplify each expression. Include absolute value bars where necessary.
$\sqrt[6]{(-6)^6}$

Amy Jiang

Numerade Educator

Problem 87

simplify each expression. Include absolute value bars where necessary.
$\sqrt[4]{(x+3)^4}$

Amy Jiang

Numerade Educator

Problem 88

simplify each expression. Include absolute value bars where necessary.
$\sqrt[4]{(x+5)^4}$

Amy Jiang

Numerade Educator

00:24

Problem 89

simplify each expression. Include absolute value bars where necessary.
$\sqrt[5]{-32(x-1)^5}$

Amy Jiang

Numerade Educator

Problem 90

simplify each expression. Include absolute value bars where necessary.
$\sqrt[5]{-32(x-2)^5}$

Amy Jiang

Numerade Educator

00:37

Problem 91

complete each table and graph the given function. Identify the function's domain and range.
$f(x)=\sqrt{x}+3$
Table Can't Copy

Amy Jiang

Numerade Educator

Problem 92

complete each table and graph the given function. Identify the function's domain and range.
$f(x)=\sqrt{x}-2$
Table Can't Copy

Amy Jiang

Numerade Educator

00:37

Problem 93

complete each table and graph the given function. Identify the function's domain and range.
$f(x)=\sqrt{x-3}$
Table Can't Copy

Amy Jiang

Numerade Educator

00:27

Problem 94

complete each table and graph the given function. Identify the function's domain and range.
$f(x)=\sqrt{4-x}$
Table Can't Copy

Amy Jiang

Numerade Educator

00:40

Problem 95

find the domain of each function.
$f(x)=\frac{\sqrt[3]{x}}{\sqrt{30-2 x}}$

Amy Jiang

Numerade Educator

01:05

Problem 96

find the domain of each function.
$f(x)=\frac{\sqrt[3]{x}}{\sqrt{80-5 x}}$

Amy Jiang

Numerade Educator

01:33

Problem 97

find the domain of each function.
$f(x)=\frac{\sqrt{x-1}}{\sqrt{3-x}}$

Amy Jiang

Numerade Educator

00:51

Problem 98

find the domain of each function.
$f(x)=\frac{\sqrt{x-2}}{\sqrt{7-x}}$

Amy Jiang

Numerade Educator

Problem 99

evaluate each expression.
$\sqrt[3]{\sqrt[4]{16}+\sqrt{625}}$

Amy Jiang

Numerade Educator

01:15

Problem 100

evaluate each expression.
$\sqrt[3]{\sqrt{\sqrt{169}}+\sqrt{9}}+\sqrt{\sqrt[3]{1000}+\sqrt[3]{216}}$

Amy Jiang

Numerade Educator

00:57

Problem 101

The function $f(x)=2.9 \sqrt{x}+20.1$ models the median height, $f(x)$, in inches, of boys who are $x$ months of age. The graph of $f$ is shown.
a. According to the model, what is the median height of boys who are 48 months, or four years, old? Use a calculator and round to the nearest tenth of an inch. The actual median height for boys at 48 months is 40.8 inches. Does the model overestimate or underestimate the actual height? By how much?
b. Use the model to find the average rate of change, in inches per month, between birth and 10 months. Round to the nearest tenth.
c. Use the model to find the average rate of change, in inches per month, between 50 and 60 months. Round to the nearest tenth. How does this compare with your answer in part (b)? How is this difference shown by the graph?

Amy Jiang

Numerade Educator

03:03

Problem 102

The function $f(x)=3.1 \sqrt{x}+19$ models the median height, $f(x)$, in inches, of girls who are $x$ months of age. The graph of $f$ is shown.

a. According to the model, what is the median height of girls who are 48 months, or four years, old? Use a calculator and round to the nearest tenth of an inch. The actual median height for girls at 48 months is 40.2 inches. Does the model overestimate or underestimate the actual height? By how much?
b. Use the model to find the average rate of change, in inches per month, between birth and 10 months. Round to the nearest tenth.
c. Use the model to find the average rate of change, in inches per month, between 50 and 60 months. Round to the nearest tenth. How does this compare with your answer in part (b)? How is this difference shown by the graph?

Karly Williams

Numerade Educator

01:35

Problem 103

A motorist is involved in an accident. A police officer measures the car's skid marks to be 245 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed limit is 50 miles per hour and the motorist tells the officer he was not speeding, should the officer believe him? Explain.

Narayan Hari

Numerade Educator

01:33

Problem 104

A motorist is involved in an accident. A police officer measures the car's skid marks to be 45 feet long. Use the function described at the bottom of the previous page to estimate the speed at which the motorist was traveling before braking. If the posted speed limit is 35 miles per hour and the motorist tells the officer she was not speeding, should the officer believe her? Explain.

Narayan Hari

Numerade Educator

00:53

Problem 105

What are the square roots of 36 ? Explain why each of these numbers is a square root.

Ashley High

Numerade Educator

Problem 106

What does the symbol $\sqrt{ }$ denote? Which of your answers in Exercise 105 is given by this symbol? Write the symbol needed to obtain the other answer.

Amy Jiang

Numerade Educator

Problem 107

Explain why $\sqrt{-1}$ is not a real number.

Amy Jiang

Numerade Educator

00:21

Problem 108

Explain how to find the domain of a square root function.

Michelle Nguyen

Numerade Educator

Problem 109

Explain how to simplify $\sqrt{a^2}$. Give an example with your explanation.

Amy Jiang

Numerade Educator

Problem 110

Explain why $\sqrt[3]{8}$ is 2 . Then describe what is meant by the cube root of a real number.

Amy Jiang

Numerade Educator

02:03

Problem 111

Describe two differences between odd and even roots.

Sheryl Ezze

Numerade Educator

01:02

Problem 112

Explain how to simplify $\sqrt[n]{a^n}$ if $n$ is even and if $n$ is odd. Give examples with your explanations.

Brittany Booher

Numerade Educator

01:35

Problem 113

Explain the meaning of the words radical, radicand, and index. Give an example with your explanation.

Christopher Stanley

Numerade Educator

04:53

Problem 114

Describe the trend in a boy's growth from birth through five years, shown in the graph for Exercise 101. Why is a square root function a useful model for the data?

Anurag Kumar

Numerade Educator

04:16

Problem 115

Use a graphing utility to graph $y_1=\sqrt{x}, y_2=\sqrt{x+4}$, and $y_3=\sqrt{x-3}$ in the same $[-5,10,1]$ by $[0,6,1]$ viewing rectangle. Describe one similarity and one difference that you observe among the graphs. Use the word shift in your response.

Jennifer White

Numerade Educator

04:16

Problem 116

Use a graphing utility to graph $y=\sqrt{x}, y=\sqrt{x}+4$, and $y=\sqrt{x}-3$ in the same $[-1,10,1]$ by $[-10,10,1]$ viewing rectangle. Describe one similarity and one difference that you observe among the graphs.

Jennifer White

Numerade Educator

Problem 117

Use a graphing utility to graph $f(x)=\sqrt{x}, g(x)=-\sqrt{x}$, $h(x)=\sqrt{-x}$, and $k(x)=-\sqrt{-x}$ in the same $[-10,10,1]$ by $[-4,4,1]$ viewing rectangle. Use the graphs to describe the domains and the ranges of functions $f, g, h$, and $k$.

Amy Jiang

Numerade Educator

03:19

Problem 118

Use a graphing utility to graph $y_1=\sqrt{x^2}$ and $y_2=-x$ in the same viewing rectangle.
a. For what values of $x$ is $\sqrt{x^2}=-x$ ?
b. For what values of $x$ is $\sqrt{x^2} \neq-x$ ?

Gavin Liang

Numerade Educator

Problem 119

$\sqrt[4]{(-8)^4}$ cannot be positive 8 because the power and the index cancel each other.

Nick Johnson

Numerade Educator

01:13

Problem 120

If I am given any real number, that number has exactly one odd root and two even roots.

Ankit Gupta

Numerade Educator

01:23

Problem 121

I need to restrict the domains of radical functions with even indices, but these restrictions are not necessary when indices are odd.

Ankit Gupta

Numerade Educator

Problem 122

Using my calculator, I determined that $5^5=3125$, so 5 must be the fifth root of 3125 .

Check back soon!

00:22

Problem 123

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The domain of $f(x)=\sqrt[3]{x-4}$ is $[4, \infty)$.

Amy Jiang

Numerade Educator

00:27

Problem 124

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If $n$ is odd and $b$ is negative, then $\sqrt[n]{b}$ is not a real number.

Amy Jiang

Numerade Educator

Problem 125

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
If $x=-2$, then $\sqrt{x^6}=x^3$.

Amy Jiang

Numerade Educator

00:18

Problem 126

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The expression $\sqrt[n]{4}$ represents increasingly larger numbers for $n=2,3,4,5,6$, and so on.

Amy Jiang

Numerade Educator

02:08

Problem 127

Write a function whose domain is $(-\infty, 5]$.

Jamie Fife

Numerade Educator

01:50

Problem 128

Let $f(x)=\sqrt{x-3}$ and $g(x)=\sqrt{x+1}$. Find the domain of $f+g$ and $\frac{f}{g}$.

Ankit Gupta

Numerade Educator

Problem 129

Simplify: $\sqrt{(2 x+3)^{10}}$.

Cory Kuzinski

Numerade Educator