Content check — Intermediate Algebra, Chapter 8: Roots and Radicals

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 8: Roots and Radicals

504 problems · 18 figures

Be Prepared 8.1 §8.1 source ↗

Simplify: ⓐ

{(−9)}^{2}

ⓑ

− 9^{2}

ⓒ

{(−9)}^{3} .

If you missed this problem, review Example 2.21.

Example 2.21 in the book ↗

Answer

ⓐ 8; ⓑ −1; ⓒ -729

Be Prepared 8.2 §8.1 source ↗

Round

3.846

to the nearest hundredth.
If you missed this problem, review Example 1.34.

Example 1.34 in the book ↗

Answer

3.85

Be Prepared 8.3 §8.1 source ↗

Simplify: ⓐ

x^{3} \cdot x^{3}

ⓑ

y^{2} \cdot y^{2} \cdot y^{2}

ⓒ

z^{3} \cdot z^{3} \cdot z^{3} \cdot z^{3} .

If you missed this problem, review Example 5.12.

Example 5.12 in the book ↗

Answer

ⓐ

x^{6}

; ⓑ

y^{6}

; ⓒ

z^{12}

Try It 8.1 §8.1 source ↗

Simplify: ⓐ

− \sqrt{64}

ⓑ

\sqrt{225} .

Answer

ⓐ

−8

ⓑ 15

Try It 8.2 §8.1 source ↗

Simplify: ⓐ

\sqrt{100}

ⓑ

− \sqrt{121} .

Answer

ⓐ 10 ⓑ

−11

Try It 8.3 §8.1 source ↗

Simplify: ⓐ

\sqrt{−169}

ⓑ

− \sqrt{81} .

Answer

ⓐ not a real number ⓑ

−9

Try It 8.4 §8.1 source ↗

Simplify: ⓐ

− \sqrt{49}

ⓑ

\sqrt{−121} .

Answer

ⓐ

−7

ⓑ not a real number

Try It 8.5 §8.1 source ↗

Simplify: ⓐ

\sqrt[3]{27}

ⓑ

\sqrt[4]{256}

ⓒ

\sqrt[5]{243} .

Answer

ⓐ 3 ⓑ 4 ⓒ 3

Try It 8.6 §8.1 source ↗

Simplify: ⓐ

\sqrt[3]{1000}

ⓑ

\sqrt[4]{16}

ⓒ

\sqrt[5]{1024} .

Answer

ⓐ 10 ⓑ 2 ⓒ 4

Try It 8.7 §8.1 source ↗

Simplify: ⓐ

\sqrt[3]{−27}

ⓑ

\sqrt[4]{−256}

ⓒ

\sqrt[5]{−32} .

Answer

ⓐ

−3

ⓑ not real ⓒ

−2

Try It 8.8 §8.1 source ↗

Simplify: ⓐ

\sqrt[3]{−216}

ⓑ

\sqrt[4]{−81}

ⓒ

\sqrt[5]{−1024} .

Answer

ⓐ

−6

ⓑ not real ⓒ

−4

Try It 8.9 §8.1 source ↗

Estimate each root between two consecutive whole numbers:

ⓐ $\sqrt{38}$ ⓑ $\sqrt[3]{93}$

Answer

ⓐ

6 <

\sqrt{38} <

7

ⓑ

4 <

\sqrt[3]{93} <

5

Try It 8.10 §8.1 source ↗

Estimate each root between two consecutive whole numbers:

ⓐ $\sqrt{84}$ ⓑ $\sqrt[3]{152}$

Answer

ⓐ

9 <

\sqrt{84} <

10

ⓑ

5 <

\sqrt[3]{152} <

6

Try It 8.11 §8.1 source ↗

Round to two decimal places:

ⓐ $\sqrt{11}$ ⓑ $\sqrt[3]{71}$ ⓒ $\sqrt[4]{127} .$

Answer

ⓐ

\approx 3.32

ⓑ

\approx 4.14

ⓒ

\approx 3.36

Try It 8.12 §8.1 source ↗

Round to two decimal places:

ⓐ $\sqrt{13}$ ⓑ $\sqrt[3]{84}$ ⓒ $\sqrt[4]{98} .$

Answer

ⓐ

\approx 3.61

ⓑ

\approx 4.38

ⓒ

\approx 3.15

Try It 8.13 §8.1 source ↗

Simplify: ⓐ

\sqrt{b^{2}}

ⓑ

\sqrt[3]{w^{3}}

ⓒ

\sqrt[4]{m^{4}}

ⓓ

\sqrt[5]{q^{5}} .

Answer

ⓐ

| b |

ⓑ w ⓒ

| m |

ⓓ q

Try It 8.14 §8.1 source ↗

Simplify: ⓐ

\sqrt{y^{2}}

ⓑ

\sqrt[3]{p^{3}}

ⓒ

\sqrt[4]{z^{4}}

ⓓ

\sqrt[5]{q^{5}} .

Answer

ⓐ

| y |

ⓑ p ⓒ

| z |

ⓓ q

Try It 8.15 §8.1 source ↗

Simplify: ⓐ

\sqrt{y^{18}}

ⓑ

\sqrt{z^{12}} .

Answer

ⓐ

| y^{9} |

ⓑ

z^{6}

Try It 8.16 §8.1 source ↗

Simplify: ⓐ

\sqrt{m^{4}}

ⓑ

\sqrt{b^{10}} .

Answer

ⓐ

m^{2}

ⓑ

| b^{5} |

Try It 8.17 §8.1 source ↗

Simplify: ⓐ

\sqrt[4]{u^{12}}

ⓑ

\sqrt[3]{v^{15}} .

Answer

ⓐ

| u^{3} |

ⓑ

v^{5}

Try It 8.18 §8.1 source ↗

Simplify: ⓐ

\sqrt[5]{c^{20}}

ⓑ

\sqrt[6]{d^{24}}

Answer

ⓐ

c^{4}

ⓑ

d^{4}

Try It 8.19 §8.1 source ↗

Simplify: ⓐ

\sqrt{64 x^{2}}

ⓑ

− \sqrt{100 p^{2}} .

Answer

ⓐ

8 | x |

ⓑ

−10 | p |

Try It 8.20 §8.1 source ↗

Simplify: ⓐ

\sqrt{169 y^{2}}

ⓑ

− \sqrt{121 y^{2}} .

Answer

ⓐ

13 | y |

ⓑ

−11 | y |

Try It 8.21 §8.1 source ↗

Simplify: ⓐ

\sqrt[3]{27 x^{27}}

ⓑ

\sqrt[4]{81 q^{28}} .

Answer

ⓐ

3 x^{9}

ⓑ

3 | q^{7} |

Try It 8.22 §8.1 source ↗

Simplify: ⓐ

\sqrt[3]{125 q^{9}}

ⓑ

\sqrt[5]{243 q^{25}} .

Answer

ⓐ

5 q^{3}

ⓑ

3 q^{5}

Try It 8.23 §8.1 source ↗

Simplify: ⓐ

\sqrt{100 a^{2} b^{2}}

ⓑ

\sqrt{144 p^{12} q^{20}}

ⓒ

\sqrt[3]{8 x^{30} y^{12}}

Answer

ⓐ

10 | a b |

ⓑ

12 p^{6} q^{10}

ⓒ

2 x^{10} y^{4}

Try It 8.24 §8.1 source ↗

Simplify: ⓐ

\sqrt{225 m^{2} n^{2}}

ⓑ

\sqrt{169 x^{10} y^{14}}

ⓒ

\sqrt[3]{27 w^{36} z^{15}}

Answer

ⓐ

15 | m n |

ⓑ

13 | x^{5} y^{7} |

ⓒ

3 w^{12} z^{5}

8.1-1 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt{64}$ ⓑ $− \sqrt{81}$

Answer

ⓐ 8 ⓑ

−9

8.1-3 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt{196}$ ⓑ $− \sqrt{1}$

Answer

ⓐ 14 ⓑ

−1

8.1-5 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt{\frac{4}{9}}$ ⓑ $− \sqrt{0.01}$

Answer

ⓐ

\frac{2}{3}

ⓑ

−0.1

8.1-7 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt{−121}$ ⓑ $− \sqrt{289}$

Answer

ⓐ not real number ⓑ

−17

8.1-9 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $− \sqrt{225}$ ⓑ $\sqrt{−9}$

Answer

ⓐ

−15

ⓑ not real number

8.1-11 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{216}$ ⓑ $\sqrt[4]{256}$

Answer

ⓐ 6 ⓑ 4

8.1-13 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{512}$ ⓑ $\sqrt[4]{81}$ ⓒ $\sqrt[5]{1}$

Answer

ⓐ 8 ⓑ 3 ⓒ 1

8.1-15 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{−8}$ ⓑ $\sqrt[4]{−81}$ ⓒ $\sqrt[5]{−32}$

Answer

ⓐ

−2

ⓑ

not real

ⓒ

−2

8.1-17 §8.1 source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{−125}$ ⓑ $\sqrt[4]{−1296}$ ⓒ $\sqrt[5]{−1024}$

Answer

ⓐ

−5

ⓑ

not real

ⓒ

−4

8.1-19 §8.1 source ↗

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

ⓐ $\sqrt{70}$ ⓑ $\sqrt[3]{71}$

Answer

ⓐ

8 <

\sqrt{70} <

9

ⓑ

4 <

\sqrt[3]{71} <

5

8.1-21 §8.1 source ↗

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

ⓐ $\sqrt{200}$ ⓑ $\sqrt[3]{137}$

Answer

ⓐ

14 <

\sqrt{200} <

15

ⓑ

5 <

\sqrt[3]{137} <

6

8.1-23 §8.1 source ↗

In the following exercises, approximate each root and round to two decimal places.

ⓐ $\sqrt{19}$ ⓑ $\sqrt[3]{89}$ ⓒ $\sqrt[4]{97}$

Answer

ⓐ

\approx 4.36

ⓑ

\approx 4.46

ⓒ

\approx 3.14

8.1-25 §8.1 source ↗

In the following exercises, approximate each root and round to two decimal places.

ⓐ $\sqrt{53}$ ⓑ $\sqrt[3]{147}$ ⓒ $\sqrt[4]{452}$

Answer

ⓐ

\approx 7.28

ⓑ

\approx 5.28

ⓒ

\approx 4.61

8.1-27 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[5]{u^{5}}$ ⓑ $\sqrt[8]{v^{8}}$

Answer

ⓐ u ⓑ

| v |

8.1-29 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[4]{y^{4}}$ ⓑ $\sqrt[7]{m^{7}}$

Answer

ⓐ

| y |

ⓑ

m

8.1-31 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt{x^{6}}$ ⓑ $\sqrt{y^{16}}$

Answer

ⓐ

| x^{3} |

ⓑ

y^{8}

8.1-33 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt{x^{24}}$ ⓑ $\sqrt{y^{22}}$

Answer

ⓐ

x^{12}

ⓑ

| y^{11} |

8.1-35 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[3]{x^{9}}$ ⓑ $\sqrt[4]{y^{12}}$

Answer

ⓐ

x^{3}

ⓑ

| y^{3} |

8.1-37 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[4]{m^{8}}$ ⓑ $\sqrt[5]{n^{20}}$

Answer

ⓐ

m^{2}

ⓑ

n^{4}

8.1-39 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt{49 x^{2}}$ ⓑ $− \sqrt{81 x^{18}}$

Answer

ⓐ

7 | x |

ⓑ

−9 | x^{9} |

8.1-41 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt{121 m^{20}}$ ⓑ $− \sqrt{64 a^{2}}$

Answer

ⓐ

11 m^{10}

ⓑ

−8 | a |

8.1-43 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[4]{16 x^{8}}$ ⓑ $\sqrt[6]{64 y^{12}}$

Answer

ⓐ

2 x^{2}

ⓑ

2 y^{2}

8.1-45 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[3]{216 a^{6}}$ ⓑ $\sqrt[5]{32 b^{20}}$

Answer

ⓐ

6 a^{2}

ⓑ

2 b^{4}

8.1-47 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt{144 x^{2} y^{2}}$ ⓑ $\sqrt{169 w^{8} y^{10}}$ ⓒ $\sqrt[3]{8 a^{51} b^{6}}$

Answer

ⓐ

12 | x y |

ⓑ

13 w^{4} | y^{5} |

ⓒ

2 a^{17} b^{2}

8.1-49 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt{121 a^{2} b^{2}}$ ⓑ $\sqrt{9 c^{8} d^{12}}$ ⓒ $\sqrt[3]{64 x^{15} y^{66}}$

Answer

ⓐ

11 | a b |

ⓑ

3 c^{4} d^{6}

ⓒ

4 x^{5} y^{22}

8.1-51 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

Why is there no real number equal to $\sqrt{−64} ?$

Answer

Answers will vary.

8.1-53 §8.1 source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

Explain what is meant by the n^th root of a number.

Answer

Answers will vary.

Be Prepared 8.4 §8.2 source ↗

Simplify:

\frac{x^{9}}{x^{4}} .

If you missed this problem, review Example 5.13.

Example 5.13 in the book ↗

Answer

x^{5}

Be Prepared 8.5 §8.2 source ↗

Simplify:

\frac{y^{3}}{y^{11}} .

If you missed this problem, review Example 5.13.

Example 5.13 in the book ↗

Answer

\frac{1}{y^{8}}

Be Prepared 8.6 §8.2 source ↗

Simplify:

{(n^{2})}^{6} .

If you missed this problem, review Example 5.17.

Example 5.17 in the book ↗

Answer

n^{12}

Try It 8.25 §8.2 source ↗

Simplify:

\sqrt{48} .

Answer

4 \sqrt{3}

Try It 8.26 §8.2 source ↗

Simplify:

\sqrt{45} .

Answer

3 \sqrt{5}

Try It 8.27 §8.2 source ↗

Simplify: ⓐ

\sqrt{288}

ⓑ

\sqrt[3]{81}

ⓒ

\sqrt[4]{64} .

Answer

ⓐ

12 \sqrt{2}

ⓑ

3 \sqrt[3]{3}

ⓒ

2 \sqrt[4]{4}

Try It 8.28 §8.2 source ↗

Simplify: ⓐ

\sqrt{432}

ⓑ

\sqrt[3]{625}

ⓒ

\sqrt[4]{729} .

Answer

ⓐ

12 \sqrt{3}

ⓑ

5 \sqrt[3]{5}

ⓒ

3 \sqrt[4]{9}

Try It 8.29 §8.2 source ↗

Simplify: ⓐ

\sqrt{b^{5}}

ⓑ

\sqrt[4]{y^{6}}

ⓒ

\sqrt[3]{z^{5}}

Answer

ⓐ

b^{2} \sqrt{b}

ⓑ

| y | \sqrt[4]{y^{2}}

ⓒ

z \sqrt[3]{z^{2}}

Try It 8.30 §8.2 source ↗

Simplify: ⓐ

\sqrt{p^{9}}

ⓑ

\sqrt[5]{y^{8}}

ⓒ

\sqrt[6]{q^{13}}

Answer

ⓐ

p^{4} \sqrt{p}

ⓑ

y \sqrt[5]{y^{3}}

ⓒ

q^{2} \sqrt[6]{q}

Try It 8.31 §8.2 source ↗

Simplify: ⓐ

\sqrt{32 y^{5}}

ⓑ

\sqrt[3]{54 p^{10}}

ⓒ

\sqrt[4]{64 q^{10}} .

Answer

ⓐ

4 y^{2} \sqrt{2 y}

ⓑ

3 p^{3} \sqrt[3]{2 p}

ⓒ

2 q^{2} \sqrt[4]{4 q^{2}}

Try It 8.32 §8.2 source ↗

Simplify: ⓐ

\sqrt{75 a^{9}}

ⓑ

\sqrt[3]{128 m^{11}}

ⓒ

\sqrt[4]{162 n^{7}} .

Answer

ⓐ

5 a^{4} \sqrt{3 a}

ⓑ

4 m^{3} \sqrt[3]{2 m^{2}}

ⓒ

3 | n | \sqrt[4]{2 n^{3}}

Try It 8.33 §8.2 source ↗

Simplify: ⓐ

\sqrt{98 a^{7} b^{5}}

ⓑ

\sqrt[3]{56 x^{5} y^{4}}

ⓒ

\sqrt[4]{32 x^{5} y^{8}} .

Answer

ⓐ

7 | a^{3} | b^{2} {\sqrt{2 a b}}^{}

ⓑ

2 x y \sqrt[3]{7 x^{2} y}

ⓒ

2 | x | y^{2} \sqrt[4]{2 x}

Try It 8.34 §8.2 source ↗

Simplify: ⓐ

\sqrt{180 m^{9} n^{11}}

ⓑ

\sqrt[3]{72 x^{6} y^{5}}

ⓒ

\sqrt[4]{80 x^{7} y^{4}} .

Answer

ⓐ

6 m^{4} | n^{5} | \sqrt{5 m n}

ⓑ

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

ⓒ

2 | x y | \sqrt[4]{5 x^{3}}

Try It 8.35 §8.2 source ↗

Simplify: ⓐ

\sqrt[3]{−64}

ⓑ

\sqrt[4]{−81} .

Answer

ⓐ

−4

ⓑ

no real number

Try It 8.36 §8.2 source ↗

Simplify: ⓐ

\sqrt[3]{−625}

ⓑ

\sqrt[4]{−324} .

Answer

ⓐ

−5 \sqrt[3]{5}

ⓑ no real number

Try It 8.37 §8.2 source ↗

Simplify: ⓐ

5 + \sqrt{75}

ⓑ

\frac{10 - \sqrt{75}}{5}

Answer

ⓐ

5 + 5 \sqrt{3}

ⓑ

2 - \sqrt{3}

Try It 8.38 §8.2 source ↗

Simplify: ⓐ

2 + \sqrt{98}

ⓑ

\frac{6 - \sqrt{45}}{3}

Answer

ⓐ

2 + 7 \sqrt{2}

ⓑ

2 - \sqrt{5}

Try It 8.39 §8.2 source ↗

Simplify: ⓐ

\sqrt{\frac{75}{48}}

ⓑ

\sqrt[3]{\frac{54}{250}}

ⓒ

\sqrt[4]{\frac{32}{162}} .

Answer

ⓐ

\frac{5}{4}

ⓑ

\frac{3}{5}

ⓒ

\frac{2}{3}

Try It 8.40 §8.2 source ↗

Simplify: ⓐ

\sqrt{\frac{98}{162}}

ⓑ

\sqrt[3]{\frac{24}{375}}

ⓒ

\sqrt[4]{\frac{4}{324}} .

Answer

ⓐ

\frac{7}{9}

ⓑ

\frac{2}{5}

ⓒ

\frac{1}{3}

Try It 8.41 §8.2 source ↗

Simplify: ⓐ

\sqrt{\frac{a^{8}}{a^{6}}}

ⓑ

\sqrt[4]{\frac{x^{7}}{x^{3}}}

ⓒ

\sqrt[4]{\frac{y^{17}}{y^{5}}} .

Answer

ⓐ

| a |

ⓑ

| x |

ⓒ

y^{3}

Try It 8.42 §8.2 source ↗

Simplify: ⓐ

\sqrt{\frac{x^{14}}{x^{10}}}

ⓑ

\sqrt[3]{\frac{m^{13}}{m^{7}}}

ⓒ

\sqrt[5]{\frac{n^{12}}{n^{2}}} .

Answer

ⓐ

x^{2}

ⓑ

m^{2}

ⓒ

n^{2}

Try It 8.43 §8.2 source ↗

Simplify:

\sqrt{\frac{24 p^{3}}{49}} .

Answer

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

Try It 8.44 §8.2 source ↗

Simplify:

\sqrt{\frac{48 x^{5}}{100}} .

Answer

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

Try It 8.45 §8.2 source ↗

Simplify: ⓐ

\sqrt{\frac{80 m^{3}}{n^{6}}}

ⓑ

\sqrt[3]{\frac{108 c^{10}}{d^{6}}}

ⓒ

\sqrt[4]{\frac{80 x^{10}}{y^{4}}} .

Answer

ⓐ

\frac{4 | m | \sqrt{5 m}}{| n^{3} |}

ⓑ

\frac{3 c^{3} \sqrt[3]{4 c}}{d^{2}}

ⓒ

\frac{2 x^{2} \sqrt[4]{5 x^{2}}}{| y |}

Try It 8.46 §8.2 source ↗

Simplify: ⓐ

\sqrt{\frac{54 u^{7}}{v^{8}}}

ⓑ

\sqrt[3]{\frac{40 r^{3}}{s^{6}}}

ⓒ

\sqrt[4]{\frac{162 m^{14}}{n^{12}}} .

Answer

ⓐ

\frac{3 u^{3} \sqrt{6 u}}{v^{4}}

ⓑ

\frac{2 r \sqrt[3]{5}}{s^{2}}

ⓒ

\frac{3 | m^{3} | \sqrt[4]{2 m^{2}}}{| n^{3} |}

Try It 8.47 §8.2 source ↗

Simplify: ⓐ

\sqrt{\frac{50 x^{5} y^{3}}{72 x^{4} y}}

ⓑ

\sqrt[3]{\frac{16 x^{5} y^{7}}{54 x^{2} y^{2}}}

ⓒ

\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}} .

Answer

ⓐ

\frac{5 | y | \sqrt{x}}{6}

ⓑ

\frac{2 x y \sqrt[3]{y^{2}}}{3}

ⓒ

\frac{| a b | \sqrt[4]{a}}{2}

Try It 8.48 §8.2 source ↗

Simplify: ⓐ

\sqrt{\frac{48 m^{7} n^{2}}{100 m^{5} n^{8}}}

ⓑ

\sqrt[3]{\frac{54 x^{7} y^{5}}{250 x^{2} y^{2}}}

ⓒ

\sqrt[4]{\frac{32 a^{9} b^{7}}{162 a^{3} b^{3}}} .

Answer

ⓐ

\frac{2 | m | \sqrt{3}}{5 | n^{3} |}

ⓑ

\frac{3 x y \sqrt[3]{x^{2}}}{5}

ⓒ

\frac{2 | a b | \sqrt[4]{a^{2}}}{3}

Try It 8.49 §8.2 source ↗

Simplify: ⓐ

\frac{\sqrt{98 z^{5}}}{\sqrt{2 z}}

ⓑ

\frac{\sqrt[3]{−500}}{\sqrt[3]{2}}

ⓒ

\frac{\sqrt[4]{486 m^{11}}}{\sqrt[4]{3 m^{5}}} .

Answer

ⓐ

7 z^{2}

ⓑ

−5 \sqrt[3]{2}

ⓒ

3 | m | \sqrt[4]{2 m^{2}}

Try It 8.50 §8.2 source ↗

Simplify: ⓐ

\frac{\sqrt{128 m^{9}}}{\sqrt{2 m}}

ⓑ

\frac{\sqrt[3]{−192}}{\sqrt[3]{3}}

ⓒ

\frac{\sqrt[4]{324 n^{7}}}{\sqrt[4]{2 n^{3}}} .

Answer

ⓐ

8 m^{4}

ⓑ

−4

ⓒ

3 | n | \sqrt[4]{2}

8.2-55 §8.2 source ↗

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

$\sqrt{27}$

Answer

3 \sqrt{3}

8.2-57 §8.2 source ↗

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

$\sqrt{125}$

Answer

5 \sqrt{5}

8.2-59 §8.2 source ↗

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

$\sqrt{147}$

Answer

7 \sqrt{3}

8.2-61 §8.2 source ↗

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

$\sqrt{800}$

Answer

20 \sqrt{2}

8.2-63 §8.2 source ↗

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

ⓐ $\sqrt[4]{32}$ ⓑ $\sqrt[5]{64}$

Answer

ⓐ

2 \sqrt[4]{2}

ⓑ

2 \sqrt[5]{2}

8.2-65 §8.2 source ↗

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

ⓐ $\sqrt[4]{64}$ ⓑ $\sqrt[3]{256}$

Answer

ⓐ

2 \sqrt[4]{4}

ⓑ

4 \sqrt[3]{4}

8.2-67 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt{y^{11}}$ ⓑ $\sqrt[3]{r^{5}}$ ⓒ $\sqrt[4]{s^{10}}$

Answer

ⓐ

| y^{5} | \sqrt[]{y}

ⓑ

r \sqrt[3]{r^{2}}

ⓒ

s^{2} \sqrt[4]{s^{2}}

8.2-69 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt{n^{21}}$ ⓑ $\sqrt[3]{q^{8}}$ ⓒ $\sqrt[8]{n^{10}}$

Answer

ⓐ

n^{10} \sqrt{n}

ⓑ

q^{2} \sqrt[3]{q^{2}}

ⓒ

| n | \sqrt[8]{n^{2}}

8.2-71 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt{125 r^{13}}$ ⓑ $\sqrt[3]{108 x^{5}}$ ⓒ $\sqrt[4]{48 y^{6}}$

Answer

ⓐ

5 r^{6} \sqrt[]{5 r}

ⓑ

3 x \sqrt[3]{4 x^{2}}

ⓒ

2 | y | \sqrt[4]{3 y^{2}}

8.2-73 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt{242 m^{23}}$ ⓑ $\sqrt[4]{405 m^{10}}$ ⓒ $\sqrt[5]{160 n^{8}}$

Answer

ⓐ

11 | m^{11} | \sqrt[]{2 m}

ⓑ

3 m^{2} \sqrt[4]{5 m^{2}}

ⓒ

2 n \sqrt[5]{5 n^{3}}

8.2-75 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt{147 m^{7} n^{11}}$ ⓑ $\sqrt[3]{48 x^{6} y^{7}}$ ⓒ $\sqrt[4]{32 x^{5} y^{4}}$

Answer

ⓐ

7 | m^{3} n^{5} | \sqrt[]{3 m n}

ⓑ

2 x^{2} y^{2} \sqrt[3]{6 y}

ⓒ

2 | x y | \sqrt[4]{2 x}

8.2-77 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt{192 q^{3} r^{7}}$ ⓑ $\sqrt[3]{54 m^{9} n^{10}}$ ⓒ $\sqrt[4]{81 a^{9} b^{8}}$

Answer

ⓐ

8 | q r^{3} | \sqrt{3 q r}

ⓑ

3 m^{3} n^{3} \sqrt[3]{2 n}

ⓒ

3 a^{2} b^{2} \sqrt[4]{a}

8.2-79 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt[3]{−864}$ ⓑ $\sqrt[4]{−256}$

Answer

ⓐ

−6 \sqrt[3]{4}

ⓑ not real

8.2-81 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt[5]{−32}$ ⓑ $\sqrt[8]{−1}$

Answer

ⓐ

−2

ⓑ not real

8.2-83 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $5 + \sqrt{12}$ ⓑ $\frac{10 - \sqrt{24}}{2}$

Answer

ⓐ

5 + 2 \sqrt{3}

ⓑ

5 - \sqrt{6}

8.2-85 §8.2 source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $1 + \sqrt{45}$ ⓑ $\frac{3 + \sqrt{90}}{3}$

Answer

ⓐ

1 + 3 \sqrt{5}

ⓑ

1 + \sqrt{10}

8.2-87 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{45}{80}}$ ⓑ $\sqrt[3]{\frac{8}{27}}$ ⓒ $\sqrt[4]{\frac{1}{81}}$

Answer

ⓐ

\frac{3}{4}

ⓑ

\frac{2}{3}

ⓒ

\frac{1}{3}

8.2-89 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{100}{36}}$ ⓑ $\sqrt[3]{\frac{81}{375}}$ ⓒ $\sqrt[4]{\frac{1}{256}}$

Answer

ⓐ

\frac{5}{3}

ⓑ

\frac{3}{5}

ⓒ

\frac{1}{4}

8.2-91 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{x^{10}}{x^{6}}}$ ⓑ $\sqrt[3]{\frac{p^{11}}{p^{2}}}$ ⓒ $\sqrt[4]{\frac{q^{17}}{q^{13}}}$

Answer

ⓐ

x^{2}

ⓑ

p^{3}

ⓒ

| q |

8.2-93 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{y^{4}}{y^{8}}}$ ⓑ $\sqrt[5]{\frac{u^{21}}{u^{11}}}$ ⓒ $\sqrt[6]{\frac{v^{30}}{v^{12}}}$

Answer

ⓐ

\frac{1}{y^{2}}

ⓑ

u^{2}

ⓒ

| v^{3} |

8.2-95 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

$\sqrt{\frac{96 x^{7}}{121}}$

Answer

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

8.2-97 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

$\sqrt{\frac{300 m^{5}}{64}}$

Answer

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

8.2-99 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

$\sqrt{\frac{98 r^{5}}{100}}$

Answer

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

8.2-101 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

$\sqrt{\frac{28 q^{6}}{225}}$

Answer

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

8.2-103 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{75 r^{9}}{s^{8}}}$ ⓑ $\sqrt[3]{\frac{54 a^{8}}{b^{3}}}$ ⓒ $\sqrt[4]{\frac{64 c^{5}}{d^{4}}}$

Answer

ⓐ

\frac{5 r^{4} \sqrt{3 r}}{s^{4}}

ⓑ

\frac{3 a^{2} \sqrt[3]{2 a^{2}}}{b}

ⓒ

\frac{2 | c | \sqrt[4]{4 c}}{| d |}

8.2-105 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{28 p^{7}}{q^{2}}}$ ⓑ $\sqrt[3]{\frac{81 s^{8}}{t^{3}}}$ ⓒ $\sqrt[4]{\frac{64 p^{15}}{q^{12}}}$

Answer

ⓐ

\frac{2 | p^{3} | \sqrt{7 p}}{| q |}

ⓑ

\frac{3 s^{2} \sqrt[3]{3 s^{2}}}{t}

ⓒ

\frac{2 | p^{3} | \sqrt[4]{4 p^{3}}}{| q^{3} |}

8.2-107 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{32 x^{5} y^{3}}{18 x^{3} y}}$ ⓑ $\sqrt[3]{\frac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}$ ⓒ $\sqrt[4]{\frac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}$

Answer

ⓐ

\frac{4 | x y |}{3}

ⓑ

\frac{y^{2} \sqrt[3]{x}}{2}

ⓒ

\frac{| a b | \sqrt[4]{a}}{2}

8.2-109 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$ ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$ ⓒ $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

Answer

ⓐ

\frac{1}{2 | p q |}

ⓑ

\frac{2 c d \sqrt[3]{d^{2}}}{5}

ⓒ

\frac{| m n |}{2}

8.2-111 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\frac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}$ ⓑ $\frac{\sqrt[4]{64}}{\sqrt[4]{2}}$ ⓒ $\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

Answer

ⓐ

\frac{3 p^{4} \sqrt{p}}{| q |}

ⓑ

2 \sqrt[4]{2}

ⓒ

2 x \sqrt[5]{2 x}

8.2-113 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\frac{\sqrt{50 m^{7}}}{\sqrt{2 m}}$ ⓑ $\sqrt[3]{\frac{1250}{2}}$ ⓒ $\sqrt[4]{\frac{486 y^{9}}{2 y^{3}}}$

Answer

ⓐ

5 | m^{3} |

ⓑ

5 \sqrt[3]{5}

ⓒ

3 | y | \sqrt[4]{3 y^{2}}

8.2-115 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

Explain why $\sqrt{x^{4}} = x^{2} .$ Then explain why $\sqrt{x^{16}} = x^{8} .$

Answer

Answers will vary.

8.2-117 §8.2 source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

Explain how you know that $\sqrt[5]{x^{10}} = x^{2} .$

Answer

Answers will vary.

Be Prepared 8.7 §8.3 source ↗

Add:

\frac{7}{15} + \frac{5}{12} .

If you missed this problem, review Example 1.28.

Example 1.28 in the book ↗

Answer

\frac{53}{60}

Be Prepared 8.8 §8.3 source ↗

Simplify:

{(4 x^{2} y^{5})}^{3} .

If you missed this problem, review Example 5.18.

Example 5.18 in the book ↗

Answer

64 x^{6} y^{15}

Be Prepared 8.9 §8.3 source ↗

Simplify:

5^{−3} .

If you missed this problem, review Example 5.14.

Example 5.14 in the book ↗

Answer

\frac{1}{125}

Try It 8.51 §8.3 source ↗

Write as a radical expression: ⓐ

t^{\frac{1}{2}}

ⓑ

m^{\frac{1}{3}}

ⓒ

r^{\frac{1}{4}} .

Answer

ⓐ

\sqrt{t}

ⓑ

\sqrt[3]{m}

ⓒ

\sqrt[4]{r}

Try It 8.52 §8.3 source ↗

Write as a radial expression: ⓐ

b^{\frac{1}{6}}

ⓑ

z^{\frac{1}{5}}

ⓒ

p^{\frac{1}{4}} .

Answer

ⓐ

\sqrt[6]{b}

ⓑ

\sqrt[5]{z}

ⓒ

\sqrt[4]{p}

Try It 8.53 §8.3 source ↗

Write with a rational exponent: ⓐ

\sqrt{10 m}

ⓑ

\sqrt[5]{3 n}

ⓒ

3 \sqrt[4]{6 y} .

Answer

ⓐ

{(10 m)}^{\frac{1}{2}}

ⓑ

{(3 n)}^{\frac{1}{5}}

ⓒ

3 {(6 y)}^{\frac{1}{4}}

Try It 8.54 §8.3 source ↗

Write with a rational exponent: ⓐ

\sqrt[7]{3 k}

ⓑ

\sqrt[4]{5 j}

ⓒ

8 \sqrt[3]{2 a} .

Answer

ⓐ

{(3 k)}^{\frac{1}{7}}

ⓑ

{(5 j)}^{\frac{1}{4}}

ⓒ

8 {(2 a)}^{\frac{1}{3}}

Try It 8.55 §8.3 source ↗

Simplify: ⓐ

36^{\frac{1}{2}}

ⓑ

8^{\frac{1}{3}}

ⓒ

16^{\frac{1}{4}} .

Answer

ⓐ 6 ⓑ 2 ⓒ 2

Try It 8.56 §8.3 source ↗

Simplify: ⓐ

100^{\frac{1}{2}}

ⓑ

27^{\frac{1}{3}}

ⓒ

81^{\frac{1}{4}} .

Answer

ⓐ 10 ⓑ 3 ⓒ 3

Try It 8.57 §8.3 source ↗

Simplify: ⓐ

{(−64)}^{- \frac{1}{2}}

ⓑ

− 64^{\frac{1}{2}}

ⓒ

{(64)}^{- \frac{1}{2}} .

Answer

ⓐ No real solution ⓑ

−8

ⓒ

\frac{1}{8}

Try It 8.58 §8.3 source ↗

Simplify: ⓐ

{(−256)}^{\frac{1}{4}}

ⓑ

− 256^{\frac{1}{4}}

ⓒ

{(256)}^{- \frac{1}{4}} .

Answer

ⓐ No real solution ⓑ

−4

ⓒ

\frac{1}{4}

Try It 8.59 §8.3 source ↗

Write with a rational exponent: ⓐ

\sqrt{x^{5}}

ⓑ

{(\sqrt[4]{3 y})}^{3}

ⓒ

\sqrt{{(\frac{2 m}{3 n})}^{5}} .

Answer

ⓐ

x^{\frac{5}{2}}

ⓑ

{(3 y)}^{\frac{3}{4}}

ⓒ

{(\frac{2 m}{3 n})}^{\frac{5}{2}}

Try It 8.60 §8.3 source ↗

Write with a rational exponent: ⓐ

\sqrt[5]{a^{2}}

ⓑ

{(\sqrt[3]{5 a b})}^{5}

ⓒ

\sqrt{{(\frac{7 x y}{z})}^{3}} .

Answer

ⓐ

a^{\frac{2}{5}}

ⓑ

{(5 a b)}^{\frac{5}{3}}

ⓒ

{(\frac{7 x y}{z})}^{\frac{3}{2}}

Try It 8.61 §8.3 source ↗

Simplify: ⓐ

27^{\frac{2}{3}}

ⓑ

81^{- \frac{3}{2}}

ⓒ

16^{- \frac{3}{4}} .

Answer

ⓐ 9 ⓑ

\frac{1}{729}

ⓒ

\frac{1}{8}

Try It 8.62 §8.3 source ↗

Simplify: ⓐ

4^{\frac{3}{2}}

ⓑ

27^{- \frac{2}{3}}

ⓒ

625^{- \frac{3}{4}} .

Answer

ⓐ 8 ⓑ

\frac{1}{9}

ⓒ

\frac{1}{125}

Try It 8.63 §8.3 source ↗

Simplify: ⓐ

{−16}^{\frac{3}{2}}

ⓑ

{−16}^{- \frac{3}{2}}

ⓒ

{(−16)}^{- \frac{3}{2}} .

Answer

ⓐ

−64

ⓑ

- \frac{1}{64}

Try It 8.64 §8.3 source ↗

Simplify: ⓐ

{−81}^{\frac{3}{2}}

ⓑ

{−81}^{- \frac{3}{2}}

ⓒ

{(−81)}^{- \frac{3}{2}} .

Answer

ⓐ

−729

ⓑ

- \frac{1}{729}

Try It 8.65 §8.3 source ↗

Simplify: ⓐ

x^{\frac{1}{6}} \cdot x^{\frac{4}{3}}

ⓑ

{(x^{6})}^{\frac{4}{3}}

ⓒ

\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}} .

Answer

ⓐ

x^{\frac{3}{2}}

ⓑ

x^{8}

ⓒ

\frac{1}{x}

Try It 8.66 §8.3 source ↗

Simplify: ⓐ

y^{\frac{3}{4}} \cdot y^{\frac{5}{8}}

ⓑ

{(m^{9})}^{\frac{2}{9}}

ⓒ

\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}} .

Answer

ⓐ

y^{\frac{11}{8}}

ⓑ

m^{2}

ⓒ

\frac{1}{d}

Try It 8.67 §8.3 source ↗

Simplify: ⓐ

{(32 x^{\frac{1}{3}})}^{\frac{3}{5}}

ⓑ

{(x^{\frac{3}{4}} y^{\frac{1}{2}})}^{\frac{2}{3}} .

Answer

ⓐ

8 x^{\frac{1}{5}}

ⓑ

x^{\frac{1}{2}} y^{\frac{1}{3}}

Try It 8.68 §8.3 source ↗

Simplify: ⓐ

{(81 n^{\frac{2}{5}})}^{\frac{3}{2}}

ⓑ

{(a^{\frac{3}{2}} b^{\frac{1}{2}})}^{\frac{4}{3}} .

Answer

ⓐ

729 n^{\frac{3}{5}}

ⓑ

a^{2} b^{\frac{2}{3}}

Try It 8.69 §8.3 source ↗

Simplify: ⓐ

\frac{m^{\frac{2}{3}} \cdot m^{- \frac{1}{3}}}{m^{- \frac{5}{3}}}

ⓑ

{(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{- \frac{1}{6}}})}^{\frac{1}{2}} .

Answer

ⓐ

m^{2}

ⓑ

\frac{5 n}{m^{\frac{1}{4}}}

Try It 8.70 §8.3 source ↗

Simplify: ⓐ

\frac{u^{\frac{4}{5}} \cdot u^{- \frac{2}{5}}}{u^{- \frac{13}{5}}}

ⓑ

{(\frac{27 x^{\frac{4}{5}} y^{\frac{1}{6}}}{x^{\frac{1}{5}} y^{- \frac{5}{6}}})}^{\frac{1}{3}} .

Answer

ⓐ

u^{3}

ⓑ

3 x^{\frac{1}{5}} y^{\frac{1}{3}}

8.3-119 §8.3 source ↗

Simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

ⓐ $x^{\frac{1}{2}}$ ⓑ $y^{\frac{1}{3}}$ ⓒ $z^{\frac{1}{4}}$

Answer

ⓐ

\sqrt{x}

ⓑ

\sqrt[3]{y}

ⓒ

\sqrt[4]{z}

8.3-121 §8.3 source ↗

Simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

ⓐ $u^{\frac{1}{5}}$ ⓑ $v^{\frac{1}{9}}$ ⓒ $w^{\frac{1}{20}}$

Answer

ⓐ

\sqrt[5]{u}

ⓑ

\sqrt[9]{v}

ⓒ

\sqrt[20]{w}

8.3-123 §8.3 source ↗

In the following exercises, write with a rational exponent.

ⓐ $\sqrt[7]{x}$ ⓑ $\sqrt[9]{y}$ ⓒ $\sqrt[5]{f}$

Answer

ⓐ

x^{\frac{1}{7}}

ⓑ

y^{\frac{1}{9}}

ⓒ

f^{\frac{1}{5}}

8.3-125 §8.3 source ↗

In the following exercises, write with a rational exponent.

ⓐ $\sqrt[3]{7 c}$ ⓑ $\sqrt[7]{12 d}$ ⓒ $2 \sqrt[4]{6 b}$

Answer

ⓐ

{(7 c)}^{\frac{1}{3}}

ⓑ

{(12 d)}^{\frac{1}{7}}

ⓒ

2 {(6 b)}^{\frac{1}{4}}

8.3-127 §8.3 source ↗

In the following exercises, write with a rational exponent.

ⓐ $\sqrt{21 p}$ ⓑ $\sqrt[4]{8 q}$ ⓒ $4 \sqrt[6]{36 r}$

Answer

ⓐ

{(21 p)}^{\frac{1}{2}}

ⓑ

{(8 q)}^{\frac{1}{4}}

ⓒ

4 {(36 r)}^{\frac{1}{6}}

8.3-129 §8.3 source ↗

In the following exercises, simplify.

ⓐ $81^{\frac{1}{2}}$ ⓑ $125^{\frac{1}{3}}$ ⓒ $64^{\frac{1}{2}}$

Answer

ⓐ 9 ⓑ 5 ⓒ 8

8.3-131 §8.3 source ↗

In the following exercises, simplify.

ⓐ $16^{\frac{1}{4}}$ ⓑ $16^{\frac{1}{2}}$ ⓒ $625^{\frac{1}{4}}$

Answer

ⓐ 2 ⓑ 4 ⓒ 5

8.3-133 §8.3 source ↗

In the following exercises, simplify.

ⓐ ${(−216)}^{\frac{1}{3}}$ ⓑ $− 216^{\frac{1}{3}}$ ⓒ ${(216)}^{- \frac{1}{3}}$

Answer

ⓐ

−6

ⓑ

−6

ⓒ

\frac{1}{6}

8.3-135 §8.3 source ↗

In the following exercises, simplify.

ⓐ ${(−81)}^{\frac{1}{4}}$ ⓑ $− 81^{\frac{1}{4}}$ ⓒ ${(81)}^{- \frac{1}{4}}$

Answer

ⓐ not real ⓑ

−3

ⓒ

\frac{1}{3}

8.3-137 §8.3 source ↗

In the following exercises, simplify.

ⓐ ${(−36)}^{\frac{1}{2}}$ ⓑ $− 36^{\frac{1}{2}}$ ⓒ ${(36)}^{- \frac{1}{2}}$

Answer

ⓐ not real ⓑ

−6

ⓒ

\frac{1}{6}

8.3-139 §8.3 source ↗

In the following exercises, simplify.

ⓐ ${(−100)}^{\frac{1}{2}}$ ⓑ $− 100^{\frac{1}{2}}$ ⓒ ${(100)}^{- \frac{1}{2}}$

Answer

ⓐ not real ⓑ

−10

ⓒ

\frac{1}{10}

8.3-141 §8.3 source ↗

Simplify Expressions with $a^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

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

Answer

ⓐ

m^{\frac{5}{2}}

ⓑ

{(3 y)}^{\frac{7}{3}}

ⓒ

{(\frac{4 x}{5 y})}^{\frac{3}{5}}

8.3-143 §8.3 source ↗

Simplify Expressions with $a^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

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

Answer

ⓐ

u^{\frac{2}{5}}

ⓑ

{(6 x)}^{\frac{5}{3}}

ⓒ

{(\frac{18 a}{5 b})}^{\frac{7}{4}}

8.3-145 §8.3 source ↗

In the following exercises, simplify.

ⓐ $64^{\frac{5}{2}}$ ⓑ $81^{\frac{−3}{2}}$ ⓒ ${(−27)}^{\frac{2}{3}}$

Answer

ⓐ 32,768 ⓑ

\frac{1}{729}

ⓒ 9

8.3-147 §8.3 source ↗

In the following exercises, simplify.

ⓐ $32^{\frac{2}{5}}$ ⓑ $27^{- \frac{2}{3}}$ ⓒ ${(−25)}^{\frac{1}{2}}$

Answer

ⓐ 4 ⓑ

\frac{1}{9}

8.3-149 §8.3 source ↗

In the following exercises, simplify.

ⓐ $− 9^{\frac{3}{2}}$ ⓑ $− 9^{- \frac{3}{2}}$ ⓒ ${(−9)}^{\frac{3}{2}}$

Answer

ⓐ

−27

ⓑ

- \frac{1}{27}

8.3-151 §8.3 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ $c^{\frac{1}{4}} \cdot c^{\frac{5}{8}}$ ⓑ ${(p^{12})}^{\frac{3}{4}}$ ⓒ $\frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}$

Answer

ⓐ

c^{\frac{7}{8}}

ⓑ

p^{9}

ⓒ

\frac{1}{r}

8.3-153 §8.3 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ $y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}$ ⓑ ${(x^{12})}^{\frac{2}{3}}$ ⓒ $\frac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}$

Answer

ⓐ

y^{\frac{5}{4}}

ⓑ

x^{8}

ⓒ

\frac{1}{m}

8.3-155 §8.3 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${(27 q^{\frac{3}{2}})}^{\frac{4}{3}}$ ⓑ ${(a^{\frac{1}{3}} b^{\frac{2}{3}})}^{\frac{3}{2}}$

Answer

ⓐ

81 q^{2}

ⓑ

a^{\frac{1}{2}} b

8.3-157 §8.3 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ ${(16 u^{\frac{1}{3}})}^{\frac{3}{4}}$ ⓑ ${(4 p^{\frac{1}{3}} q^{\frac{1}{2}})}^{\frac{3}{2}}$

Answer

ⓐ

8 u^{\frac{1}{4}}

ⓑ

8 p^{\frac{1}{2}} q^{\frac{3}{4}}

8.3-159 §8.3 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ $\frac{r^{\frac{5}{2}} \cdot r^{- \frac{1}{2}}}{r^{- \frac{3}{2}}}$ ⓑ ${(\frac{36 s^{\frac{1}{5}} t^{- \frac{3}{2}}}{s^{- \frac{9}{5}} t^{\frac{1}{2}}})}^{\frac{1}{2}}$

Answer

ⓐ

r^{\frac{7}{2}}

ⓑ

\frac{6 s}{t}

8.3-161 §8.3 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

ⓐ $\frac{c^{\frac{5}{3}} \cdot c^{- \frac{1}{3}}}{c^{- \frac{2}{3}}}$ ⓑ ${(\frac{8 x^{\frac{5}{3}} y^{- \frac{1}{2}}}{27 x^{- \frac{4}{3}} y^{\frac{5}{2}}})}^{\frac{1}{3}}$

Answer

ⓐ

c^{2}

ⓑ

\frac{2 x}{3 y}

8.3-163 §8.3 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

Show two different algebraic methods to simplify $4^{\frac{3}{2}} .$ Explain all your steps.

Answer

Answers will vary.

Be Prepared 8.10 §8.4 source ↗

Add:

3 x^{2} +

9 x -

5 -

(x^{2} - 2 x + 3) .

If you missed this problem, review Example 5.5.

Example 5.5 in the book ↗

Answer

2 x^{2} + 11 x - 8

Be Prepared 8.11 §8.4 source ↗

Simplify:

(2 + a) (4 - a) .

If you missed this problem, review Example 5.28.

Example 5.28 in the book ↗

Answer

8 + 2 a - a^{2}

Be Prepared 8.12 §8.4 source ↗

Simplify:

{(9 - 5 y)}^{2} .

If you missed this problem, review Example 5.31.

Example 5.31 in the book ↗

Answer

81 - 90 y + 25 y^{2}

Try It 8.71 §8.4 source ↗

Simplify: ⓐ

8 \sqrt{2} - 9 \sqrt{2}

ⓑ

4 \sqrt[3]{x} + 7 \sqrt[3]{x}

ⓒ

3 \sqrt[4]{x} - 5 \sqrt[4]{y} .

Answer

ⓐ

− \sqrt{2}

ⓑ

11 \sqrt[3]{x}

ⓒ

3 \sqrt[4]{x} - 5 \sqrt[4]{y}

Try It 8.72 §8.4 source ↗

Simplify: ⓐ

5 \sqrt{3} - 9 \sqrt{3}

ⓑ

5 \sqrt[3]{y} + 3 \sqrt[3]{y}

ⓒ

5 \sqrt[4]{m} - 2 \sqrt[3]{m} .

Answer

ⓐ

−4 \sqrt{3}

ⓑ

8 \sqrt[3]{y}

ⓒ

5 \sqrt[4]{m} - 2 \sqrt[3]{m}

Try It 8.73 §8.4 source ↗

Simplify: ⓐ

\sqrt{7 x} - 7 \sqrt{7 x} + 4 \sqrt{7 x}

ⓑ

4 \sqrt[4]{5 x y} +

2 \sqrt[4]{5 x y} -

7 \sqrt[4]{5 x y} .

Answer

ⓐ

−2 \sqrt{7 x}

ⓑ

− \sqrt[4]{5 x y}

Try It 8.74 §8.4 source ↗

Simplify: ⓐ

4 \sqrt{3 y} - 7 \sqrt{3 y} + 2 \sqrt{3 y}

ⓑ

6 \sqrt[3]{7 m n} +

\sqrt[3]{7 m n} -

4 \sqrt[3]{7 m n} .

Answer

ⓐ

− \sqrt{3 y}

ⓑ

3 \sqrt[3]{7 m n}

Try It 8.75 §8.4 source ↗

Simplify: ⓐ

\sqrt{18} + 6 \sqrt{2}

ⓑ

6 \sqrt[3]{16} - 2 \sqrt[3]{250}

ⓒ

\frac{2}{3} \sqrt[3]{81} -

\frac{1}{2} \sqrt[3]{24} .

Answer

ⓐ

9 \sqrt{2}

ⓑ

2 \sqrt[3]{2}

ⓒ

\sqrt[3]{3}

Try It 8.76 §8.4 source ↗

Simplify: ⓐ

\sqrt{27} + 4 \sqrt{3}

ⓑ

4 \sqrt[3]{5} - 7 \sqrt[3]{40}

ⓒ

\frac{1}{2} \sqrt[3]{128} -

\frac{5}{3} \sqrt[3]{54} .

Answer

ⓐ

7 \sqrt{3}

ⓑ

−10 \sqrt[3]{5}

ⓒ

−3 \sqrt[3]{2}

Try It 8.77 §8.4 source ↗

Simplify: ⓐ

\sqrt{32 m^{7}} - \sqrt{50 m^{7}}

ⓑ

\sqrt[3]{135 x^{7}} -

\sqrt[3]{40 x^{7}} .

Answer

ⓐ

− m^{3} \sqrt{2 m}

ⓑ

x^{2} \sqrt[3]{5 x}

Try It 8.78 §8.4 source ↗

Simplify: ⓐ

\sqrt{27 p^{3}} - \sqrt{48 p^{3}}

ⓑ

\sqrt[3]{256 y^{5}} -

\sqrt[3]{32 y^{5}} .

Answer

ⓐ

− p \sqrt{3 p}

ⓑ

4 y \sqrt[3]{4 y^{2}} -

2 y \sqrt[3]{4 n^{2}}

Try It 8.79 §8.4 source ↗

Simplify: ⓐ

(3 \sqrt{2}) (2 \sqrt{30})

ⓑ

(2 \sqrt[3]{18}) (−3 \sqrt[3]{6}) .

Answer

ⓐ

12 \sqrt{15}

ⓑ

−18 \sqrt[3]{4}

Try It 8.80 §8.4 source ↗

Simplify: ⓐ

(3 \sqrt{3}) (3 \sqrt{6})

ⓑ

(−4 \sqrt[3]{9}) (3 \sqrt[3]{6}) .

Answer

ⓐ

27 \sqrt{2}

ⓑ

−36 \sqrt[3]{2}

Try It 8.81 §8.4 source ↗

Simplify: ⓐ

(6 \sqrt{6 x^{2}}) (8 \sqrt{30 x^{4}})

ⓑ

(−4 \sqrt[4]{12 y^{3}}) (− \sqrt[4]{8 y^{3}}) .

Answer

ⓐ

288 x^{3} \sqrt{5}

ⓑ

8 y \sqrt[4]{6 y^{2}}

Try It 8.82 §8.4 source ↗

Simplify: ⓐ

(2 \sqrt{6 y^{4}}) (12 \sqrt{30 y})

ⓑ

(−4 \sqrt[4]{9 a^{3}}) (3 \sqrt[4]{27 a^{2}}) .

Answer

ⓐ

144 y^{2} \sqrt{5 y}

ⓑ

−36 a \sqrt[4]{3 a}

Try It 8.83 §8.4 source ↗

Simplify: ⓐ

\sqrt{6} (1 + 3 \sqrt{6})

ⓑ

\sqrt[3]{4} (−2 - \sqrt[3]{6}) .

Answer

ⓐ

18 + \sqrt{6}

ⓑ

−2 \sqrt[3]{4} - 2 \sqrt[3]{3}

Try It 8.84 §8.4 source ↗

Simplify: ⓐ

\sqrt{8} (2 - 5 \sqrt{8})

ⓑ

\sqrt[3]{3} (− \sqrt[3]{9} - \sqrt[3]{6}) .

Answer

ⓐ

−40 + 4 \sqrt{2}

ⓑ

−3 - \sqrt[3]{18}

Try It 8.85 §8.4 source ↗

Simplify: ⓐ

(6 - 3 \sqrt{7}) (3 + 4 \sqrt{7})

ⓑ

(\sqrt[3]{x} - 2) (\sqrt[3]{x} - 3) .

Answer

ⓐ

−66 + 15 \sqrt{7}

ⓑ

\sqrt[3]{x^{2}} - 5 \sqrt[3]{x} + 6

Try It 8.86 §8.4 source ↗

Simplify: ⓐ

(2 - 3 \sqrt{11}) (4 - \sqrt{11})

ⓑ

(\sqrt[3]{x} + 1) (\sqrt[3]{x} + 3) .

Answer

ⓐ

41 - 14 \sqrt{11}

ⓑ

\sqrt[3]{x^{2}} + 4 \sqrt[3]{x} + 3

Try It 8.87 §8.4 source ↗

Simplify:

(5 \sqrt{3} - \sqrt{7}) (\sqrt{3} + 2 \sqrt{7})

Answer

1 + 9 \sqrt{21}

Try It 8.88 §8.4 source ↗

Simplify:

(\sqrt{6} - 3 \sqrt{8}) (2 \sqrt{6} + \sqrt{8})

Answer

−12 - 20 \sqrt{3}

Try It 8.89 §8.4 source ↗

Simplify: ⓐ

{(10 + \sqrt{2})}^{2}

ⓑ

{(1 + 3 \sqrt{6})}^{2} .

Answer

ⓐ

102 + 20 \sqrt{2}

ⓑ

55 + 6 \sqrt{6}

Try It 8.90 §8.4 source ↗

Simplify: ⓐ

{(6 - \sqrt{5})}^{2}

ⓑ

{(9 - 2 \sqrt{10})}^{2} .

Answer

ⓐ

41 - 12 \sqrt{5}

ⓑ

121 - 36 \sqrt{10}

Try It 8.91 §8.4 source ↗

Simplify:

(3 - 2 \sqrt{5}) (3 + 2 \sqrt{5})

Answer

−11

Try It 8.92 §8.4 source ↗

Simplify:

(4 + 5 \sqrt{7}) (4 - 5 \sqrt{7}) .

Answer

−159

8.4-165 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

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

Answer

ⓐ

3 \sqrt{2}

ⓑ

7 \sqrt[3]{m}

ⓒ

6 \sqrt[4]{m}

8.4-167 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

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

Answer

ⓐ

9 \sqrt{5}

ⓑ

12 \sqrt[3]{a}

ⓒ

6 \sqrt[4]{2 z}

8.4-169 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

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

Answer

ⓐ

4 \sqrt{2 a}

ⓑ 0

8.4-171 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

ⓐ $8 \sqrt{3 c} + 2 \sqrt{3 c} - 9 \sqrt{3 c}$ ⓑ $2 \sqrt[3]{4 p q} -$ $5 \sqrt[3]{4 p q} +$ $4 \sqrt[3]{4 p q}$

Answer

ⓐ

\sqrt{3 c}

ⓑ

\sqrt[3]{4 p q}

8.4-173 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

ⓐ $\sqrt{27} - \sqrt{75}$ ⓑ $\sqrt[3]{40} - \sqrt[3]{320}$ ⓒ $\frac{1}{2} \sqrt[4]{32} +$ $\frac{2}{3} \sqrt[4]{162}$

Answer

ⓐ

−2 \sqrt{3}

ⓑ

−2 \sqrt[3]{5}

ⓒ

3 \sqrt[4]{2}

8.4-175 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

ⓐ $\sqrt{48} + \sqrt{27}$ ⓑ $\sqrt[3]{54} + \sqrt[3]{128}$ ⓒ $6 \sqrt[4]{5} - \frac{3}{2} \sqrt[4]{80}$

Answer

ⓐ

7 \sqrt{3}

ⓑ

7 \sqrt[3]{2}

ⓒ

3 \sqrt[4]{5}

8.4-177 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

ⓐ $\sqrt{72 a^{5}} - \sqrt{50 a^{5}}$ ⓑ $9 \sqrt[4]{80 p^{4}} -$ $6 \sqrt[4]{405 p^{4}}$

Answer

ⓐ

a^{2} \sqrt{2 a}

ⓑ 0

8.4-179 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

ⓐ $\sqrt{80 c^{7}} - \sqrt{20 c^{7}}$ ⓑ $2 \sqrt[4]{162 r^{10}} +$ $4 \sqrt[4]{32 r^{10}}$

Answer

ⓐ

2 c^{3} \sqrt{5 c}

ⓑ

14 r^{2} \sqrt[4]{2 r^{2}}

8.4-181 §8.4 source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

$3 \sqrt{128 y^{2}} +$ $4 y \sqrt{162} -$ $8 \sqrt{98 y^{2}}$

Answer

4 y \sqrt{2}

8.4-183 §8.4 source ↗

Multiply Radical Expressions

In the following exercises, simplify.

ⓐ $(−2 \sqrt[]{3}) (3 \sqrt[]{18})$ ⓑ $(8 \sqrt[3]{4}) (−4 \sqrt[3]{18})$

Answer

ⓐ

−18 \sqrt{6}

ⓑ

−64 \sqrt[3]{9}

8.4-185 §8.4 source ↗

Multiply Radical Expressions

In the following exercises, simplify.

ⓐ $(5 \sqrt[]{6}) (− \sqrt[]{12})$ ⓑ $(−2 \sqrt[4]{18}) (− \sqrt[4]{9})$

Answer

ⓐ

−30 \sqrt{2}

ⓑ

6 \sqrt[4]{2}

8.4-187 §8.4 source ↗

Multiply Radical Expressions

In the following exercises, simplify.

ⓐ $(4 \sqrt{12 z^{3}}) (3 \sqrt{9 z})$ ⓑ $(5 \sqrt[3]{3 x^{3}}) (3 \sqrt[3]{18 x^{3}})$

Answer

ⓐ

72 z^{2} \sqrt{3}

ⓑ

45 x^{2} \sqrt[3]{2}

8.4-189 §8.4 source ↗

Multiply Radical Expressions

In the following exercises, simplify.

ⓐ $(−2 \sqrt{7 z^{3}}) (3 \sqrt{14 z^{8}})$ ⓑ $(2 \sqrt[4]{8 y^{2}}) (−2 \sqrt[4]{12 y^{3}})$

Answer

ⓐ

−42 z^{5} \sqrt{2 z}

ⓑ

−8 y \sqrt[4]{6 y}

8.4-191 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ $\sqrt{7} (5 + 2 \sqrt{7})$ ⓑ $\sqrt[3]{6} (4 + \sqrt[3]{18})$

Answer

ⓐ

14 + 5 \sqrt{7}

ⓑ

4 \sqrt[3]{6} + 3 \sqrt[3]{4}

8.4-193 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ $\sqrt{11} (−3 + 4 \sqrt{11})$ ⓑ $\sqrt[4]{3} (\sqrt[4]{54} + \sqrt[4]{18})$

Answer

ⓐ

44 - 3 \sqrt{11}

ⓑ

3 \sqrt[4]{2} + \sqrt[4]{54}

8.4-195 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

$(7 + \sqrt{3}) (9 - \sqrt{3})$

Answer

60 + 2 \sqrt{3}

8.4-197 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ $(9 - 3 \sqrt{2}) (6 + 4 \sqrt{2})$ ⓑ $(\sqrt[3]{x} - 3) (\sqrt[3]{x} + 1)$

Answer

ⓐ

30 + 18 \sqrt{2}

ⓑ

\sqrt[3]{x^{2}} - 2 \sqrt[3]{x} - 3

8.4-199 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ $(1 + 3 \sqrt{10}) (5 - 2 \sqrt{10})$ ⓑ $(2 \sqrt[3]{x} + 6) (\sqrt[3]{x} + 1)$

Answer

ⓐ

−55 + 13 \sqrt{10}

ⓑ

2 \sqrt[3]{x^{2}} + 8 \sqrt[3]{x} + 6

8.4-201 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

$(\sqrt{3} + \sqrt{10}) (\sqrt{3} + 2 \sqrt{10})$

Answer

23 + 3 \sqrt{30}

8.4-203 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

$(2 \sqrt{7} - 5 \sqrt{11}) (4 \sqrt{7} + 9 \sqrt{11})$

Answer

−439 - 2 \sqrt{77}

8.4-205 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

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

Answer

ⓐ

14 + 6 \sqrt{5}

ⓑ

79 - 20 \sqrt{3}

8.4-207 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ ${(9 - \sqrt{6})}^{2}$ ⓑ ${(10 + 3 \sqrt{7})}^{2}$

Answer

ⓐ

87 - 18 \sqrt{6}

ⓑ

163 + 60 \sqrt{7}

8.4-209 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

$(4 + \sqrt{2}) (4 - \sqrt{2})$

Answer

8.4-211 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

$(4 + 9 \sqrt{3}) (4 - 9 \sqrt{3})$

Answer

−227

8.4-213 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

$(12 - 5 \sqrt{5}) (12 + 5 \sqrt{5})$

Answer

19

8.4-215 §8.4 source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

$(\sqrt[3]{3 x} + 2) (\sqrt[3]{3 x} - 2)$

Answer

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

8.4-217 §8.4 source ↗

Mixed Practice

$\frac{2}{3} \sqrt{27} + \frac{3}{4} \sqrt{48}$

Answer

5 \sqrt{3}

8.4-219 §8.4 source ↗

Mixed Practice

$\frac{5}{6} \sqrt{162} +$ $\frac{3}{16} \sqrt{128}$

Answer

9 \sqrt{2}

8.4-221 §8.4 source ↗

Mixed Practice

$\frac{1}{2} \sqrt[4]{80} -$ $\frac{2}{3} \sqrt[4]{405}$

Answer

− \sqrt[4]{5}

8.4-223 §8.4 source ↗

Mixed Practice

$5 \sqrt{12 c^{4}} - 3 \sqrt{27 c^{6}}$

Answer

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

8.4-225 §8.4 source ↗

Mixed Practice

$\frac{3}{5} \sqrt{75} - \frac{1}{4} \sqrt{48}$

Answer

2 \sqrt{3}

8.4-227 §8.4 source ↗

Mixed Practice

$8 \sqrt[3]{64 q^{6}} -$ $3 \sqrt[3]{125 q^{6}}$

Answer

17 q^{2}

8.4-229 §8.4 source ↗

Mixed Practice

$\sqrt{3} \cdot \sqrt{21}$

Answer

3 \sqrt{7}

8.4-231 §8.4 source ↗

Mixed Practice

$(7 \sqrt[3]{4}) (−3 \sqrt[3]{18})$

Answer

−42 \sqrt[3]{9}

8.4-233 §8.4 source ↗

Mixed Practice

${(\sqrt{29})}^{2}$

Answer

8.4-235 §8.4 source ↗

Mixed Practice

$(−4 + \sqrt{17}) (−3 + \sqrt{17})$

Answer

29 - 7 \sqrt{17}

8.4-237 §8.4 source ↗

Mixed Practice

${(6 - 3 \sqrt{2})}^{2}$

Answer

54 - 36 \sqrt{2}

8.4-239 §8.4 source ↗

Mixed Practice

$\sqrt[3]{3} (2 \sqrt[3]{9} + \sqrt[3]{18})$

Answer

6 + 3 \sqrt[3]{2}

8.4-241 §8.4 source ↗

Mixed Practice

Explain when a radical expression is in simplest form.

Answer

Answers will vary.

8.4-243 §8.4 source ↗

Mixed Practice

ⓐ Explain why ${(− \sqrt{n})}^{2}$ is always non-negative, for $n \geq 0 .$
ⓑ Explain why $- {(\sqrt{n})}^{2}$ is always non-positive, for $n \geq 0.$

Answer

Answers will vary.

Be Prepared 8.13 §8.5 source ↗

Simplify:

\frac{30}{48} .

If you missed this problem, review Example 1.24.

Example 1.24 in the book ↗

Answer

\frac{5}{8}

Be Prepared 8.14 §8.5 source ↗

Simplify:

x^{2} \cdot x^{4} .

If you missed this problem, review Example 5.12.

Example 5.12 in the book ↗

Answer

x^{6}

Be Prepared 8.15 §8.5 source ↗

Multiply:

(7 + 3 x) (7 - 3 x) .

If you missed this problem, review Example 5.32.

Example 5.32 in the book ↗

Answer

49 - 9 x^{2}

Try It 8.93 §8.5 source ↗

Simplify: ⓐ

\frac{\sqrt{50 s^{3}}}{\sqrt{128 s}}

ⓑ

\frac{\sqrt[3]{56 a^{}}}{\sqrt[3]{7 a^{4}}} .

Answer

ⓐ

\frac{5 s}{8}

ⓑ

\frac{2}{a}

Try It 8.94 §8.5 source ↗

Simplify: ⓐ

\frac{\sqrt{75 q^{5}}}{\sqrt{108 q}}

ⓑ

\frac{\sqrt[3]{72 b^{2}}}{\sqrt[3]{9 b^{5}}} .

Answer

ⓐ

\frac{5 q^{2}}{6}

ⓑ

\frac{2}{b}

Try It 8.95 §8.5 source ↗

Simplify: ⓐ

\frac{\sqrt{162 x^{10} y^{2}}}{\sqrt{2 x^{6} y^{6}}}

ⓑ

\frac{\sqrt[3]{−128 x^{2} y^{−1}}}{\sqrt[3]{2 x^{−1} y^{2}}} .

Answer

ⓐ

\frac{9 x^{2}}{y^{2}}

ⓑ

\frac{−4 x}{y}

Try It 8.96 §8.5 source ↗

Simplify: ⓐ

\frac{\sqrt{300 m^{3} n^{7}}}{\sqrt{3 m^{5} n}}

ⓑ

\frac{\sqrt[3]{−81 p q^{−1}}}{\sqrt[3]{3 p^{−2} q^{5}}} .

Answer

ⓐ

\frac{10 n^{3}}{m}

ⓑ

\frac{−3 p}{q^{2}}

Try It 8.97 §8.5 source ↗

Simplify:

\frac{\sqrt{64 x^{4} y^{5}}}{\sqrt{2 x y^{3}}} .

Answer

4 x y \sqrt{2 x}

Try It 8.98 §8.5 source ↗

Simplify:

\frac{\sqrt{96 a^{5} b^{4}}}{\sqrt{2 a^{3} b}} .

Answer

4 a b \sqrt{3 b}

Try It 8.99 §8.5 source ↗

Simplify: ⓐ

\frac{5}{\sqrt{3}}

ⓑ

\sqrt{\frac{3}{32}}

ⓒ

\frac{2}{\sqrt{2 x}} .

Answer

ⓐ

\frac{5 \sqrt{3}}{3}

ⓑ

\frac{\sqrt{6}}{8}

ⓒ

\frac{\sqrt{2 x}}{x}

Try It 8.100 §8.5 source ↗

Simplify: ⓐ

\frac{6}{\sqrt{5}}

ⓑ

\sqrt{\frac{7}{18}}

ⓒ

\frac{5}{\sqrt{5 x}} .

Answer

ⓐ

\frac{6 \sqrt{5}}{5}

ⓑ

\frac{\sqrt{14}}{6}

ⓒ

\frac{\sqrt{5 x}}{x}

Try It 8.101 §8.5 source ↗

Simplify: ⓐ

\frac{1}{\sqrt[3]{7}}

ⓑ

\sqrt[3]{\frac{5}{12}}

ⓒ

\frac{5}{\sqrt[3]{9 y}} .

Answer

ⓐ

\frac{\sqrt[3]{49}}{7}

ⓑ

\frac{\sqrt[3]{90}}{6}

ⓒ

\frac{5 \sqrt[3]{3 y^{2}}}{3 y}

Try It 8.102 §8.5 source ↗

Simplify: ⓐ

\frac{1}{\sqrt[3]{2}}

ⓑ

\sqrt[3]{\frac{3}{20}}

ⓒ

\frac{2}{\sqrt[3]{25 n}} .

Answer

ⓐ

\frac{\sqrt[3]{4}}{2}

ⓑ

\frac{\sqrt[3]{150}}{10}

ⓒ

\frac{2 \sqrt[3]{5 n^{2}}}{5 n}

Try It 8.103 §8.5 source ↗

Simplify: ⓐ

\frac{1}{\sqrt[4]{3}}

ⓑ

\sqrt[4]{\frac{3}{64}}

ⓒ

\frac{3}{\sqrt[4]{125 x}} .

Answer

ⓐ

\frac{\sqrt[4]{27}}{3}

ⓑ

\frac{\sqrt[4]{12}}{4}

ⓒ

\frac{3 \sqrt[4]{5 x^{3}}}{5 x}

Try It 8.104 §8.5 source ↗

Simplify: ⓐ

\frac{1}{\sqrt[4]{5}}

ⓑ

\sqrt[4]{\frac{7}{128}}

ⓒ

\frac{4}{\sqrt[4]{4 x}}

Answer

ⓐ

\frac{\sqrt[4]{125}}{5}

ⓑ

\frac{\sqrt[4]{14}}{4}

ⓒ

\frac{2 \sqrt[4]{4 x^{3}}}{x}

Try It 8.105 §8.5 source ↗

Simplify:

\frac{3}{1 - \sqrt{5}} .

Answer

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

Try It 8.106 §8.5 source ↗

Simplify:

\frac{2}{4 - \sqrt{6}} .

Answer

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

Try It 8.107 §8.5 source ↗

Simplify:

\frac{\sqrt{5}}{\sqrt{x} + \sqrt{2}} .

Answer

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

Try It 8.108 §8.5 source ↗

Simplify:

\frac{\sqrt{10}}{\sqrt{y} - \sqrt{3}} .

Answer

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

Try It 8.109 §8.5 source ↗

Simplify:

\frac{\sqrt{p} + \sqrt{2}}{\sqrt{p} - \sqrt{2}} .

Answer

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

Try It 8.110 §8.5 source ↗

Simplify:

\frac{\sqrt{q} - \sqrt{10}}{\sqrt{q} + \sqrt{10}}

Answer

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

8.5-245 §8.5 source ↗

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{128}}{\sqrt{72}}$ ⓑ $\frac{\sqrt[3]{128}}{\sqrt[3]{54}}$

Answer

ⓐ

\frac{4}{3}

ⓑ

\frac{4}{3}

8.5-247 §8.5 source ↗

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{200 m^{5}}}{\sqrt{98 m}}$ ⓑ $\frac{\sqrt[3]{54 y^{2}}}{\sqrt[3]{2 y^{5}}}$

Answer

ⓐ

\frac{10 m^{2}}{7}

ⓑ

\frac{3}{y}

8.5-249 §8.5 source ↗

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{75 r^{3}}}{\sqrt{108 r^{7}}}$ ⓑ $\frac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{81 x^{4}}}$

Answer

ⓐ

\frac{5}{6 r^{2}}

ⓑ

\frac{2 x}{3}

8.5-251 §8.5 source ↗

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}}$ ⓑ $\frac{\sqrt[3]{−16 a^{4} b^{−2}}}{\sqrt[3]{2 a^{−2} b^{}}}$

Answer

ⓐ

\frac{6 p}{q^{2}}

ⓑ

- \frac{2 a^{2}}{b}

8.5-253 §8.5 source ↗

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{320 m n^{−5}}}{\sqrt{45 m^{−7} n^{3}}}$ ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

Answer

ⓐ

\frac{8 m^{4}}{3 n^{4}}

ⓑ

- \frac{2 x^{2}}{3 y^{2}}

8.5-255 §8.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{56 x^{5} y^{4}}}{\sqrt{2 x y^{3}}}$

Answer

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

8.5-257 §8.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt[3]{48 a^{3} b^{6}}}{\sqrt[3]{3 a^{- 1} b^{3}}}$

Answer

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

8.5-259 §8.5 source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{10}{\sqrt{6}}$ ⓑ $\sqrt{\frac{4}{27}}$ ⓒ $\frac{10}{\sqrt{5 x}}$

Answer

ⓐ

\frac{5 \sqrt{6}}{3}

ⓑ

\frac{2 \sqrt{3}}{9}

ⓒ

\frac{2 \sqrt{5 x}}{x}

8.5-261 §8.5 source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{6}{\sqrt{7}}$ ⓑ $\sqrt{\frac{8}{45}}$ ⓒ $\frac{12}{\sqrt{3 p}}$

Answer

ⓐ

\frac{6 \sqrt{7}}{7}

ⓑ

\frac{2 \sqrt{10}}{15}

ⓒ

\frac{4 \sqrt{3 p}}{p}

8.5-263 §8.5 source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{1}{\sqrt[3]{5}}$ ⓑ $\sqrt[3]{\frac{5}{24}}$ ⓒ $\frac{4}{\sqrt[3]{36 a}}$

Answer

ⓐ

\frac{\sqrt[3]{25}}{5}

ⓑ

\frac{\sqrt[3]{45}}{6}

ⓒ

\frac{2 \sqrt[3]{6 a^{2}}}{3 a}

8.5-265 §8.5 source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{1}{\sqrt[3]{11}}$ ⓑ $\sqrt[3]{\frac{7}{54}}$ ⓒ $\frac{3}{\sqrt[3]{3 x^{2}}}$

Answer

ⓐ

\frac{\sqrt[3]{121}}{11}

ⓑ

\frac{\sqrt[3]{28}}{6}

ⓒ

\frac{\sqrt[3]{9 x}}{x}

8.5-267 §8.5 source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{1}{\sqrt[4]{7}}$ ⓑ $\sqrt[4]{\frac{5}{32}}$ ⓒ $\frac{4}{\sqrt[4]{4 x^{2}}}$

Answer

ⓐ

\frac{\sqrt[4]{343}}{7}

ⓑ

\frac{\sqrt[4]{40}}{4}

ⓒ

\frac{2 \sqrt[4]{4 x^{2}}}{x}

8.5-269 §8.5 source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{1}{\sqrt[4]{9}}$ ⓑ $\sqrt[4]{\frac{25}{128}}$ ⓒ $\frac{6}{\sqrt[4]{27 a}}$

Answer

ⓐ

\frac{\sqrt[4]{9}}{3}

ⓑ

\frac{\sqrt[4]{50}}{4}

ⓒ

\frac{2 \sqrt[4]{3 a^{3}}}{a}

8.5-271 §8.5 source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

$\frac{8}{1 - \sqrt{5}}$

Answer

−2 (1 + \sqrt{5})

8.5-273 §8.5 source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

$\frac{6}{3 - \sqrt{7}}$

Answer

3 (3 + \sqrt{7})

8.5-275 §8.5 source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

$\frac{\sqrt{3}}{\sqrt{m} - \sqrt{5}}$

Answer

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

8.5-277 §8.5 source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

$\frac{\sqrt{2}}{\sqrt{x} - \sqrt{6}}$

Answer

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

8.5-279 §8.5 source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

$\frac{\sqrt{r} + \sqrt{5}}{\sqrt{r} - \sqrt{5}}$

Answer

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

8.5-281 §8.5 source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

$\frac{\sqrt{x} + \sqrt{8}}{\sqrt{x} - \sqrt{8}}$

Answer

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

8.5-283 §8.5 source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

ⓐ Simplify $\sqrt{\frac{27}{3}}$ and explain all your steps.
ⓑ Simplify $\sqrt{\frac{27}{5}}$ and explain all your steps.
ⓒ Why are the two methods of simplifying square roots different?

Answer

Answers will vary.

8.5-285 §8.5 source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

Explain why multiplying $\sqrt{2 x} - 3$ by its conjugate results in an expression with no radicals.

Answer

Answers will vary.

Be Prepared 8.16 §8.6 source ↗

Simplify:

{(y - 3)}^{2} .

If you missed this problem, review Example 5.31.

Example 5.31 in the book ↗

Answer

y^{2} - 6 y + 9

Be Prepared 8.17 §8.6 source ↗

Solve:

2 x - 5 = 0 .

If you missed this problem, review Example 2.2.

Example 2.2 in the book ↗

Answer

x = \frac{5}{2}

Be Prepared 8.18 §8.6 source ↗

Solve

n^{2} - 6 n + 8 = 0 .

If you missed this problem, review Example 6.45.

Example 6.45 in the book ↗

Answer

n = 2 or n = 4

Try It 8.111 §8.6 source ↗

Solve:

\sqrt{3 m + 2} - 5 = 0 .

Answer

m = \frac{23}{3}

Try It 8.112 §8.6 source ↗

Solve:

\sqrt{10 z + 1} - 2 = 0 .

Answer

z = \frac{3}{10}

Try It 8.113 §8.6 source ↗

Solve:

\sqrt{2 r - 3} + 5 = 0 .

Answer

no solution

Try It 8.114 §8.6 source ↗

Solve:

\sqrt{7 s - 3} + 2 = 0 .

Answer

no solution

Try It 8.115 §8.6 source ↗

Solve:

\sqrt{x - 2} + 2 = x .

Answer

x = 2, x = 3

Try It 8.116 §8.6 source ↗

Solve:

\sqrt{y - 5} + 5 = y .

Answer

y = 5, y = 6

Try It 8.117 §8.6 source ↗

Solve:

\sqrt[3]{4 x - 3} + 8 = 5

Answer

x = −6

Try It 8.118 §8.6 source ↗

Solve:

\sqrt[3]{6 x - 10} + 1 = −3

Answer

x = −9

Try It 8.119 §8.6 source ↗

Solve:

{(9 x + 9)}^{\frac{1}{4}} -

2 =

1 .

Answer

x = 8

Try It 8.120 §8.6 source ↗

Solve:

{(4 x - 8)}^{\frac{1}{4}} +

5 =

7 .

Answer

x = 6

Try It 8.121 §8.6 source ↗

Solve:

\sqrt{m + 9} - m + 3 = 0 .

Answer

m = 7

Try It 8.122 §8.6 source ↗

Solve:

\sqrt{n + 1} - n + 1 = 0 .

Answer

n = 3

Try It 8.123 §8.6 source ↗

Solve:

2 \sqrt{4 a + 4} -

16 =

16 .

Answer

a = 63

Try It 8.124 §8.6 source ↗

Solve:

3 \sqrt{2 b + 3} -

25 =

50 .

Answer

b = 311

Try It 8.125 §8.6 source ↗

Solve:

\sqrt[3]{5 x - 4} =

\sqrt[3]{2 x + 5} .

Answer

x = 3

Try It 8.126 §8.6 source ↗

Solve:

\sqrt[3]{7 x + 1} =

\sqrt[3]{2 x - 5} .

Answer

x = - \frac{6}{5}

Try It 8.127 §8.6 source ↗

Solve:

3 - \sqrt{x} = \sqrt{x - 3} .

Answer

x = 4

Try It 8.128 §8.6 source ↗

Solve:

\sqrt{x} + 2 = \sqrt{x + 16} .

Answer

x = 9

Try It 8.129 §8.6 source ↗

Solve:

\sqrt{x - 1} + 2 = \sqrt{2 x + 6}

Answer

x = 5

Try It 8.130 §8.6 source ↗

Solve:

\sqrt{x} + 2 = \sqrt{3 x + 4}

Answer

x = 0 x = 4

Try It 8.131 §8.6 source ↗

A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula

t = \frac{\sqrt{h}}{4}

to find how many seconds it took for the package to reach the ground.

Answer

9 seconds

Try It 8.132 §8.6 source ↗

A window washer dropped a squeegee from a platform 196 feet above the sidewalk Use the formula

t = \frac{\sqrt{h}}{4}

to find how many seconds it took for the squeegee to reach the sidewalk.

Answer

3.5

seconds

Try It 8.133 §8.6 source ↗

An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula

s = \sqrt{24 d}

to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

Answer

42.7

feet

Try It 8.134 §8.6 source ↗

The skid marks of a vehicle involved in an accident were 122 feet long. Use the formula

s = \sqrt{24 d}

to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Answer

54.1

feet

8.6-287 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{5 x - 6} = 8$

Answer

x = 14

8.6-289 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{5 x + 1} = −3$

Answer

no solution

8.6-291 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt[3]{2 x} = −2$

Answer

x = −4

8.6-293 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{2 m - 3} - 5 = 0$

Answer

m = 14

8.6-295 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{6 v - 2} - 10 = 0$

Answer

v = 17

8.6-297 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{4 m + 2} + 2 = 6$

Answer

m = \frac{7}{2}

8.6-299 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{2 u - 3} + 2 = 0$

Answer

no solution

8.6-301 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{u - 3} + 3 = u$

Answer

u = 3, u = 4

8.6-303 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{r - 1} = r - 1$

Answer

r = 1, r = 2

8.6-305 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt[3]{6 x + 4} = 4$

Answer

x = 10

8.6-307 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt[3]{4 x + 5} - 2 = −5$

Answer

x = −8

8.6-309 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

${(6 x + 1)}^{\frac{1}{2}} -$ $3 =$ $4$

Answer

x = 8

8.6-311 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

${(8 x + 5)}^{\frac{1}{3}} +$ $2 =$ $−1$

Answer

x = −4

8.6-313 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

${(12 x - 3)}^{\frac{1}{4}} -$ $5 =$ $−2$

Answer

x = 7

8.6-315 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{x + 1} - x + 1 = 0$

Answer

x = 3

8.6-317 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{z + 100} - z = −10$

Answer

z = 21

8.6-319 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$3 \sqrt{2 x - 3} - 20 = 7$

Answer

x = 42

8.6-321 §8.6 source ↗

Solve Radical Equations

In the following exercises, solve.

$2 \sqrt{8 r + 1} - 8 = 2$

Answer

r = 3

8.6-323 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{3 u + 7} = \sqrt{5 u + 1}$

Answer

u = 3

8.6-325 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{8 + 2 r} = \sqrt{3 r + 10}$

Answer

r = −2

8.6-327 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt[3]{5 x - 1} = \sqrt[3]{x + 3}$

Answer

x = 1

8.6-329 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt[3]{2 x^{2} + 9 x - 18} =$ $\sqrt[3]{x^{2} + 3 x - 2}$

Answer

x = −8, x = 2

8.6-331 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{a} + 2 = \sqrt{a + 4}$

Answer

a = 0

8.6-333 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{u} + 1 = \sqrt{u + 4}$

Answer

u = \frac{9}{4}

8.6-335 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{a + 5} - \sqrt{a} = 1$

Answer

a = 4

8.6-337 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{2 x + 1} = 1 + \sqrt{x}$

Answer

x = 0 x = 4

8.6-339 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{2 x - 1} - \sqrt{x - 1} = 1$

Answer

x = 1 x = 5

8.6-341 §8.6 source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{x + 7} - \sqrt{x - 5} = 2$

Answer

x = 9

8.6-343 §8.6 source ↗

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula $s = \sqrt{A}$ to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.

Answer

8.7

feet

8.6-345 §8.6 source ↗

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Gravity A hang glider dropped his cell phone from a height of 350 feet. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the cell phone to reach the ground.

Answer

4.7

seconds

8.6-347 §8.6 source ↗

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Accident investigation The skid marks for a car involved in an accident measured 216 feet. Use the formula $s = \sqrt{24 d}$ to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

Answer

72 feet

8.6-349 §8.6 source ↗

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Explain why an equation of the form $\sqrt{x} + 1 = 0$ has no solution.

Answer

Answers will vary.

Be Prepared 8.19 §8.7 source ↗

Solve:

1 - 2 x \geq 0 .

If you missed this problem, review Example 2.50.

Example 2.50 in the book ↗

Answer

(− \infty, \frac{1}{2}]

Be Prepared 8.20 §8.7 source ↗

For

f (x) = 3 x - 4,

evaluate

f (2),

f (−1),

f (0) .

If you missed this problem, review Example 3.48.

Example 3.48 in the book ↗

Answer

f (2) =

2,

f (−1) =

−7,

f (0) =

−4

Be Prepared 8.21 §8.7 source ↗

Graph

f (x) = \sqrt{x} .

State the domain and range of the function in interval notation.
If you missed this problem, review Example 3.56.

Example 3.56 in the book ↗

Answer

domain: [0, \infty);

range: [0, \infty)

Try It 8.135 §8.7 source ↗

For the function

f (x) = \sqrt{3 x - 2},

find ⓐ

f (6)

ⓑ

f (0) .

Answer

ⓐ

f (6) = 4

ⓑ no value at

x = 0

Try It 8.136 §8.7 source ↗

For the function

g (x) = \sqrt{5 x + 5},

find ⓐ

g (4)

ⓑ

g (−3) .

Answer

ⓐ

g (4) = 5

ⓑ no value at

f (−3)

Try It 8.137 §8.7 source ↗

For the function

g (x) = \sqrt[3]{3 x - 4},

find ⓐ

g (4)

ⓑ

g (1) .

Answer

ⓐ

g (4) = 2

ⓑ

g (1) = −1

Try It 8.138 §8.7 source ↗

For the function

h (x) = \sqrt[3]{5 x - 2},

find ⓐ

h (2)

ⓑ

h (−5) .

Answer

ⓐ

h (2) = 2

ⓑ

h (−5) = −3

Try It 8.139 §8.7 source ↗

For the function

f (x) = \sqrt[4]{3 x + 4},

find ⓐ

f (4)

ⓑ

f (−1) .

Answer

ⓐ

f (4) = 2

ⓑ

f (−1) = 1

Try It 8.140 §8.7 source ↗

For the function

g (x) = \sqrt[4]{5 x + 1},

find ⓐ

g (16)

ⓑ

g (3) .

Answer

ⓐ

g (16) = 3

ⓑ

g (3) = 2

Try It 8.141 §8.7 source ↗

Find the domain of the function,

f (x) = \sqrt{6 x - 5} .

Write the domain in interval notation.

Answer

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

Try It 8.142 §8.7 source ↗

Find the domain of the function,

f (x) = \sqrt{4 - 5 x} .

Write the domain in interval notation.

Answer

(− \infty, \frac{4}{5}]

Try It 8.143 §8.7 source ↗

Find the domain of the function,

f (x) = \sqrt{\frac{4}{x + 3}} .

Write the domain in interval notation.

Answer

(−3, \infty)

Try It 8.144 §8.7 source ↗

Find the domain of the function,

h (x) = \sqrt{\frac{9}{x - 5}} .

Write the domain in interval notation.

Answer

(5, \infty)

Try It 8.145 §8.7 source ↗

Find the domain of the function,

f (x) =

\sqrt[3]{3 x^{2} - 1} .

Write the domain in interval notation.

Answer

(− \infty, \infty)

Try It 8.146 §8.7 source ↗

Find the domain of the function,

g (x) = \sqrt[3]{5 x - 4} .

Write the domain in interval notation.

Answer

(− \infty, \infty)

Try It 8.147 §8.7 source ↗

For the function

f (x) = \sqrt{x + 2},

Answer

ⓐ domain: $[−2, \infty)$
ⓑ

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).

Try It 8.148 §8.7 source ↗

For the function

f (x) = \sqrt{x - 2},

Answer

ⓐ domain: $[2, \infty)$
ⓑ

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).

Try It 8.149 §8.7 source ↗

For the function $f (x) = − \sqrt[3]{x},$

Answer

ⓐ domain: $(- \infty, \infty)$
ⓑ

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).

Try It 8.150 §8.7 source ↗

For the function $f (x) = \sqrt[3]{x - 2},$

Answer

ⓐ domain: $(- \infty, \infty)$
ⓑ

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).

8.7-351 §8.7 source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

$f (x) = \sqrt{4 x - 4},$ find ⓐ $f (5)$ ⓑ $f (0) .$

Answer

ⓐ

f (5) = 4

ⓑ no value at

x = 0

8.7-353 §8.7 source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

$g (x) = \sqrt{6 x + 1},$ find ⓐ $g (4)$ ⓑ $g (8) .$

Answer

ⓐ

g (4) = 5

ⓑ

g (8) = 7

8.7-355 §8.7 source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

$F (x) = \sqrt{3 - 2 x},$ find ⓐ $F (1)$ ⓑ $F (−11) .$

Answer

ⓐ

F (1) = 1

ⓑ

F (−11) = 5

8.7-357 §8.7 source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

$G (x) = \sqrt{5 x - 1},$ find ⓐ $G (5)$ ⓑ $G (2) .$

Answer

ⓐ

G (5) = 2 \sqrt{6}

ⓑ

G (2) = 3

8.7-359 §8.7 source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

$g (x) = \sqrt[3]{2 x - 4},$ find ⓐ $g (6)$ ⓑ $g (−2) .$

Answer

ⓐ

g (6) = 2

ⓑ

g (−2) = −2

8.7-361 §8.7 source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

$h (x) = \sqrt[3]{x^{2} - 4},$ find ⓐ $h (−2)$ ⓑ $h (6) .$

Answer

ⓐ

h (−2) = 0

ⓑ

h (6) = 2 \sqrt[3]{4}

8.7-363 §8.7 source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

For the function $f (x) = \sqrt[4]{2 x^{3}},$ find ⓐ $f (0)$ ⓑ $f (2) .$

Answer

ⓐ

f (0) = 0

ⓑ

f (2) = 2

8.7-365 §8.7 source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

For the function $g (x) = \sqrt[4]{4 - 4 x},$ find ⓐ $g (1)$ ⓑ $g (−3) .$

Answer

ⓐ

g (1) = 0

ⓑ

g (−3) = 2

8.7-367 §8.7 source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$f (x) = \sqrt{3 x - 1}$

Answer

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

8.7-369 §8.7 source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$g (x) = \sqrt{2 - 3 x}$

Answer

(− \infty, \frac{2}{3}]

8.7-371 §8.7 source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

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

Answer

(2, \infty)

8.7-373 §8.7 source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$f (x) = \sqrt{\frac{x + 3}{x - 2}}$

Answer

(− \infty, −3] \cup (2, \infty)

8.7-375 §8.7 source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$g (x) = \sqrt[3]{8 x - 1}$

Answer

(− \infty, \infty)

8.7-377 §8.7 source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$f (x) =$ $\sqrt[3]{4 x^{2} - 16}$

Answer

(− \infty, \infty)

8.7-379 §8.7 source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$F (x) = \sqrt[4]{8 x + 3}$

Answer

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

8.7-381 §8.7 source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$G (x) = \sqrt[5]{2 x - 1}$

Answer

(− \infty, \infty)

8.7-383 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$f (x) = \sqrt{x + 1}$

Answer

ⓐ domain: $[−1, \infty)$
ⓑ

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).

8.7-385 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$g (x) = \sqrt{x + 4}$

Answer

ⓐ domain: $[−4, \infty)$
ⓑ

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).

8.7-387 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$f (x) = \sqrt{x} + 2$

Answer

ⓐ domain: $[0, \infty)$
ⓑ

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).

8.7-389 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$g (x) = 2 \sqrt{x}$

Answer

ⓐ domain: $[0, \infty)$
ⓑ

8.7-391 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$f (x) = \sqrt{3 - x}$

Answer

ⓐ domain: $(− \infty, 3]$
ⓑ

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).

8.7-393 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$g (x) = − \sqrt{x}$

Answer

ⓐ domain: $[0, \infty)$
ⓑ

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 negative 8 to 0. The function has a starting point at (0, 0) and goes through the points (1, negative 1) and (4, negative 2).

8.7-395 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$f (x) = \sqrt[3]{x + 1}$

Answer

ⓐ domain: $(− \infty, \infty)$
ⓑ

The figure shows a cube root function graph on the x y-coordinate plane. The x-axis of the plane runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The function has a center point at (negative 1, 0) and goes through the points (negative 2, negative 1) and (0, 1).

8.7-397 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$g (x) = \sqrt[3]{x + 2}$

Answer

ⓐ domain: $(− \infty, \infty)$
ⓑ

8.7-399 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$f (x) = \sqrt[3]{x} + 3$

Answer

ⓐ domain: $(− \infty, \infty)$
ⓑ

8.7-401 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$g (x) = \sqrt[3]{x}$

Answer

ⓐ domain: $(− \infty, \infty)$
ⓑ

8.7-403 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$f (x) = 2 \sqrt[3]{x}$

Answer

ⓐ domain: $(− \infty, \infty)$
ⓑ

8.7-405 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

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

Answer

Answers will vary.

8.7-407 §8.7 source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

Explain why $y = \sqrt[3]{x}$ is a function.

Answer

Answers will vary.

Be Prepared 8.22 §8.8 source ↗

Given the numbers

−4,

-

\sqrt{7},

0. \bar{5},

\frac{7}{3},

3,

\sqrt{81},

Example 1.42 in the book ↗

Answer

ⓐ

−4,

0. \bar{5},

\frac{7}{3},

3,

\sqrt{81};

ⓑ

\sqrt{7};

ⓒ

−4,

-

\sqrt{7},

0. \bar{5},

\frac{7}{3},

3,

\sqrt{81}

Be Prepared 8.23 §8.8 source ↗

Multiply:

(x - 3) (2 x + 5) .

If you missed this problem, review Example 5.28.

Example 5.28 in the book ↗

Answer

2 x^{2} - x - 15

Be Prepared 8.24 §8.8 source ↗

Rationalize the denominator:

\frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}} .

If you missed this problem, review Example 5.32.

Example 5.32 in the book ↗

Answer

\frac{5 + \sqrt{15}}{2}

Try It 8.151 §8.8 source ↗

Write each expression in terms of i and simplify if possible:

Answer

ⓐ

9 i

ⓑ

\sqrt{5} i

ⓒ

3 \sqrt{2} i

Try It 8.152 §8.8 source ↗

Write each expression in terms of i and simplify if possible:

Answer

ⓐ

6 i

ⓑ

\sqrt{3} i

ⓒ

3 \sqrt{3} i

Try It 8.153 §8.8 source ↗

Add:

\sqrt{−8} + \sqrt{−32} .

Answer

6 \sqrt{2} i

Try It 8.154 §8.8 source ↗

Add:

\sqrt{−27} + \sqrt{−48} .

Answer

7 \sqrt{3} i

Try It 8.155 §8.8 source ↗

Simplify: ⓐ

(2 + 7 i) +

(4 - 2 i)

ⓑ

(8 - 4 i) -

(2 - i) .

Answer

ⓐ

6 + 5 i

ⓑ

6 - 3 i

Try It 8.156 §8.8 source ↗

Simplify: ⓐ

(3 - 2 i) +

(−5 - 4 i)

ⓑ

(4 + 3 i) -

(2 - 6 i) .

Answer

ⓐ

−2 - 6 i

ⓑ

2 + 9 i

Try It 8.157 §8.8 source ↗

Multiply:

4 i (5 - 3 i) .

Answer

12 + 20 i

Try It 8.158 §8.8 source ↗

Multiply:

−3 i (2 + 4 i) .

Answer

12 - 6 i

Try It 8.159 §8.8 source ↗

Multiply:

(5 - 3 i) (−1 - 2 i) .

Answer

−11 - 7 i

Try It 8.160 §8.8 source ↗

Multiply:

(−4 - 3 i) (2 + i) .

Answer

−5 - 10 i

Try It 8.161 §8.8 source ↗

Multiply using the Binomial Squares pattern:

{(−2 - 5 i)}^{2} .

Answer

−21 + 20 i

Try It 8.162 §8.8 source ↗

Multiply using the Binomial Squares pattern:

{(−5 + 4 i)}^{2} .

Answer

9 - 40 i

Try It 8.163 §8.8 source ↗

Multiply:

\sqrt{−49} \cdot \sqrt{−4} .

Answer

−14

Try It 8.164 §8.8 source ↗

Multiply:

\sqrt{−36} \cdot \sqrt{−81} .

Answer

−54

Try It 8.165 §8.8 source ↗

Multiply:

(4 - \sqrt{−12}) (3 - \sqrt{−48}) .

Answer

−12 - 22 \sqrt{3} i

Try It 8.166 §8.8 source ↗

Multiply:

(−2 + \sqrt{−8}) (3 - \sqrt{−18}) .

Answer

6 + 12 \sqrt{2} i

Try It 8.167 §8.8 source ↗

Multiply:

(4 - 3 i) \cdot (4 + 3 i) .

Answer

Try It 8.168 §8.8 source ↗

Multiply:

(−2 + 5 i) \cdot (−2 - 5 i) .

Answer

Try It 8.169 §8.8 source ↗

Multiply using the Product of Complex Conjugates Pattern:

(3 - 10 i) (3 + 10 i) .

Answer

109

Try It 8.170 §8.8 source ↗

Multiply using the Product of Complex Conjugates Pattern:

(−5 + 4 i) (−5 - 4 i) .

Answer

Try It 8.171 §8.8 source ↗

Divide:

\frac{2 + 5 i}{5 - 2 i} .

Answer

Try It 8.172 §8.8 source ↗

Divide:

\frac{1 + 6 i}{6 - i} .

Answer

Try It 8.173 §8.8 source ↗

Divide, writing the answer in standard form:

\frac{4}{1 - 4 i} .

Answer

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

Try It 8.174 §8.8 source ↗

Divide, writing the answer in standard form:

\frac{−2}{−1 + 2 i} .

Answer

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

Try It 8.175 §8.8 source ↗

Divide:

\frac{3 + 3 i}{2 i} .

Answer

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

Try It 8.176 §8.8 source ↗

Divide:

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

Answer

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

Try It 8.177 §8.8 source ↗

Simplify:

i^{75} .

Answer

− i

Try It 8.178 §8.8 source ↗

Simplify:

i^{92} .

Answer

1

8.8-409 §8.8 source ↗

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

Answer

ⓐ

4 i

ⓑ

\sqrt{11} i

ⓒ

2 \sqrt{2} i

8.8-411 §8.8 source ↗

Evaluate the Square Root of a Negative Number

In the following exercises, write each expression in terms of i and simplify if possible.

Answer

ⓐ

10 i

ⓑ

\sqrt{13} i

ⓒ

3 \sqrt{5} i

8.8-413 §8.8 source ↗

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\sqrt{−75} + \sqrt{−48}$

Answer

9 \sqrt{3} i

8.8-415 §8.8 source ↗

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$\sqrt{−50} + \sqrt{−18}$

Answer

8 \sqrt{2} i

8.8-417 §8.8 source ↗

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$(1 + 3 i) +$ $(7 + 4 i)$

Answer

8 + 7 i

8.8-419 §8.8 source ↗

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$(8 - i) +$ $(6 + 3 i)$

Answer

14 + 2 i

8.8-421 §8.8 source ↗

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$(1 - 4 i) -$ $(3 - 6 i)$

Answer

−2 + 2 i

8.8-423 §8.8 source ↗

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$(6 + i) -$ $(−2 - 4 i)$

Answer

8 + 5 i

8.8-425 §8.8 source ↗

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$(5 - \sqrt{−36}) +$ $(2 - \sqrt{−49})$

Answer

7 - 13 i

8.8-427 §8.8 source ↗

Add or Subtract Complex Numbers In the following exercises, add or subtract.

$(−7 - \sqrt{−50}) -$ $(−32 - \sqrt{−18})$

Answer

25 - 2 \sqrt{2} i

8.8-429 §8.8 source ↗

Multiply Complex Numbers

In the following exercises, multiply.

$4 i (5 - 3 i)$

Answer

12 + 20 i

8.8-431 §8.8 source ↗

Multiply Complex Numbers

In the following exercises, multiply.

$−6 i (−3 - 2 i)$

Answer

−12 + 18 i

8.8-433 §8.8 source ↗

Multiply Complex Numbers

In the following exercises, multiply.

$(4 + 3 i) (−5 + 6 i)$

Answer

−38 + + 9 i

8.8-435 §8.8 source ↗

Multiply Complex Numbers

In the following exercises, multiply.

$(−3 + 3 i) (−2 - 7 i)$

Answer

27 + 15 i

8.8-437 §8.8 source ↗

In the following exercises, multiply using the Product of Binomial Squares Pattern.

${(3 + 4 i)}^{2}$

Answer

−7 + 24 i

8.8-439 §8.8 source ↗

In the following exercises, multiply using the Product of Binomial Squares Pattern.

${(−2 - 3 i)}^{2}$

Answer

−5 + 12 i

8.8-441 §8.8 source ↗

In the following exercises, multiply.

$\sqrt{−25} \cdot \sqrt{−36}$

Answer

−30

8.8-443 §8.8 source ↗

In the following exercises, multiply.

$\sqrt{−9} \cdot \sqrt{−100}$

Answer

−30

8.8-445 §8.8 source ↗

In the following exercises, multiply.

$(−2 - \sqrt{−27}) (4 - \sqrt{−48})$

Answer

−44 - 4 i \sqrt{3}

8.8-447 §8.8 source ↗

In the following exercises, multiply.

$(2 + \sqrt{−8}) (−4 + \sqrt{−18})$

Answer

−20 - 2 \sqrt{2} i

8.8-449 §8.8 source ↗

In the following exercises, multiply.

$(2 - i) (2 + i)$

Answer

8.8-451 §8.8 source ↗

In the following exercises, multiply.

$(7 - 2 i) (7 + 2 i)$

Answer

8.8-453 §8.8 source ↗

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

$(7 - i) (7 + i)$

Answer

8.8-455 §8.8 source ↗

In the following exercises, multiply using the Product of Complex Conjugates Pattern.

$(9 - 2 i) (9 + 2 i)$

Answer

8.8-457 §8.8 source ↗

Divide Complex Numbers

In the following exercises, divide.

$\frac{3 + 4 i}{4 - 3 i}$

Answer

8.8-459 §8.8 source ↗

Divide Complex Numbers

In the following exercises, divide.

$\frac{2 + i}{3 - 4 i}$

Answer

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

8.8-461 §8.8 source ↗

Divide Complex Numbers

In the following exercises, divide.

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

Answer

\frac{6}{13} + \frac{9}{13} i

8.8-463 §8.8 source ↗

Divide Complex Numbers

In the following exercises, divide.

$\frac{−4}{3 - 2 i}$

Answer

- \frac{12}{13} - \frac{8}{13} i

8.8-465 §8.8 source ↗

Divide Complex Numbers

In the following exercises, divide.

$\frac{1 + 4 i}{3 i}$

Answer

\frac{4}{3} - \frac{1}{3} i

8.8-467 §8.8 source ↗

Divide Complex Numbers

In the following exercises, divide.

$\frac{−2 - 3 i}{4 i}$

Answer

- \frac{3}{4} + \frac{1}{2} i

8.8-469 §8.8 source ↗

Simplify Powers of i

In the following exercises, simplify.

$i^{41}$

Answer

8.8-471 §8.8 source ↗

Simplify Powers of i

In the following exercises, simplify.

$i^{66}$

Answer

−1

8.8-473 §8.8 source ↗

Simplify Powers of i

In the following exercises, simplify.

$i^{128}$

Answer

8.8-475 §8.8 source ↗

Simplify Powers of i

In the following exercises, simplify.

$i^{137}$

Answer

8.8-477 §8.8 source ↗

Simplify Powers of i

In the following exercises, simplify.

Explain the relationship between real numbers and complex numbers.

Answer

Answers will vary.

8.8-479 §8.8 source ↗

Simplify Powers of i

In the following exercises, simplify.

Why is $\sqrt{−64} = 8 i$ but $\sqrt[3]{−64} = −4 .$

Answer

Answers will vary.

8.R-481 §8.R source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

ⓐ $\sqrt{225}$ ⓑ $− \sqrt{16}$

Answer

ⓐ 15 ⓑ

−4

8.R-483 §8.R source ↗

Simplify Expressions with Roots

In the following exercises, simplify.

Answer

ⓐ 2 ⓑ 3 ⓒ 3

8.R-485 §8.R source ↗

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

ⓐ $\sqrt{68}$ ⓑ $\sqrt[3]{84}$

Answer

ⓐ

8 <

\sqrt{68} <

9

ⓑ

4 <

\sqrt[3]{84} <

5

8.R-487 §8.R source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[3]{a^{3}}$
ⓑ $\sqrt[7]{b^{7}}$

Answer

ⓐ a ⓑ

b

8.R-489 §8.R source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[4]{m^{8}}$
ⓑ $\sqrt[5]{n^{20}}$

Answer

ⓐ

m^{2}

ⓑ

n^{4}

8.R-491 §8.R source ↗

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

ⓐ $\sqrt[3]{216 a^{6}}$
ⓑ $\sqrt[5]{32 b^{20}}$

Answer

ⓐ

6 a^{2}

ⓑ

2 b^{4}

8.R-493 §8.R source ↗

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

$\sqrt{125}$

Answer

5 \sqrt{5}

8.R-495 §8.R source ↗

Use the Product Property to Simplify Radical Expressions

In the following exercises, use the Product Property to simplify radical expressions.

ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$

Answer

ⓐ

5 \sqrt[3]{5}

ⓑ

2 \sqrt[6]{2}

8.R-497 §8.R source ↗

In the following exercises, simplify using absolute value signs as needed.

Answer

ⓐ

4 | s^{7} | \sqrt[]{5 s}

ⓑ

2 a \sqrt[5]{3 a^{2}}

ⓒ

2 | b | \sqrt[6]{2 b}

8.R-499 §8.R source ↗

In the following exercises, simplify using absolute value signs as needed.

ⓐ $\sqrt[5]{−32}$
ⓑ $\sqrt[8]{−1}$

Answer

ⓐ

−2

ⓑ not real

8.R-501 §8.R source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

Answer

ⓐ

\frac{6}{7}

ⓑ

\frac{2}{3}

ⓒ

\frac{1}{2}

8.R-503 §8.R source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

$\sqrt{\frac{300 m^{5}}{64}}$

Answer

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

8.R-505 §8.R source ↗

Use the Quotient Property to Simplify Radical Expressions

In the following exercises, use the Quotient Property to simplify square roots.

ⓐ $\sqrt{\frac{27 p^{2} q}{108 p^{4} q^{3}}}$
ⓑ $\sqrt[3]{\frac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$
ⓒ $\sqrt[6]{\frac{2 m^{9} n^{7}}{128 m^{3} n}}$

Answer

ⓐ

\frac{1}{2 | p q |}

ⓑ

\frac{2 c d}{5} \sqrt[3]{d^{2}}

ⓒ

\frac{| m n |}{2}

8.R-507 §8.R source ↗

Simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

Answer

ⓐ

\sqrt{r}

ⓑ

\sqrt[3]{s}

ⓒ

\sqrt[4]{t}

8.R-509 §8.R source ↗

In the following exercises, simplify.

Answer

ⓐ 5 ⓑ 3 ⓒ 2

8.R-511 §8.R source ↗

In the following exercises, simplify.

Answer

ⓐ

−2

ⓑ

\frac{1}{3}

ⓒ

−5

8.R-513 §8.R source ↗

In the following exercises, simplify.

Answer

ⓐ 125 ⓑ

\frac{1}{27}

8.R-515 §8.R source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Answer

ⓐ

6^{3}

ⓑ

b^{9}

ⓒ

\frac{1}{w}

8.R-517 §8.R source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

Answer

ⓐ

4 \sqrt{2}

ⓑ

9 \sqrt[3]{p}

ⓒ

2 \sqrt[3]{x}

8.R-519 §8.R source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

Answer

ⓐ

7 \sqrt{3}

ⓑ

7 \sqrt[3]{2}

ⓒ

3 \sqrt[4]{5}

8.R-521 §8.R source ↗

Add and Subtract Radical Expressions

In the following exercises, simplify.

$3 \sqrt{75 y^{2}} +$ $8 y \sqrt{48} -$ $\sqrt{300 y^{2}}$

Answer

37 y \sqrt{3}

8.R-523 §8.R source ↗

Multiply Radical Expressions

In the following exercises, simplify.

ⓐ $(3 \sqrt{2 x^{3}}) (7 \sqrt{18 x^{2}})$
ⓑ $(−6 \sqrt[3]{20 a^{2}}) (−2 \sqrt[3]{16 a^{3}})$

Answer

ⓐ

126 x^{2} \sqrt{x}

ⓑ

48 a \sqrt[3]{5 a^{2}}

8.R-525 §8.R source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ $(3 - 2 \sqrt{7}) (5 - 4 \sqrt{7})$
ⓑ $(\sqrt[3]{x} - 5) (\sqrt[3]{x} - 3)$

Answer

ⓐ

71 - 22 \sqrt{7}

ⓑ

\sqrt[3]{x^{2}} - 8 \sqrt[3]{x} + 15

8.R-527 §8.R source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

ⓐ ${(4 + \sqrt{11})}^{2}$
ⓑ ${(3 - 2 \sqrt{5})}^{2}$

Answer

ⓐ

27 + 8 \sqrt{11}

ⓑ

29 - 12 \sqrt{5}

8.R-529 §8.R source ↗

Use Polynomial Multiplication to Multiply Radical Expressions

In the following exercises, multiply.

$(\sqrt[3]{3 x} + 2) (\sqrt[3]{3 x} - 2)$

Answer

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

8.R-531 §8.R source ↗

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{320 m n^{−5}}}{\sqrt{45 m^{−7} n^{3}}}$
ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

Answer

ⓐ

\frac{8 m^{4}}{3 n^{4}}

ⓑ

- \frac{2 x^{2}}{3 y^{2}}

8.R-533 §8.R source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Answer

ⓐ

\frac{\sqrt[3]{121}}{11}

ⓑ

\frac{\sqrt[3]{28}}{6}

ⓒ

\frac{\sqrt[3]{9 x}}{x}

8.R-535 §8.R source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

$\frac{7}{2 - \sqrt{6}}$

Answer

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

8.R-537 §8.R source ↗

Rationalize a Two Term Denominator

In the following exercises, simplify.

$\frac{\sqrt{x} + \sqrt{8}}{\sqrt{x} - \sqrt{8}}$

Answer

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

8.R-539 §8.R source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{5 x + 1} = −3$

Answer

no solution

8.R-541 §8.R source ↗

Solve Radical Equations

In the following exercises, solve.

$\sqrt{u - 3} + 3 = u$

Answer

u = 3, u = 4

8.R-543 §8.R source ↗

Solve Radical Equations

In the following exercises, solve.

${(8 x + 5)}^{\frac{1}{3}} +$ $2 =$ $−1$

Answer

x = −4

8.R-545 §8.R source ↗

Solve Radical Equations

In the following exercises, solve.

$2 \sqrt{8 r + 1} - 8 = 2$

Answer

r = 3

8.R-547 §8.R source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt[3]{2 x^{2} + 9 x - 18} =$ $\sqrt[3]{x^{2} + 3 x - 2}$

Answer

x = −8, x = 2

8.R-549 §8.R source ↗

Solve Radical Equations with Two Radicals

In the following exercises, solve.

$\sqrt{x + 1} - \sqrt{x - 2} = 1$

Answer

x = 3

8.R-551 §8.R source ↗

Use Radicals in Applications

In the following exercises, solve. Round approximations to one decimal place.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula $s = \sqrt{24 d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Answer

64.8

feet

8.R-553 §8.R source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

$G (x) = \sqrt{5 x - 1},$ find
ⓐ $G (5)$
ⓑ $G (2)$

Answer

ⓐ

G (5) = 2 \sqrt{6}

ⓑ

G (2) = 3

8.R-555 §8.R source ↗

Evaluate a Radical Function

In the following exercises, evaluate each function.

For the function
$g (x) = \sqrt[4]{4 - 4 x},$ find
ⓐ $g (1)$
ⓑ $g (−3)$

Answer

ⓐ

g (1) = 0

ⓑ

g (−3) = 2

8.R-557 §8.R source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$F (x) = \sqrt{\frac{x + 3}{x - 2}}$

Answer

(2, \infty)

8.R-559 §8.R source ↗

Find the Domain of a Radical Function

In the following exercises, find the domain of the function and write the domain in interval notation.

$F (x) = \sqrt[4]{10 - 7 x}$

Answer

- \infty, \frac{10}{7}

8.R-561 §8.R source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$g (x) = 2 \sqrt{x}$

Answer

ⓐ domain: $[0, \infty)$
ⓑ

8.R-563 §8.R source ↗

Graph Radical Functions

In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$f (x) = \sqrt[3]{x} + 3$

Answer

ⓐ domain: $(− \infty, \infty)$
ⓑ

8.R-565 §8.R source ↗

Add or Subtract Complex Numbers

In the following exercises, add or subtract.

$\sqrt{−50} + \sqrt{−18}$

Answer

8 \sqrt{2} i

8.R-567 §8.R source ↗

Add or Subtract Complex Numbers

In the following exercises, add or subtract.

$(6 + i) -$ $(−2 - 4 i)$

Answer

8 + 5 i

8.R-569 §8.R source ↗

Multiply Complex Numbers

In the following exercises, multiply.

$(−2 - 5 i) (−4 + 3 i)$

Answer

23 + 14 i

8.R-571 §8.R source ↗

Multiply Complex Numbers

In the following exercises, multiply.

$\sqrt{−4} \cdot \sqrt{−16}$

Answer

−8

8.R-573 §8.R source ↗

In the following exercises, multiply using the Product of Binomial Squares Pattern.

${(−2 - 3 i)}^{2}$

Answer

−5 + 12 i

8.R-575 §8.R source ↗

Divide Complex Numbers

In the following exercises, divide.

$\frac{2 + i}{3 - 4 i}$

Answer

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

8.R-577 §8.R source ↗

Simplify Powers of i

In the following exercises, simplify.

$i^{48}$

Answer

8.T-579 §8.T source ↗

In the following exercises, simplify using absolute values as necessary.

$\sqrt[3]{125 x^{9}}$

Answer

5 x^{3}

8.T-581 §8.T source ↗

In the following exercises, simplify using absolute values as necessary.

$\sqrt[3]{72 x^{8} y^{4}}$

Answer

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

8.T-583 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

ⓐ $256^{- \frac{1}{4}}$ ⓑ $− 49^{\frac{3}{2}}$

Answer

ⓐ

\frac{1}{4}

ⓑ

−343

8.T-585 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

$\frac{x^{- \frac{1}{4}} \cdot x^{\frac{5}{4}}}{x^{- \frac{3}{4}}}$

Answer

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

8.T-587 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

$\sqrt{48 x^{5}} - \sqrt{75 x^{5}}$

Answer

− x^{2} \sqrt{3 x}

8.T-589 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

$2 \sqrt{12 x^{5}} \cdot 3 \sqrt{6 x^{3}}$

Answer

36 x^{4} \sqrt{2}

8.T-591 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

$(4 - 3 \sqrt{3}) (5 + 2 \sqrt{3})$

Answer

2 - 7 \sqrt{3}

8.T-593 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

$\frac{\sqrt{245 x y^{−4}}}{\sqrt{45 x^{−4} y^{3}}}$

Answer

\frac{7 x^{2} \sqrt{x}}{3 | y^{3} | \sqrt{y}}

8.T-595 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

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

Answer

3 (2 - \sqrt{3})

8.T-597 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

$−4 i (−2 - 3 i)$

Answer

−12 + 8 i

8.T-599 §8.T source ↗

In the following exercises, simplify. Assume all variables are positive.

$i^{172}$

Answer

1

8.T-601 §8.T source ↗

In the following exercises, solve.

$\sqrt{x + 5} + 1 = x$

Answer

x = 4

8.T-603 §8.T source ↗

In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.

$g (x) = \sqrt{x + 2}$

Answer

ⓐ domain: $[−2, \infty)$
ⓑ

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Problems and answers are adapted from Intermediate Algebra 2e by Lynn Marecek and Andrea Honeycutt Mathis, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.