# Algebra 2

## Educators

Problem 1

Find all the real square roots of each number.
$$225$$

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Problem 2

Find all the real square roots of each number.
$$0.0049$$

Problem 3

Find all the real square roots of each number.
$$-\frac{1}{121}$$

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Problem 4

Find all the real square roots of each number.
$$\frac{64}{169}$$

Problem 5

Find all the real cube roots of each number.
$$-64$$

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Problem 6

Find all the real cube roots of each number.
$$0.125$$

Problem 7

Find all the real cube roots of each number.
$$-\frac{27}{216}$$

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Problem 8

Find all the real cube roots of each number.
$$0.000343$$

Problem 9

Find all the real fourth roots of each number.
$$16$$

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Problem 10

Find all the real fourth roots of each number.
$$-16$$

Problem 11

Find all the real fourth roots of each number.
$$0.0081$$

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Problem 12

Find all the real fourth roots of each number.
$$\frac{10,000}{81}$$

Problem 13

Find each real-number root.
$$\sqrt{36}$$

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Problem 14

Find each real-number root.
$$-\sqrt{36}$$

Problem 15

Find each real-number root.
$$\sqrt{-36}$$

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Problem 16

Find each real-number root.
$$\sqrt{0.36}$$

Problem 17

Find each real-number root.
$$-\sqrt[3]{64}$$

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Problem 18

Find each real-number root.
$$\sqrt[3]{-64}$$

Problem 19

Find each real-number root.
$$-\sqrt[4]{81}$$

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Problem 20

Find each real-number root.
$$\sqrt[4]{-81}$$

Problem 21

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt{16 x^{2}}$$

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Problem 22

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt{0.25 x^{6}}$$

Problem 23

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt{x^{8} y^{18}}$$

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Problem 24

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt{64 b^{48}}$$

Problem 25

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[3]{-64 a^{3}}$$

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Problem 26

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[3]{27 y^{6}}$$

Problem 27

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[4]{x^{8} y^{12}}$$

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Problem 28

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[5]{32 y^{10}}$$

Problem 29

Geometry The formula for the volume of a sphere is $V=\frac{4}{3} \pi r^{3} .$ Find the radius to the nearest hundredth of a sphere with each volume.
$$10 \mathrm{in} .^{3}$$

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Problem 30

Geometry The formula for the volume of a sphere is $V=\frac{4}{3} \pi r^{3} .$ Find the radius to the nearest hundredth of a sphere with each volume.
$$20 \mathrm{ft}^{3}$$

Problem 31

Geometry The formula for the volume of a sphere is $V=\frac{4}{3} \pi r^{3} .$ Find the radius to the nearest hundredth of a sphere with each volume.
$$0.45 \mathrm{cm}^{3}$$

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Problem 32

Geometry The formula for the volume of a sphere is $V=\frac{4}{3} \pi r^{3} .$ Find the radius to the nearest hundredth of a sphere with each volume.
$$0.002 \mathrm{mm}^{3}$$

Problem 33

Find the two real-number solutions of each equation.
$$x^{2}=100$$

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Problem 34

Find the two real-number solutions of each equation.
$$x^{4}=1$$

Problem 35

Find the two real-number solutions of each equation.
$$x^{2}=0.25$$

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Problem 36

Find the two real-number solutions of each equation.
$$x^{4}=\frac{16}{81}$$

Problem 37

Arrange the numbers $\sqrt[3]{-64},-\sqrt[3]{-64}, \sqrt{64},$ and $\sqrt[6]{64}$ in order from least to greatest.

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Problem 38

Boat Building Boat builders share an old rule of thumb for sailboats. The maximum speed $K$ in knots is 1.35 times the square root of the length $L$ in feet of the boat's waterline.
a. A customer is planning to order a sailboat with a maximum speed of 8 knots. How long should the waterline be?
b. How much longer would the waterline have to be to achieve a maximum speed of 10 knots?

Problem 39

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[3]{0.125}$$

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Problem 40

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[3]{\frac{8}{216}}$$

Problem 41

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[4]{0.0016}$$

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Problem 42

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[4]{\frac{1}{256}}$$

Problem 43

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[4]{16 c^{4}}$$

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Problem 44

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[3]{81 x^{3} y^{6}}$$

Problem 45

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt{144 x^{3} y^{4} z^{5}}$$

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Problem 46

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[5]{y^{20}}$$

Problem 47

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[5]{-y^{20}}$$

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Problem 48

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[5]{k^{15}}$$

Problem 49

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[5]{-k^{15}}$$

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Problem 50

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt{(x+3)^{2}}$$

Problem 51

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt{(x+1)^{4}}$$

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Problem 52

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[2 n]{x^{2 n}}$$

Problem 53

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[2 n]{x^{4 n}}$$

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Problem 54

Simplify each radical expression. Use absolute value symbols when needed.
$$\sqrt[2 n]{x^{6 n}}$$

Problem 55

Open-Ended Find three radical expressions that simplify to $-2 x^{2}$

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Problem 56

Critical Thinking For what positive integers $n$ is each of the statements true?.
a. If $x^{n}=b,$ then $x$ is an $n$ th root of $b$
b. If $x^{n}=b,$ then $x=\sqrt[n]{b}$

Problem 57

Writing Is 10 a first root of 10$?$ Explain.

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Problem 58

Tell whether each equation is true for all, some, or no values of the variable. Explain your answers.
$$\sqrt{x^{4}}=x^{2}$$

Problem 59

Tell whether each equation is true for all, some, or no values of the variable. Explain your answers.
$$\sqrt{x^{6}}=x^{3}$$

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Problem 60

Tell whether each equation is true for all, some, or no values of the variable. Explain your answers.
$$\sqrt[3]{x^{8}}=x^{2}$$

Problem 61

Tell whether each equation is true for all, some, or no values of the variable. Explain your answers.
$$\sqrt[3]{x^{3}}=|x|$$

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Problem 62

Simplify each radical expression. $n$ is an even number.
$$\sqrt[n]{m^{n}}$$

Problem 63

Simplify each radical expression. $n$ is an even number.
$$\sqrt[n]{m^{2 n}}$$

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Problem 64

Simplify each radical expression. $n$ is an even number.
$$\sqrt[n]{m^{3 n}}$$

Problem 65

Simplify each radical expression. $n$ is an even number.
$$\sqrt[n]{m^{4 n}}$$

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Problem 66

Simplify each radical expression. $n$ is an odd number.
$$\sqrt[n]{m^{n}}$$

Problem 67

Simplify each radical expression. $n$ is an odd number.
$$\sqrt[n]{m^{2 n}}$$

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Problem 68

Simplify each radical expression. $n$ is an odd number.
$$\sqrt[n]{m^{3 n}}$$

Problem 69

Simplify each radical expression. $n$ is an odd number.
$$\sqrt[n]{m^{4 n}}$$

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Problem 70

Which equation has more than one real-number solution?
$$\begin{array}{llll}{\text { A. } x^{2}=0} & {\text { B. } x^{2}=1} & {\text { C. } x^{2}=-1} & {\text { D. } x^{3}=-1}\end{array}$$

Problem 71

Which number is greatest?
$$\begin{array}{lllll}{\text { F. } \sqrt{0.5}} & {\text { G. } \sqrt[3]{0.5}} & {\text { H. } \sqrt[4]{0.5}} & {\text { J. } \sqrt[5]{0.5}}\end{array}$$

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Problem 72

Which statement is NOT true?
$$\begin{array}{ll}{\text { A. }-3=-\sqrt{9}} & {\text { B. }-3=-\sqrt{-9}} \\ {\text { C. }-3=\sqrt[3]{-27}} & {\text { D. }-3=-\sqrt[4]{81}}\end{array}$$

Problem 73

Absolute value symbols are needed when you simplify some of these expressions. Which ones are they?
$$\text { I. }\sqrt[3]{-x^{3} y^{6}} \quad \text { II. } \sqrt{x^{2} y^{4}} \quad \text { III. } \sqrt[4]{x^{8} y^{12}} \quad \text { IV. } \sqrt{x^{4} y^{6}}$$
F. II and III only
H. II, III and IV only
G. II and IV only
J. II, III and IV

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Problem 74

For what values of $x$ and $y$ does $\sqrt{x^{2} y^{4}}$ equal $\sqrt[3]{x^{3}} y^{6} ?$ Explain your answer.

Problem 75

Expand each binomial.
$$(x+y)^{5}$$

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Problem 76

Expand each binomial.
$$(2-3 y)^{4}$$

Problem 77

Expand each binomial.
$$(3 x-5)^{6}$$

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Problem 78

Expand each binomial.
$$(2 a-b)^{7}$$

Problem 79

Write each function in factored form. Check by multiplying.
$$y=4 x^{3}-49 x$$

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Problem 80

Write each function in factored form. Check by multiplying.
$$y=81 x^{2}+36 x+4$$

Problem 81

Write each function in factored form. Check by multiplying.
$$y=4 x^{3}+8 x^{2}+4 x$$

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Problem 82

Write each function in factored form. Check by multiplying.
$$y=12 x^{3}+14 x^{2}+2 x$$

Problem 83

Rewrite each equation in vertex form.
$$y=3 x^{2}-7$$

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Problem 84

Rewrite each equation in vertex form.
$$y=-2 x^{2}+x-10$$

$$y=\frac{x^{2}}{4}+2 x-1$$