Content check — Elementary Algebra, Chapter 9: Roots and Radicals

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

Chapter 9: Roots and Radicals

652 problems · 13 figures

Be Prepared 9.1 §9.1 source ↗

Simplify: ⓐ

9^{2}

ⓑ

{(−9)}^{2}

ⓒ

− 9^{2}

.
If you missed this problem, review Example 1.50.

Example 1.50 in the book ↗

Answer

ⓐ

81

ⓑ

81

ⓒ

- 81

Be Prepared 9.2 §9.1 source ↗

Round 3.846 to the nearest hundredth.
If you missed this problem, review Example 1.94.

Example 1.94 in the book ↗

Answer

3.85

Be Prepared 9.3 §9.1 source ↗

For each number, identify whether it is a real number or not a real number:
ⓐ

− \sqrt{100}

ⓑ

\sqrt{−100}

.
If you missed this problem, review Example 1.113.

Example 1.113 in the book ↗

Answer

ⓐ

- \sqrt{100}

ⓑ

\sqrt{- 100}

Try It 9.1 §9.1 source ↗

Simplify: ⓐ

− \sqrt{49}

ⓑ

\sqrt{225}

Answer

ⓐ

−7

ⓑ

15

Try It 9.2 §9.1 source ↗

Simplify: ⓐ

\sqrt{64}

ⓑ

− \sqrt{121}

Answer

ⓐ

8

ⓑ

−11

Try It 9.3 §9.1 source ↗

Simplify: ⓐ

\sqrt{−196}

ⓑ

− \sqrt{81}

Answer

ⓐ not a real number ⓑ

−9

Try It 9.4 §9.1 source ↗

Simplify: ⓐ

− \sqrt{49}

ⓑ

\sqrt{−121}

Answer

ⓐ

−7

ⓑ not a real number

Try It 9.5 §9.1 source ↗

Simplify: ⓐ

\sqrt{9} + \sqrt{16}

ⓑ

\sqrt{9 + 16}

Answer

ⓐ 7 ⓑ 5

Try It 9.6 §9.1 source ↗

Simplify: ⓐ

\sqrt{64 + 225}

ⓑ

\sqrt{64} + \sqrt{225}

Answer

ⓐ 17 ⓑ

23

Try It 9.7 §9.1 source ↗

Estimate the square root

\sqrt{38}

between two consecutive whole numbers.

Answer

6 <

\sqrt{38} <

7

Try It 9.8 §9.1 source ↗

Estimate the square root

\sqrt{84}

between two consecutive whole numbers.

Answer

9 <

\sqrt{84} <

10

Try It 9.9 §9.1 source ↗

Round

\sqrt{11}

to two decimal places.

Answer

\approx 3.32

Try It 9.10 §9.1 source ↗

Round

\sqrt{13}

to two decimal places.

Answer

\approx 3.61

Try It 9.11 §9.1 source ↗

Simplify: ⓐ

\sqrt{y^{8}}

ⓑ

\sqrt{z^{12}}

Answer

ⓐ

y^{4}

ⓑ

z^{6}

Try It 9.12 §9.1 source ↗

Simplify: ⓐ

\sqrt{m^{4}}

ⓑ

\sqrt{b^{10}}

Answer

ⓐ

m^{2}

ⓑ

b^{5}

Try It 9.13 §9.1 source ↗

Simplify:

\sqrt{64 x^{2}}

Answer

8 x

Try It 9.14 §9.1 source ↗

Simplify:

\sqrt{169 y^{2}}

Answer

13 y

Try It 9.15 §9.1 source ↗

Simplify:

− \sqrt{121 y^{2}}

Answer

−11 y

Try It 9.16 §9.1 source ↗

Simplify:

− \sqrt{100 p^{2}}

Answer

−10 p

Try It 9.17 §9.1 source ↗

Simplify:

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

Answer

10 a b

Try It 9.18 §9.1 source ↗

Simplify:

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

Answer

15 m n

Try It 9.19 §9.1 source ↗

Simplify:

\sqrt{49 x^{30}}

Answer

7 x^{15}

Try It 9.20 §9.1 source ↗

Simplify:

\sqrt{81 w^{36}}

Answer

9 w^{18}

Try It 9.21 §9.1 source ↗

Simplify:

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

Answer

13 x^{5} y^{7}

Try It 9.22 §9.1 source ↗

Simplify:

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

Answer

12 p^{6} q^{10}

9.1-1 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{36}$

Answer

9.1-3 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{64}$

Answer

9.1-5 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{9}$

Answer

9.1-7 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{100}$

Answer

9.1-9 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$− \sqrt{4}$

Answer

−2

9.1-11 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$− \sqrt{1}$

Answer

−1

9.1-13 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{−121}$

Answer

not a real number

9.1-15 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{−9}$

Answer

not a real number

9.1-17 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{9 + 16}$

Answer

9.1-19 §9.1 source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{9} + \sqrt{16}$

Answer

9.1-21 §9.1 source ↗

Estimate Square Roots

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

$\sqrt{70}$

Answer

8 <

\sqrt{70} <

9

9.1-23 §9.1 source ↗

Estimate Square Roots

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

$\sqrt{200}$

Answer

14 <

\sqrt{200} <

15

9.1-25 §9.1 source ↗

Approximate Square Roots

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

$\sqrt{19}$

Answer

4.36

9.1-27 §9.1 source ↗

Approximate Square Roots

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

$\sqrt{53}$

Answer

7.28

9.1-29 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{y^{2}}$

Answer

y

9.1-31 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{a^{14}}$

Answer

a^{7}

9.1-33 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{49 x^{2}}$

Answer

7 x

9.1-35 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{121 m^{20}}$

Answer

11 m^{10}

9.1-37 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{81 x^{36}}$

Answer

9 x^{18}

9.1-39 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$− \sqrt{81 x^{18}}$

Answer

−9 x^{9}

9.1-41 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$− \sqrt{64 a^{2}}$

Answer

−8 a

9.1-43 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{144 x^{2} y^{2}}$

Answer

12 x y

9.1-45 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{169 w^{8} y^{10}}$

Answer

13 w^{4} y^{5}

9.1-47 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{9 c^{8} d^{12}}$

Answer

3 c^{4} d^{6}

9.1-49 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Decorating Denise wants to have a square accent of designer tiles in her new shower. She can afford to buy 625 square centimeters of the designer tiles. How long can a side of the accent be?

Answer

25 centimeters

9.1-51 §9.1 source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

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

Answer

Answers will vary.

Be Prepared 9.4 §9.2 source ↗

Simplify:

\frac{80}{176}

.
If you missed this problem, review Example 1.65.

Example 1.65 in the book ↗

Answer

\frac{5}{11}

Be Prepared 9.5 §9.2 source ↗

Simplify:

\frac{n^{9}}{n^{3}}

.
If you missed this problem, review Example 6.59.

Example 6.59 in the book ↗

Answer

n^{6}

Be Prepared 9.6 §9.2 source ↗

Simplify:

\frac{q^{4}}{q^{12}}

.
If you missed this problem, review Example 6.60.

Example 6.60 in the book ↗

Answer

\frac{1}{q^{8}}

Try It 9.23 §9.2 source ↗

Simplify:

\sqrt{48}

Answer

4 \sqrt{3}

Try It 9.24 §9.2 source ↗

Simplify:

\sqrt{45}

Answer

3 \sqrt{5}

Try It 9.25 §9.2 source ↗

Simplify:

\sqrt{288}

Answer

12 \sqrt{2}

Try It 9.26 §9.2 source ↗

Simplify:

\sqrt{432}

Answer

12 \sqrt{3}

Try It 9.27 §9.2 source ↗

Simplify:

\sqrt{b^{5}}

Answer

b^{2} \sqrt{b}

Try It 9.28 §9.2 source ↗

Simplify:

\sqrt{p^{9}}

Answer

p^{4} \sqrt{p}

Try It 9.29 §9.2 source ↗

Simplify:

\sqrt{16 x^{7}}

Answer

4 x^{3} \sqrt{x}

Try It 9.30 §9.2 source ↗

Simplify:

\sqrt{49 v^{9}}

Answer

7 v^{4} \sqrt{v}

Try It 9.31 §9.2 source ↗

Simplify:

\sqrt{32 y^{5}}

Answer

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

Try It 9.32 §9.2 source ↗

Simplify:

\sqrt{75 a^{9}}

Answer

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

Try It 9.33 §9.2 source ↗

Simplify:

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

Answer

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

Try It 9.34 §9.2 source ↗

Simplify:

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

Answer

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

Try It 9.35 §9.2 source ↗

Simplify:

5 + \sqrt{75}

Answer

5 + 5 \sqrt{3}

Try It 9.36 §9.2 source ↗

Simplify:

2 + \sqrt{98}

Answer

2 + 7 \sqrt{2}

Try It 9.37 §9.2 source ↗

Simplify:

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

Answer

2 - \sqrt{3}

Try It 9.38 §9.2 source ↗

Simplify:

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

Answer

2 - \sqrt{5}

Try It 9.39 §9.2 source ↗

Simplify:

\sqrt{\frac{25}{16}}

Answer

\frac{5}{4}

Try It 9.40 §9.2 source ↗

Simplify:

\sqrt{\frac{49}{81}}

Answer

\frac{7}{9}

Try It 9.41 §9.2 source ↗

Simplify:

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

Answer

\frac{5}{4}

Try It 9.42 §9.2 source ↗

Simplify:

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

Answer

\frac{7}{9}

Try It 9.43 §9.2 source ↗

Simplify:

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

Answer

a

Try It 9.44 §9.2 source ↗

Simplify:

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

Answer

x^{2}

Try It 9.45 §9.2 source ↗

Simplify:

\sqrt{\frac{75 x^{5}}{3 x}}

Answer

5 x^{2}

Try It 9.46 §9.2 source ↗

Simplify:

\sqrt{\frac{72 z^{12}}{2 z^{10}}}

Answer

6 z

Try It 9.47 §9.2 source ↗

Simplify:

\sqrt{\frac{19}{49}}

Answer

\frac{\sqrt{19}}{7}

Try It 9.48 §9.2 source ↗

Simplify:

\sqrt{\frac{28}{81}}

Answer

\frac{2 \sqrt{7}}{9}

Try It 9.49 §9.2 source ↗

Simplify:

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

Answer

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

Try It 9.50 §9.2 source ↗

Simplify:

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

Answer

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

Try It 9.51 §9.2 source ↗

Simplify:

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

Answer

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

Try It 9.52 §9.2 source ↗

Simplify:

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

Answer

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

Try It 9.53 §9.2 source ↗

Simplify:

\sqrt{\frac{64 x^{7}}{9 x^{3}}}

Answer

\frac{8 x^{2}}{3}

Try It 9.54 §9.2 source ↗

Simplify:

\sqrt{\frac{16 a^{9}}{100 a^{5}}}

Answer

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

Try It 9.55 §9.2 source ↗

Simplify:

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

Answer

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

Try It 9.56 §9.2 source ↗

Simplify:

\sqrt{\frac{48 m^{7} n^{2}}{125 m^{5} n^{9}}}

Answer

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

9.2-53 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{27}$

Answer

3 \sqrt{3}

9.2-55 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{125}$

Answer

5 \sqrt{5}

9.2-57 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{200}$

Answer

10 \sqrt{2}

9.2-59 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{450}$

Answer

15 \sqrt{2}

9.2-61 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{800}$

Answer

20 \sqrt{2}

9.2-63 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{675}$

Answer

15 \sqrt{3}

9.2-65 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{x^{7}}$

Answer

x^{3} \sqrt{x}

9.2-67 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{p^{3}}$

Answer

p \sqrt{p}

9.2-69 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{m^{13}}$

Answer

m^{6} \sqrt{m}

9.2-71 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{r^{25}}$

Answer

r^{12} \sqrt{r}

9.2-73 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{49 n^{17}}$

Answer

7 n^{8} \sqrt{n}

9.2-75 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{81 r^{15}}$

Answer

9 r^{7} \sqrt{r}

9.2-77 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{98 m^{5}}$

Answer

7 m^{2} \sqrt{2 m}

9.2-79 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{125 r^{13}}$

Answer

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

9.2-81 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{200 p^{13}}$

Answer

10 p^{6} \sqrt{2 p}

9.2-83 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{242 m^{23}}$

Answer

11 m^{11} \sqrt{2 m}

9.2-85 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{147 m^{7} n^{11}}$

Answer

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

9.2-87 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{75 r^{13} s^{9}}$

Answer

5 r^{6} s^{4} \sqrt{3 r s}

9.2-89 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{300 p^{9} q^{11}}$

Answer

10 p^{4} q^{5} \sqrt{3 p q}

9.2-91 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{242 m^{13} n^{21}}$

Answer

11 m^{6} n^{10} \sqrt{2 m n}

9.2-93 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$5 + \sqrt{12}$

Answer

5 + 2 \sqrt{3}

9.2-95 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$1 + \sqrt{45}$

Answer

1 + 3 \sqrt{5}

9.2-97 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\frac{10 - \sqrt{24}}{2}$

Answer

5 - \sqrt{6}

9.2-99 §9.2 source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\frac{3 + \sqrt{90}}{3}$

Answer

1 + \sqrt{10}

9.2-101 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{49}{64}}$

Answer

\frac{7}{8}

9.2-103 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{121}{16}}$

Answer

\frac{11}{4}

9.2-105 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{72}{98}}$

Answer

\frac{6}{7}

9.2-107 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{45}{125}}$

Answer

\frac{3}{5}

9.2-109 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{x^{10}}{x^{6}}}$

Answer

x^{2}

9.2-111 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{y^{4}}{y^{8}}}$

Answer

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

9.2-113 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{200 x^{7}}{2 x^{3}}}$

Answer

10 x^{2}

9.2-115 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{96 p^{9}}{6 p}}$

Answer

4 p^{4}

9.2-117 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{36}{35}}$

Answer

\frac{6}{\sqrt{35}}

9.2-119 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{20}{81}}$

Answer

\frac{2 \sqrt{5}}{9}

9.2-121 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

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

Answer

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

9.2-123 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

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

Answer

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

9.2-125 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

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

Answer

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

9.2-127 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

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

Answer

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

9.2-129 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{75 r^{9}}{s^{8}}}$

Answer

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

9.2-131 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{28 p^{7}}{q^{2}}}$

Answer

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

9.2-133 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{100 x^{5}}{36 x^{3}}}$

Answer

\frac{5 x}{3}

9.2-135 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{121 p^{5}}{81 p^{2}}}$

Answer

\frac{11 p \sqrt{p}}{9}

9.2-137 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{32 x^{5} y^{3}}{18 x^{3} y}}$

Answer

\frac{4 x y}{3}

9.2-139 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{27 p^{2} q}{108 p^{5} q^{3}}}$

Answer

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

9.2-141 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

ⓐ Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify $\sqrt{288}$ to determine the length and the width of his garden. Round to the nearest tenth of a foot.

ⓑ Suppose Elliott decides to reduce the size of his square garden so that he can create a 5-foot-wide walking path on the north and east sides of the garden. Simplify $\sqrt{288} - 5$ to determine the length and width of the new garden. Round to the nearest tenth of a foot.

Answer

ⓐ

17.0 feet

ⓑ

12.0 feet

9.2-143 §9.2 source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

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

Answer

Answers will vary.

Be Prepared 9.7 §9.3 source ↗

Add: ⓐ

3 x + 9 x

ⓑ

5 m + 5 n

.
If you missed this problem, review Example 1.24.

Example 1.24 in the book ↗

Answer

ⓐ

12 x

ⓑ

5 m + 5 n

Be Prepared 9.8 §9.3 source ↗

Simplify:

\sqrt{50 x^{3}}

.
If you missed this problem, review Example 9.16.

Example 9.16 in the book ↗

Answer

5 x \sqrt{2 x}

Try It 9.57 §9.3 source ↗

Simplify:

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

Answer

− \sqrt{2}

Try It 9.58 §9.3 source ↗

Simplify:

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

Answer

−4 \sqrt{3}

Try It 9.59 §9.3 source ↗

Simplify:

2 \sqrt{x} + 7 \sqrt{x}

Answer

9 \sqrt{x}

Try It 9.60 §9.3 source ↗

Simplify:

5 \sqrt{u} + 3 \sqrt{u}

Answer

8 \sqrt{u}

Try It 9.61 §9.3 source ↗

Simplify:

7 \sqrt{p} - 6 \sqrt{q}

Answer

7 \sqrt{p} - 6 \sqrt{q}

Try It 9.62 §9.3 source ↗

Simplify:

6 \sqrt{a} - 3 \sqrt{b}

Answer

6 \sqrt{a} - 3 \sqrt{b}

Try It 9.63 §9.3 source ↗

Simplify:

4 \sqrt{11} + 2 \sqrt{11} + 3 \sqrt{11}

Answer

9 \sqrt{11}

Try It 9.64 §9.3 source ↗

Simplify:

6 \sqrt{10} + 2 \sqrt{10} + 3 \sqrt{10}

Answer

11 \sqrt{10}

Try It 9.65 §9.3 source ↗

Simplify:

5 \sqrt{5} - 4 \sqrt{5} + 2 \sqrt{6}

Answer

\sqrt{5} + 2 \sqrt{6}

Try It 9.66 §9.3 source ↗

Simplify:

3 \sqrt{7} - 8 \sqrt{7} + 2 \sqrt{5}

Answer

−5 \sqrt{7} + 2 \sqrt{5}

Try It 9.67 §9.3 source ↗

Simplify:

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

Answer

−2 \sqrt{7 x}

Try It 9.68 §9.3 source ↗

Simplify:

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

Answer

− \sqrt{3 y}

Try It 9.69 §9.3 source ↗

Simplify:

\sqrt{5 x y} +

4 \sqrt{5 x y} -

7 \sqrt{5 x y}

Answer

−2 \sqrt{5 x y}

Try It 9.70 §9.3 source ↗

Simplify:

3 \sqrt{7 m n} +

\sqrt{7 m n} -

4 \sqrt{7 m n}

Answer

0

Try It 9.71 §9.3 source ↗

Simplify:

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

Answer

9 \sqrt{2}

Try It 9.72 §9.3 source ↗

Simplify:

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

Answer

7 \sqrt{3}

Try It 9.73 §9.3 source ↗

Simplify:

\sqrt{32} - \sqrt{18}

Answer

\sqrt{2}

Try It 9.74 §9.3 source ↗

Simplify:

\sqrt{20} - \sqrt{45}

Answer

− \sqrt{5}

Try It 9.75 §9.3 source ↗

Simplify:

4 \sqrt{27} - 3 \sqrt{12}

Answer

6 \sqrt{3}

Try It 9.76 §9.3 source ↗

Simplify:

3 \sqrt{20} - 7 \sqrt{45}

Answer

−15 \sqrt{5}

Try It 9.77 §9.3 source ↗

Simplify:

\frac{2}{3} \sqrt{108} - \frac{5}{7} \sqrt{147}

Answer

− \sqrt{3}

Try It 9.78 §9.3 source ↗

Simplify:

\frac{3}{5} \sqrt{200} - \frac{3}{4} \sqrt{128}

Answer

0

Try It 9.79 §9.3 source ↗

Simplify:

\frac{2}{5} \sqrt{32} - \frac{1}{3} \sqrt{8}

Answer

\frac{14}{15} \sqrt{2}

Try It 9.80 §9.3 source ↗

Simplify:

\frac{1}{3} \sqrt{80} - \frac{1}{4} \sqrt{125}

Answer

\frac{1}{12} \sqrt{5}

Try It 9.81 §9.3 source ↗

Simplify:

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

Answer

− m^{3} \sqrt{2 m}

Try It 9.82 §9.3 source ↗

Simplify:

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

Answer

− p \sqrt{3 p}

Try It 9.83 §9.3 source ↗

Simplify:

5 \sqrt{32 x^{2}} - 3 \sqrt{48 x^{2}}

Answer

20 x \sqrt{2} - 12 x \sqrt{3}

Try It 9.84 §9.3 source ↗

Simplify:

7 \sqrt{48 y^{2}} - 4 \sqrt{72 y^{2}}

Answer

28 y \sqrt{3} - 24 y \sqrt{2}

Try It 9.85 §9.3 source ↗

Simplify:

3 \sqrt{12 x^{2}} -

2 x \sqrt{48} +

4 \sqrt{27 x^{2}}

Answer

10 x \sqrt{3}

Try It 9.86 §9.3 source ↗

Simplify:

3 \sqrt{18 x^{2}} -

6 x \sqrt{32} +

2 \sqrt{50 x^{2}}

Answer

−5 x \sqrt{2}

9.3-145 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$8 \sqrt{2} - 5 \sqrt{2}$

Answer

3 \sqrt{2}

9.3-147 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$3 \sqrt{5} + 6 \sqrt{5}$

Answer

9 \sqrt{5}

9.3-149 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$9 \sqrt{7} - 10 \sqrt{7}$

Answer

− \sqrt{7}

9.3-151 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$7 \sqrt{y} + 2 \sqrt{y}$

Answer

9 \sqrt{y}

9.3-153 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$\sqrt{a} - 4 \sqrt{a}$

Answer

−3 \sqrt{a}

9.3-155 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$5 \sqrt{c} + 2 \sqrt{c}$

Answer

7 \sqrt{c}

9.3-157 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$8 \sqrt{a} - 2 \sqrt{b}$

Answer

8 \sqrt{a} - 2 \sqrt{b}

9.3-159 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$5 \sqrt{m} + \sqrt{n}$

Answer

5 \sqrt{m} + \sqrt{n}

9.3-161 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$8 \sqrt{7} + 2 \sqrt{7} + 3 \sqrt{7}$

Answer

13 \sqrt{7}

9.3-163 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$3 \sqrt{11} + 2 \sqrt{11} - 8 \sqrt{11}$

Answer

−3 \sqrt{11}

9.3-165 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$3 \sqrt{3} - 8 \sqrt{3} + 7 \sqrt{5}$

Answer

−5 \sqrt{3} + 7 \sqrt{5}

9.3-167 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

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

Answer

8 \sqrt{2} - 3 \sqrt{5}

9.3-169 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

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

Answer

4 \sqrt{2 a}

9.3-171 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$8 \sqrt{3 c} + 2 \sqrt{3 c} - 9 \sqrt{3 c}$

Answer

\sqrt{3 c}

9.3-173 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

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

Answer

4 \sqrt{3 a b}

9.3-175 §9.3 source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$2 \sqrt{p q} - 5 \sqrt{p q} + 4 \sqrt{p q}$

Answer

\sqrt{p q}

9.3-177 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{50} + 4 \sqrt{2}$

Answer

9 \sqrt{2}

9.3-179 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{80} - 3 \sqrt{5}$

Answer

\sqrt{5}

9.3-181 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{27} - \sqrt{75}$

Answer

−2 \sqrt{3}

9.3-183 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{48} + \sqrt{27}$

Answer

7 \sqrt{3}

9.3-185 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$2 \sqrt{50} - 3 \sqrt{72}$

Answer

−8 \sqrt{2}

9.3-187 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$2 \sqrt{12} + 3 \sqrt{48}$

Answer

16 \sqrt{3}

9.3-189 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\frac{2}{3} \sqrt{72} + \frac{1}{5} \sqrt{50}$

Answer

5 \sqrt{2}

9.3-191 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\frac{1}{2} \sqrt{20} - \frac{2}{3} \sqrt{45}$

Answer

− \sqrt{5}

9.3-193 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\frac{1}{6} \sqrt{27} - \frac{3}{8} \sqrt{48}$

Answer

− \sqrt{3}

9.3-195 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\frac{1}{4} \sqrt{98} - \frac{1}{3} \sqrt{128}$

Answer

- \frac{11}{12} \sqrt{2}

9.3-197 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{72 a^{5}} - \sqrt{50 a^{5}}$

Answer

a^{2} \sqrt{2 a}

9.3-199 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{80 c^{7}} - \sqrt{20 c^{7}}$

Answer

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

9.3-201 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$9 \sqrt{80 p^{4}} - 6 \sqrt{98 p^{4}}$

Answer

36 p^{2} \sqrt{5} - 42 p^{2} \sqrt{2}

9.3-203 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$2 \sqrt{50 r^{8}} + 4 \sqrt{54 r^{8}}$

Answer

10 r^{4} \sqrt{2} + 12 r^{4} \sqrt{6}

9.3-205 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$3 \sqrt{20 x^{2}} -$ $4 \sqrt{45 x^{2}} +$ $5 x \sqrt{80}$

Answer

14 x \sqrt{5}

9.3-207 §9.3 source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

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

Answer

4 y \sqrt{2}

9.3-209 §9.3 source ↗

Mixed Practice

$2 \sqrt{8} + 6 \sqrt{8} - 5 \sqrt{8}$

Answer

6 \sqrt{2}

9.3-211 §9.3 source ↗

Mixed Practice

$\sqrt{175 k^{4}} - \sqrt{63 k^{4}}$

Answer

2 k^{2} \sqrt{7}

9.3-213 §9.3 source ↗

Mixed Practice

$2 \sqrt{363} - 2 \sqrt{300}$

Answer

2 \sqrt{3}

9.3-215 §9.3 source ↗

Mixed Practice

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

Answer

\sqrt{2}

9.3-217 §9.3 source ↗

Mixed Practice

$8 \sqrt{13} - 4 \sqrt{13} - 3 \sqrt{13}$

Answer

\sqrt{13}

9.3-219 §9.3 source ↗

Mixed Practice

$\sqrt{80 a^{5}} - \sqrt{45 a^{5}}$

Answer

a^{2} \sqrt{5 a}

9.3-221 §9.3 source ↗

Mixed Practice

$21 \sqrt{19} - 2 \sqrt{19}$

Answer

19 \sqrt{19}

9.3-223 §9.3 source ↗

Mixed Practice

$\frac{5}{6} \sqrt{27} + \frac{5}{8} \sqrt{48}$

Answer

5 \sqrt{3}

9.3-225 §9.3 source ↗

Mixed Practice

$\sqrt{75} - \sqrt{108}$

Answer

− \sqrt{3}

9.3-227 §9.3 source ↗

Mixed Practice

$4 \sqrt{24 x^{2}} -$ $\sqrt{54 x^{2}} +$ $3 x \sqrt{6}$

Answer

8 x \sqrt{6}

9.3-229 §9.3 source ↗

Mixed Practice

A decorator decides to use square tiles as an accent strip in the design of a new shower, but she wants to rotate the tiles to look like diamonds. She will use 9 large tiles that measure 8 inches on a side and 8 small tiles that measure 2 inches on a side. Determine the width of the accent strip by simplifying the expression $9 (8 \sqrt{2}) + 8 (2 \sqrt{2})$ . (Round to the nearest tenth of an inch.)

Answer

124.5 inches

9.3-231 §9.3 source ↗

Mixed Practice

Explain the difference between like radicals and unlike radicals. Make sure your answer makes sense for radicals containing both numbers and variables.

Answer

Answers will vary.

Be Prepared 9.9 §9.4 source ↗

Simplify:

(3 u) (8 v)

.
If you missed this problem, review Example 6.26.

Example 6.26 in the book ↗

Answer

24 u v

Be Prepared 9.10 §9.4 source ↗

Simplify:

6 (12 - 7 n)

.
If you missed this problem, review Example 6.28.

Example 6.28 in the book ↗

Answer

72 - 42 n

Be Prepared 9.11 §9.4 source ↗

Simplify:

(2 + a) (4 - a)

.
If you missed this problem, review Example 6.39.

Example 6.39 in the book ↗

Answer

8 + 2 a + a^{2}

Try It 9.87 §9.4 source ↗

Simplify: ⓐ

\sqrt{3} \cdot \sqrt{6}

ⓑ

(2 \sqrt{6}) (3 \sqrt{12})

Answer

ⓐ

3 \sqrt{2}

ⓑ

36 \sqrt{2}

Try It 9.88 §9.4 source ↗

Simplify: ⓐ

\sqrt{5} \cdot \sqrt{10}

ⓑ

(6 \sqrt{3}) (5 \sqrt{6})

Answer

ⓐ

5 \sqrt{2}

ⓑ

90 \sqrt{2}

Try It 9.89 §9.4 source ↗

Simplify:

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

Answer

12 \sqrt{15}

Try It 9.90 §9.4 source ↗

Simplify:

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

Answer

27 \sqrt{2}

Try It 9.91 §9.4 source ↗

Simplify: ⓐ

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

ⓑ

(\sqrt{2 y^{3}}) (\sqrt{50 y^{2}})

Answer

ⓐ

3 x^{2} \sqrt{2}

ⓑ

10 y^{2} \sqrt{y}

Try It 9.92 §9.4 source ↗

Simplify: ⓐ

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

ⓑ

(\sqrt{12 y^{2}}) (\sqrt{3 y^{5}})

Answer

ⓐ

2 x^{3} \sqrt{3}

ⓑ

6 y^{3} \sqrt{y}

Try It 9.93 §9.4 source ↗

Simplify:

(6 \sqrt{2 x^{2}}) (8 \sqrt{45 x^{4}})

Answer

144 x^{3} \sqrt{10}

Try It 9.94 §9.4 source ↗

Simplify:

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

Answer

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

Try It 9.95 §9.4 source ↗

Simplify: ⓐ

{(\sqrt{12})}^{2}

ⓑ

{(− \sqrt{15})}^{2}

Answer

ⓐ 12 ⓑ

15

Try It 9.96 §9.4 source ↗

Simplify: ⓐ

{(\sqrt{16})}^{2}

ⓑ

{(− \sqrt{20})}^{2}

Answer

ⓐ 16 ⓑ 20

Try It 9.97 §9.4 source ↗

Simplify: ⓐ

(6 \sqrt{11}) (5 \sqrt{11})

ⓑ

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

Answer

ⓐ 330 ⓑ 200

Try It 9.98 §9.4 source ↗

Simplify: ⓐ

(3 \sqrt{7}) (10 \sqrt{7})

ⓑ

{(−4 \sqrt{6})}^{2}

Answer

ⓐ 210 ⓑ 96

Try It 9.99 §9.4 source ↗

Simplify: ⓐ

2 (3 - \sqrt{5})

ⓑ

\sqrt{3} (2 - \sqrt{18})

Answer

ⓐ

6 - 2 \sqrt{5}

ⓑ

2 \sqrt{3} - 3 \sqrt{6}

Try It 9.100 §9.4 source ↗

Simplify: ⓐ

6 (2 + \sqrt{6})

ⓑ

\sqrt{7} (1 + \sqrt{14})

Answer

ⓐ

12 + 6 \sqrt{6}

ⓑ

\sqrt{7} + 7 \sqrt{2}

Try It 9.101 §9.4 source ↗

Simplify: ⓐ

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

ⓑ

\sqrt{12} (\sqrt{3} + \sqrt{24})

Answer

ⓐ

18 + \sqrt{6}

ⓑ

6 + 12 \sqrt{2}

Try It 9.102 §9.4 source ↗

Simplify: ⓐ

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

ⓑ

\sqrt{14} (\sqrt{2} + \sqrt{42})

Answer

ⓐ

−40 + 4 \sqrt{2}

ⓑ

2 \sqrt{7} + 14 \sqrt{3}

Try It 9.103 §9.4 source ↗

Simplify:

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

Answer

−3 + 2 \sqrt{6}

Try It 9.104 §9.4 source ↗

Simplify:

(4 - \sqrt{10}) (2 + \sqrt{10})

Answer

−2 + 2 \sqrt{10}

Try It 9.105 §9.4 source ↗

Simplify:

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

Answer

−66 + 15 \sqrt{7}

Try It 9.106 §9.4 source ↗

Simplify:

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

Answer

41 - 14 \sqrt{11}

Try It 9.107 §9.4 source ↗

Simplify:

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

Answer

1 + 9 \sqrt{21}

Try It 9.108 §9.4 source ↗

Simplify:

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

Answer

−12 - 20 \sqrt{3}

Try It 9.109 §9.4 source ↗

Simplify:

(6 - 5 \sqrt{m}) (2 + 3 \sqrt{m})

Answer

12 + 8 \sqrt{m} - 15 m

Try It 9.110 §9.4 source ↗

Simplify:

(10 + 3 \sqrt{n}) (1 - 5 \sqrt{n})

Answer

10 - 47 \sqrt{n} - 15 n

Try It 9.111 §9.4 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

102 + 20 \sqrt{2}

ⓑ

55 + 6 \sqrt{6}

Try It 9.112 §9.4 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

41 - 12 \sqrt{5}

ⓑ

121 - 36 \sqrt{10}

Try It 9.113 §9.4 source ↗

Simplify:

{(2 + 5 \sqrt{m})}^{2}

Answer

4 + 20 \sqrt{m} + 25 m

Try It 9.114 §9.4 source ↗

Simplify:

{(3 - 4 \sqrt{n})}^{2}

Answer

9 - 24 \sqrt{n} + 16 n

Try It 9.115 §9.4 source ↗

Simplify:

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

Answer

1

Try It 9.116 §9.4 source ↗

Simplify:

(1 + \sqrt{5}) (1 - \sqrt{5})

Answer

−4

Try It 9.117 §9.4 source ↗

Simplify:

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

Answer

−11

Try It 9.118 §9.4 source ↗

Simplify:

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

Answer

−159

9.4-233 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

ⓐ $\sqrt{2} \cdot \sqrt{8}$ ⓑ $(3 \sqrt{3}) (2 \sqrt{18})$

Answer

ⓐ

4

ⓑ

18 \sqrt{6}

9.4-235 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

ⓐ $\sqrt{7} \cdot \sqrt{14}$ ⓑ $(4 \sqrt{8}) (5 \sqrt{8})$

Answer

ⓐ

7 \sqrt{2}

ⓑ 160

9.4-237 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

$(5 \sqrt{2}) (3 \sqrt{6})$

Answer

30 \sqrt{3}

9.4-239 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

$(−2 \sqrt{3}) (3 \sqrt{18})$

Answer

−18 \sqrt{6}

9.4-241 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

$(5 \sqrt{6}) (− \sqrt{12})$

Answer

−30 \sqrt{2}

9.4-243 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

$(−2 \sqrt{7}) (−2 \sqrt{14})$

Answer

28 \sqrt{2}

9.4-245 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

ⓐ $(\sqrt{15 y}) (\sqrt{5 y^{3}})$ ⓑ $(\sqrt{2 n^{2}}) (\sqrt{18 n^{3}})$

Answer

ⓐ

5 y^{2} \sqrt{3}

ⓑ

6 n^{2} \sqrt{n}

9.4-247 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

ⓐ $(\sqrt{16 y^{2}}) (\sqrt{8 y^{4}})$ ⓑ $(\sqrt{11 s^{6}}) (\sqrt{11 s})$

Answer

ⓐ

8 y^{3} \sqrt{2}

ⓑ

11 s^{3} \sqrt{s}

9.4-249 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

$(2 \sqrt{5 b^{3}}) (4 \sqrt{15 b})$

Answer

40 b^{2} \sqrt{3}

9.4-251 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

$(5 \sqrt{2 d^{7}}) (3 \sqrt{50 d^{3}})$

Answer

150 d^{5}

9.4-253 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

$3 \sqrt{4 y^{4}} 3 \sqrt{9 y^{5}}$

Answer

54 y^{4} \sqrt{y}

9.4-255 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

$(4 \sqrt{2 k^{5}}) (−3 \sqrt{32 k^{6}})$

Answer

−96 k^{5} \sqrt{k}

9.4-257 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

ⓐ ${(\sqrt{11})}^{2}$ ⓑ ${(− \sqrt{21})}^{2}$

Answer

ⓐ 11 ⓑ 21

9.4-259 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

ⓐ ${(\sqrt{23})}^{2}$
ⓑ ${(− \sqrt{3})}^{2}$

Answer

ⓐ 23 ⓑ 3

9.4-261 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

ⓐ $(2 \sqrt{13}) (−9 \sqrt{13})$ ⓑ ${(6 \sqrt{5})}^{2}$

Answer

ⓐ −234 ⓑ 180

9.4-263 §9.4 source ↗

Multiply Square Roots

In the following exercises, simplify.

ⓐ $(−7 \sqrt{5}) (−3 \sqrt{10})$ ⓑ ${(−2 \sqrt{14})}^{2}$

Answer

ⓐ

105 \sqrt{2}

ⓑ 56

9.4-265 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

ⓐ $4 (6 - \sqrt{11})$ ⓑ $\sqrt{2} (5 - \sqrt{12})$

Answer

ⓐ

24 - 4 \sqrt{11}

ⓑ

5 \sqrt{2} - 2 \sqrt{6}

9.4-267 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

ⓐ $7 (−2 - \sqrt{11})$ ⓑ $\sqrt{7} (6 - \sqrt{14})$

Answer

ⓐ

−14 - 7 \sqrt{11}

ⓑ

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

9.4-269 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

ⓐ $\sqrt{11} (8 + 4 \sqrt{11})$ ⓑ $\sqrt{3} (\sqrt{12} + \sqrt{27})$

Answer

ⓐ

44 + 8 \sqrt{11}

ⓑ

15

9.4-271 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

ⓐ $\sqrt{2} (−5 + 9 \sqrt{2})$ ⓑ $\sqrt{7} (\sqrt{3} - \sqrt{21})$

Answer

ⓐ

18 - 5 \sqrt{2}

ⓑ

\sqrt{21} - 7 \sqrt{3}

9.4-273 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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

Answer

60 + 2 \sqrt{3}

9.4-275 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(9 - \sqrt{2}) (6 + \sqrt{2})$

Answer

52 + 3 \sqrt{2}

9.4-277 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(5 - \sqrt{7}) (4 - \sqrt{7})$

Answer

27 - 9 \sqrt{7}

9.4-279 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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

Answer

−62 + 55 \sqrt{5}

9.4-281 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(\sqrt{11} + \sqrt{5}) (\sqrt{11} + 6 \sqrt{5})$

Answer

41 + 7 \sqrt{55}

9.4-283 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(4 \sqrt{6} + 7 \sqrt{13}) (8 \sqrt{6} - 3 \sqrt{13})$

Answer

−81 + 44 \sqrt{78}

9.4-285 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(9 - \sqrt{w}) (2 + \sqrt{w})$

Answer

18 + 7 \sqrt{w} - w

9.4-287 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(6 + 5 \sqrt{n}) (11 + 3 \sqrt{n})$

Answer

66 + 73 \sqrt{n} + 15 n

9.4-289 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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

Answer

ⓐ

27 + 8 \sqrt{11}

ⓑ

29 - 12 \sqrt{5}

9.4-291 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

ⓐ ${(5 - \sqrt{10})}^{2}$ ⓑ ${(8 + 3 \sqrt{2})}^{2}$

Answer

ⓐ

35 - 10 \sqrt{10}

ⓑ

82 + 48 \sqrt{2}

9.4-293 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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

Answer

9.4-295 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(7 + \sqrt{10}) (7 - \sqrt{10})$

Answer

9.4-297 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(1 + 8 \sqrt{2}) (1 - 8 \sqrt{2})$

Answer

−127

9.4-299 §9.4 source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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

Answer

9.4-301 §9.4 source ↗

Mixed Practice

In the following exercises, simplify.

$(4 \sqrt{6}) (− \sqrt{18})$

Answer

−24 \sqrt{3}

9.4-303 §9.4 source ↗

Mixed Practice

In the following exercises, simplify.

$(−5 \sqrt{7}) (6 \sqrt{21})$

Answer

−210 \sqrt{3}

9.4-305 §9.4 source ↗

Mixed Practice

In the following exercises, simplify.

$(\sqrt{35 y^{3}}) (\sqrt{7 y^{3}})$

Answer

7 y^{3} \sqrt{5}

9.4-307 §9.4 source ↗

Mixed Practice

In the following exercises, simplify.

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

Answer

29

9.4-309 §9.4 source ↗

Mixed Practice

In the following exercises, simplify.

$(−4 + \sqrt{17}) (−3 + \sqrt{17})$

Answer

29 - 7 \sqrt{17}

9.4-311 §9.4 source ↗

Mixed Practice

In the following exercises, simplify.

An artist wants to make a small monument in the shape of a square base topped by a right triangle, as shown below. The square base will be adjacent to one leg of the triangle. The other leg of the triangle will measure 2 feet and the hypotenuse will be 5 feet.

ⓐ Use the Pythagorean Theorem to find the length of a side of the square base. Round your answer to the nearest tenth of a foot.
This figure shows a marble sculpture in the form of a square with a right triangle resting on top of it. The sides of the square are x inches long, the legs of the triangle are x and two inches long, and the hypotenuse of the triangle is five inches long.

ⓑ Find the exact area of the face of the square base.

Answer

ⓐ

4.6 feet

ⓑ

21 sq. feet

9.4-313 §9.4 source ↗

Mixed Practice

In the following exercises, simplify.

A garden will be made so as to contain two square sections, one section with side length $\sqrt{5} + \sqrt{6}$ yards and one section with side length $\sqrt{2} + \sqrt{3}$ yards. Simplify ${\sqrt{5} + \sqrt{6}}^{2} + {\sqrt{2} + \sqrt{3}}^{2}$ to determine the total area of the garden. Round your answer to the nearest tenth.

Answer

31.9 square yards

9.4-315 §9.4 source ↗

Mixed Practice

In the following exercises, simplify.

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

ⓑ Explain why $- {\sqrt{n}}^{2}$ is always negative, for $n \geq 0$ .

Answer

ⓐ when squaring a negative, it becomes a positive ⓑ since the negative is not included in the parenthesis, it is not squared, and remains negative

Be Prepared 9.12 §9.5 source ↗

Find a fraction equivalent to

\frac{5}{8}

with denominator 48.
If you missed this problem, review Example 1.64.

Example 1.64 in the book ↗

Answer

\frac{30}{48}

Be Prepared 9.13 §9.5 source ↗

Simplify:

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

.
If you missed this problem, review Example 9.48.

Example 9.48 in the book ↗

Answer

5

Be Prepared 9.14 §9.5 source ↗

Multiply:

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

.
If you missed this problem, review Example 6.54.

Example 6.54 in the book ↗

Answer

49 - 9 x^{2}

Try It 9.119 §9.5 source ↗

Simplify:

\frac{\sqrt{32}}{8}

Answer

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

Try It 9.120 §9.5 source ↗

Simplify:

\frac{\sqrt{75}}{15}

Answer

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

Try It 9.121 §9.5 source ↗

Simplify:

\frac{8 - \sqrt{40}}{10}

Answer

\frac{4 - \sqrt{10}}{5}

Try It 9.122 §9.5 source ↗

Simplify:

\frac{10 - \sqrt{75}}{20}

Answer

\frac{2 - \sqrt{3}}{4}

Try It 9.123 §9.5 source ↗

Simplify:

\frac{\sqrt{48}}{\sqrt{108}}

Answer

\frac{2}{3}

Try It 9.124 §9.5 source ↗

Simplify:

\frac{\sqrt{96}}{\sqrt{54}}

Answer

\frac{4}{3}

Try It 9.125 §9.5 source ↗

Simplify:

\frac{\sqrt{12 r^{3}}}{\sqrt{6 r}}

Answer

r \sqrt{2}

Try It 9.126 §9.5 source ↗

Simplify:

\frac{\sqrt{14 p^{9}}}{\sqrt{2 p^{5}}}

Answer

p^{2} \sqrt{7}

Try It 9.127 §9.5 source ↗

Simplify:

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

Answer

\frac{5 s}{8}

Try It 9.128 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.129 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.130 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.131 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.132 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.133 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.134 §9.5 source ↗

Simplify:

- \frac{9}{4 \sqrt{3}}

Answer

- \frac{3 \sqrt{3}}{4}

Try It 9.135 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.136 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.137 §9.5 source ↗

Simplify:

\sqrt{\frac{3}{27}}

Answer

\frac{1}{3}

Try It 9.138 §9.5 source ↗

Simplify:

\sqrt{\frac{10}{50}}

Answer

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

Try It 9.139 §9.5 source ↗

Simplify:

\frac{2}{2 + \sqrt{3}}

Answer

\frac{2 (2 - \sqrt{3})}{1}

Try It 9.140 §9.5 source ↗

Simplify:

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

Answer

\frac{5 (5 - \sqrt{3})}{22}

Try It 9.141 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.142 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.143 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.144 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.145 §9.5 source ↗

Simplify:

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

Answer

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

Try It 9.146 §9.5 source ↗

Simplify:

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

Answer

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

9.5-317 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{27}}{6}$

Answer

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

9.5-319 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{72}}{9}$

Answer

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

9.5-321 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{2 - \sqrt{32}}{8}$

Answer

\frac{1 - 2 \sqrt{2}}{4}

9.5-323 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{6 + \sqrt{45}}{6}$

Answer

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

9.5-325 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{80}}{\sqrt{125}}$

Answer

\frac{4}{5}

9.5-327 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{128}}{\sqrt{72}}$

Answer

\frac{4}{3}

9.5-329 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{8 x^{6}}}{\sqrt{2 x^{2}}}$ ⓑ $\frac{\sqrt{200 m^{5}}}{\sqrt{98 m}}$

Answer

ⓐ

2 x^{2}

ⓑ

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

9.5-331 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{75 r^{3}}}{\sqrt{108 r}}$

Answer

\frac{5 r}{6}

9.5-333 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}}$

Answer

\frac{6 p \sqrt{102}}{q^{2}}

9.5-335 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{320 m n^{5}}}{\sqrt{45 m^{7} n^{3}}}$

Answer

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

9.5-337 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{98}}{14}$

Answer

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

9.5-339 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{5 + \sqrt{125}}{15}$

Answer

\frac{1 + \sqrt{5}}{3}

9.5-341 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{96}}{\sqrt{150}}$

Answer

\frac{4}{5}

9.5-343 §9.5 source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{26 y^{7}}}{\sqrt{2 y}}$

Answer

y^{3} \sqrt{13}

9.5-345 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

$\frac{10}{\sqrt{6}}$

Answer

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

9.5-347 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

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

Answer

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

9.5-349 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

$\frac{3}{\sqrt{13}}$

Answer

\frac{3 \sqrt{13}}{13}

9.5-351 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

$\frac{10}{3 \sqrt{10}}$

Answer

\frac{\sqrt{10}}{3}

9.5-353 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

$\frac{4}{9 \sqrt{5}}$

Answer

\frac{4 \sqrt{5}}{45}

9.5-355 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

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

Answer

- \frac{3 \sqrt{3}}{2}

9.5-357 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

$\sqrt{\frac{3}{20}}$

Answer

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

9.5-359 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

$\sqrt{\frac{7}{40}}$

Answer

\frac{\sqrt{70}}{20}

9.5-361 §9.5 source ↗

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

$\sqrt{\frac{19}{175}}$

Answer

\frac{\sqrt{133}}{35}

9.5-363 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

ⓐ $\frac{3}{3 + \sqrt{11}}$ ⓑ $\frac{8}{1 - \sqrt{5}}$

Answer

ⓐ

\frac{3 (3 - \sqrt{11})}{−2}

ⓑ

−2 (1 + \sqrt{5})

9.5-365 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

ⓐ $\frac{5}{5 + \sqrt{6}}$ ⓑ $\frac{6}{3 - \sqrt{7}}$

Answer

ⓐ

\frac{5 (5 - \sqrt{6})}{19}

ⓑ

3 (3 + \sqrt{7})

9.5-367 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

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

Answer

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

9.5-369 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

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

Answer

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

9.5-371 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

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

Answer

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

9.5-373 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

$\frac{\sqrt{150 x^{2} y^{6}}}{\sqrt{6 x^{4} y^{2}}}$

Answer

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

9.5-375 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

$\frac{15}{\sqrt{5}}$

Answer

3 \sqrt{5}

9.5-377 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

$\sqrt{\frac{8}{54}}$

Answer

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

9.5-379 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

$\frac{3}{5 + \sqrt{5}}$

Answer

\frac{3 (5 - \sqrt{5})}{20}

9.5-381 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

$\frac{\sqrt{2}}{\sqrt{x} - \sqrt{3}}$

Answer

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

9.5-383 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

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

Answer

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

9.5-385 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

A supply kit is dropped from an airplane flying at an altitude of 250 feet. Simplify $\sqrt{\frac{250}{16}}$ to determine how many seconds it takes for the supply kit to reach the ground.

Answer

\frac{5 \sqrt{10}}{4} seconds

9.5-387 §9.5 source ↗

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

ⓐ Simplify $\sqrt{\frac{27}{3}}$ and explain all your steps.

ⓑ Simplify $\sqrt{\frac{27}{5}}$ and explain all your steps.

Answer

Answers will vary.

Be Prepared 9.15 §9.6 source ↗

Simplify: ⓐ

\sqrt{9}

ⓑ

9^{2}

.
If you missed this problem, review Example 9.1 and Example 1.19.

Example 9.1 in the book ↗

Example 1.19 in the book ↗

Answer

ⓐ

3

ⓑ

81

Be Prepared 9.16 §9.6 source ↗

Solve:

5 (x + 1) -

4 =

3 (2 x - 7)

.
If you missed this problem, review Example 2.42.

Example 2.42 in the book ↗

Answer

22

Be Prepared 9.17 §9.6 source ↗

Solve:

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

.
If you missed this problem, review Example 7.73.

Example 7.73 in the book ↗

Answer

n = 2 or n = 4

Try It 9.147 §9.6 source ↗

For the equation $\sqrt{x + 6} = x$ :

ⓐ Is $x = −2$ a solution? ⓑ Is $x = 3$ a solution?

Answer

ⓐ no ⓑ

yes

Try It 9.148 §9.6 source ↗

For the equation $\sqrt{− x + 2} = x$ :

ⓐ Is $x = −2$ a solution? ⓑ Is $x = 1$ a solution?

Answer

ⓐ no ⓑ

yes

Try It 9.149 §9.6 source ↗

Solve:

\sqrt{3 x - 5} = 5

Answer

10

Try It 9.150 §9.6 source ↗

Solve:

\sqrt{4 x + 8} = 6

Answer

7

Try It 9.151 §9.6 source ↗

Solve:

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

Answer

\frac{23}{3}

Try It 9.152 §9.6 source ↗

Solve:

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

Answer

\frac{3}{10}

Try It 9.153 §9.6 source ↗

Solve:

\sqrt{3 p + 3} + 3 = 5

Answer

\frac{1}{3}

Try It 9.154 §9.6 source ↗

Solve:

\sqrt{5 q + 1} + 4 = 6

Answer

\frac{3}{5}

Try It 9.155 §9.6 source ↗

Solve:

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

Answer

no solution

Try It 9.156 §9.6 source ↗

Solve:

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

Answer

no solution

Try It 9.157 §9.6 source ↗

Solve:

\sqrt{x - 2} + 2 = x

Answer

2, 3

Try It 9.158 §9.6 source ↗

Solve:

\sqrt{y - 5} + 5 = y

Answer

5, 6

Try It 9.159 §9.6 source ↗

Solve:

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

Answer

7

Try It 9.160 §9.6 source ↗

Solve:

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

Answer

3

Try It 9.161 §9.6 source ↗

Solve:

2 \sqrt{4 a + 2} - 16 = 16

Answer

\frac{127}{2}

Try It 9.162 §9.6 source ↗

Solve:

3 \sqrt{6 b + 3} - 25 = 50

Answer

\frac{311}{3}

Try It 9.163 §9.6 source ↗

Solve:

\sqrt{2 x - 5} = \sqrt{5 x + 3}

Answer

no solution

Try It 9.164 §9.6 source ↗

Solve:

\sqrt{7 y + 1} = \sqrt{2 y - 5}

Answer

no solution

Try It 9.165 §9.6 source ↗

Solve:

\sqrt{x} + 3 = \sqrt{x + 5}

Answer

no solution

Try It 9.166 §9.6 source ↗

Solve:

\sqrt{m} + 5 = \sqrt{m + 16}

Answer

no solution

Try It 9.167 §9.6 source ↗

Solve:

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

Answer

no solution

Try It 9.168 §9.6 source ↗

Solve:

\sqrt{n - 4} + 5 = \sqrt{3 n + 3}

Answer

no solution

Try It 9.169 §9.6 source ↗

Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 370 square feet. Use the formula

s = \sqrt{A}

to find the length of each side of her lawn. Round your answer to the nearest tenth of a foot.

Answer

19.2 feet

Try It 9.170 §9.6 source ↗

Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of 2704 square centimeters. Use the formula

s = \sqrt{A}

to find the length of each side of his mosaic. Round your answer to the nearest tenth of a centimeter.

Answer

52.0 cm

Try It 9.171 §9.6 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 9.172 §9.6 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 9.173 §9.6 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 miles per hour

Try It 9.174 §9.6 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 miles per hour

9.6-389 §9.6 source ↗

Solve Radical Equations

In the following exercises, check whether the given values are solutions.

For the equation $\sqrt{x + 12} = x$ : ⓐ Is $x = 4$ a solution? ⓑ Is $x = −3$ a solution?

Answer

ⓐ

yes

ⓑ

no

9.6-391 §9.6 source ↗

Solve Radical Equations

In the following exercises, check whether the given values are solutions.

For the equation $\sqrt{t + 6} = t$ : ⓐ Is $t = −2$ a solution? ⓑ Is $t = 3$ a solution?

Answer

ⓐ no ⓑ yes

9.6-393 §9.6 source ↗

In the following exercises, solve.

$\sqrt{5 y + 1} = 4$

Answer

9.6-395 §9.6 source ↗

In the following exercises, solve.

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

Answer

9.6-397 §9.6 source ↗

In the following exercises, solve.

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

Answer

9.6-399 §9.6 source ↗

In the following exercises, solve.

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

Answer

17

9.6-401 §9.6 source ↗

In the following exercises, solve.

$\sqrt{5 q + 3} - 4 = 0$

Answer

\frac{13}{5}

9.6-403 §9.6 source ↗

In the following exercises, solve.

$\sqrt{6 n + 1} + 4 = 8$

Answer

\frac{5}{2}

9.6-405 §9.6 source ↗

In the following exercises, solve.

$\sqrt{5 v - 2} + 5 = 0$

Answer

no solution

9.6-407 §9.6 source ↗

In the following exercises, solve.

$\sqrt{2 m + 1} + 4 = 0$

Answer

no solution

9.6-409 §9.6 source ↗

In the following exercises, solve.

ⓐ $\sqrt{v - 10} + 10 = v$
ⓑ $\sqrt{y + 4} - y + 2 = 0$

Answer

ⓐ

10, 11

ⓑ

5

9.6-411 §9.6 source ↗

In the following exercises, solve.

ⓐ $\sqrt{s - 8} - s = −8$
ⓑ $\sqrt{w + 25} - w + 5 = 0$

Answer

ⓐ

8, 9

ⓑ

11

9.6-413 §9.6 source ↗

In the following exercises, solve.

$2 \sqrt{5 x + 1} - 8 = 0$

Answer

3

9.6-415 §9.6 source ↗

In the following exercises, solve.

$3 \sqrt{7 y + 1} - 10 = 8$

Answer

5

9.6-417 §9.6 source ↗

In the following exercises, solve.

$\sqrt{4 v + 3} = \sqrt{v - 6}$

Answer

not a real number

9.6-419 §9.6 source ↗

In the following exercises, solve.

$\sqrt{12 c + 6} = \sqrt{10 - 4 c}$

Answer

\frac{1}{4}

9.6-421 §9.6 source ↗

In the following exercises, solve.

ⓐ $\sqrt{r} + 6 = \sqrt{r + 8}$
ⓑ $\sqrt{s - 3} + 2 = \sqrt{s + 4}$

Answer

ⓐ

no solution

ⓑ

\frac{57}{16}

9.6-423 §9.6 source ↗

In the following exercises, solve.

ⓐ $\sqrt{x} + 10 = \sqrt{x + 2}$
ⓑ $\sqrt{y - 2} + 2 = \sqrt{2 y + 4}$

Answer

ⓐ

no solution

ⓑ 6

9.6-425 §9.6 source ↗

In the following exercises, solve.

$\sqrt{8 u + 1} + 9 = 0$

Answer

no solution

9.6-427 §9.6 source ↗

In the following exercises, solve.

$\sqrt{d} - 2 = \sqrt{d - 20}$

Answer

9.6-429 §9.6 source ↗

In the following exercises, solve.

$\sqrt{9 p + 9} = \sqrt{10 p - 6}$

Answer

9.6-431 §9.6 source ↗

Use Square Roots in Applications

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

Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula $s = \sqrt{A}$ to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.

Answer

11.4 feet

9.6-433 §9.6 source ↗

Use Square Roots in Applications

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

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the flare to reach the water.

Answer

8 seconds

9.6-435 §9.6 source ↗

Use Square Roots in Applications

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

Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4000 feet above the Colorado River. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the hammer to reach the river.

Answer

15.8 seconds

9.6-437 §9.6 source ↗

Use Square Roots 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 miles per hour

9.6-439 §9.6 source ↗

Use Square Roots 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 117 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

53.0 miles per hour

9.6-441 §9.6 source ↗

Use Square Roots in Applications

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

ⓐ Solve the equation $\sqrt{r + 4} - r + 2 = 0$ .

ⓑ Explain why one of the “solutions” that was found was not actually a solution to the equation.

Answer

Answers will vary.

Be Prepared 9.18 §9.7 source ↗

Simplify:

y^{5} y^{4}

.
If you missed this problem, review Example 6.18.

Example 6.18 in the book ↗

Answer

y^{9}

Be Prepared 9.19 §9.7 source ↗

Simplify:

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

.
If you missed this problem, review Example 6.22.

Example 6.22 in the book ↗

Answer

n^{12}

Be Prepared 9.20 §9.7 source ↗

Simplify:

\frac{x^{8}}{x^{3}}

.
If you missed this problem, review Example 6.59.

Example 6.59 in the book ↗

Answer

x^{5}

Try It 9.175 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{27}

ⓑ

\sqrt[4]{256}

ⓒ

\sqrt[5]{243}

Answer

ⓐ 3 ⓑ 4 ⓒ 3

Try It 9.176 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{1000}

ⓑ

\sqrt[4]{16}

ⓒ

\sqrt[5]{32}

Answer

ⓐ 10 ⓑ 2 ⓒ 2

Try It 9.177 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{−125}

ⓑ

\sqrt[4]{−16}

ⓒ

\sqrt[5]{−32}

Answer

ⓐ

−5

ⓑ not real ⓒ

−2

Try It 9.178 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{−216}

ⓑ

\sqrt[4]{−81}

ⓒ

\sqrt[5]{−1024}

Answer

ⓐ

−6

ⓑ not real ⓒ

−4

Try It 9.179 §9.7 source ↗

Simplify: ⓐ

\sqrt{b^{2}}

ⓑ

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

ⓒ

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

ⓓ

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

Answer

ⓐ

| b |

ⓑ

w

ⓒ

| m |

ⓓ

q

Try It 9.180 §9.7 source ↗

Simplify: ⓐ

\sqrt{y^{2}}

ⓑ

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

ⓒ

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

ⓓ

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

Answer

ⓐ

| y |

ⓑ

p

ⓒ

| z |

ⓓ

q

Try It 9.181 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

u^{3}

ⓑ

v^{5}

Try It 9.182 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

c^{4}

ⓑ

d^{4}

Try It 9.183 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

3 x^{9}

ⓑ

3 | q^{7} |

Try It 9.184 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

5 p^{3}

ⓑ

3 q^{5}

Try It 9.185 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

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

ⓑ

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

Try It 9.186 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

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

ⓑ

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

Try It 9.187 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{81}

ⓑ

\sqrt[4]{64}

Answer

ⓐ

3 \sqrt[3]{3}

ⓑ

2 \sqrt[4]{4}

Try It 9.188 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{625}

ⓑ

\sqrt[4]{729}

Answer

ⓐ

5 \sqrt[3]{5}

ⓑ

3 \sqrt[4]{9}

Try It 9.189 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

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

ⓑ

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

Try It 9.190 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

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

ⓑ

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

Try It 9.191 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{−108}

ⓑ

\sqrt[4]{−48}

Answer

ⓐ

−3 \sqrt[3]{4}

ⓑ

not real

Try It 9.192 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{−625}

ⓑ

\sqrt[4]{−324}

Answer

ⓐ

−5 \sqrt[3]{5}

ⓑ

not real

Try It 9.193 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

| x |

ⓑ

y^{3}

Try It 9.194 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

m^{2}

ⓑ

n^{2}

Try It 9.195 §9.7 source ↗

Simplify: ⓐ

\frac{\sqrt[3]{−532}}{\sqrt[3]{2}}

ⓑ

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

Answer

ⓐ

- \sqrt[3]{266}

ⓑ

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

Try It 9.196 §9.7 source ↗

Simplify: ⓐ

\frac{\sqrt[3]{−192}}{\sqrt[3]{3}}

ⓑ

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

Answer

ⓐ

−4

ⓑ

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

Try It 9.197 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

\sqrt[4]{\frac{80 x^{10}}{y^{5}}}

Answer

ⓐ

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

ⓑ

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

Try It 9.198 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

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

Answer

ⓐ

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

ⓑ

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

Try It 9.199 §9.7 source ↗

Simplify: ⓐ

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

ⓑ

3 \sqrt[3]{9} - \sqrt[3]{9}

Answer

ⓐ

2 \sqrt[5]{3 x}

ⓑ

2 \sqrt[3]{9}

Try It 9.200 §9.7 source ↗

Simplify: ⓐ

\sqrt[4]{10 y} + \sqrt[4]{10 y}

ⓑ

5 \sqrt[6]{32} - 3 \sqrt[6]{32}

Answer

ⓐ

2 \sqrt[4]{10 y}

ⓑ

2 \sqrt[6]{32}

Try It 9.201 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{192} - \sqrt[3]{81}

ⓑ

\sqrt[4]{32} + \sqrt[4]{512}

Answer

ⓐ

\sqrt[3]{3}

ⓑ

6 \sqrt[4]{2}

Try It 9.202 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{128} - \sqrt[3]{250}

ⓑ

\sqrt[5]{64} + \sqrt[5]{486}

Answer

ⓐ

− \sqrt[3]{2}

ⓑ

5 \sqrt[5]{2}

Try It 9.203 §9.7 source ↗

Simplify: ⓐ

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

\sqrt[3]{−108 y^{8}}

ⓑ

\sqrt[4]{243 r^{11}} +

\sqrt[4]{768 r^{10}}

Answer

ⓐ

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

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

ⓑ

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

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

Try It 9.204 §9.7 source ↗

Simplify: ⓐ

\sqrt[3]{40 z^{7}} -

\sqrt[3]{−135 z^{4}}

ⓑ

\sqrt[4]{80 s^{13}} +

\sqrt[4]{1280 s^{6}}

Answer

ⓐ

2 z^{2} \sqrt[3]{5 z} +

3 z \sqrt[3]{5 z}

ⓑ

2 | s^{3} | \sqrt[4]{5 s} +

4 | s | \sqrt[4]{5 s^{2}}

9.7-443 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{27}$ ⓑ $\sqrt[4]{16}$ ⓒ $\sqrt[5]{243}$

Answer

ⓐ

3

ⓑ

2

ⓒ

3

9.7-445 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

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

Answer

ⓐ

5

ⓑ

6

ⓒ

4

9.7-447 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{−64}$ ⓑ $\sqrt[4]{−16}$ ⓒ $\sqrt[5]{−243}$

Answer

ⓐ

−4

ⓑ

not real

ⓒ

−3

9.7-449 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

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

Answer

ⓐ

−8

ⓑ not a real number ⓒ

−1

9.7-451 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{a^{3}}$

ⓑ

Answer

ⓐ

a

ⓑ

| b |

9.7-453 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[8]{k^{8}}$ ⓑ $\sqrt[6]{p^{6}}$

Answer

ⓐ

| k |

ⓑ

| p |

9.7-455 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[5]{a^{10}}$ ⓑ $\sqrt[3]{b^{27}}$

Answer

ⓐ

a^{2}

ⓑ

b^{9}

9.7-457 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[6]{r^{12}}$ ⓑ $\sqrt[3]{s^{30}}$

Answer

ⓐ

r^{2}

ⓑ

s^{10}

9.7-459 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{−8 c^{9}}$ ⓑ $\sqrt[3]{125 d^{15}}$

Answer

ⓐ

−2 c^{3}

ⓑ

5 d^{5}

9.7-461 §9.7 source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[7]{128 r^{14}}$ ⓑ $\sqrt[4]{81 s^{24}}$

Answer

ⓐ

2 r^{2}

ⓑ

3 s^{6}

9.7-463 §9.7 source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[5]{u^{7}}$ ⓑ $\sqrt[6]{v^{11}}$

Answer

ⓐ

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

ⓑ

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

9.7-465 §9.7 source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[5]{p^{8}}$ ⓑ $\sqrt[3]{q^{8}}$

Answer

ⓐ

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

ⓑ

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

9.7-467 §9.7 source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{625}$ ⓑ $\sqrt[6]{128}$

Answer

ⓐ

5 \sqrt[3]{5}

ⓑ

2 \sqrt[6]{2}

9.7-469 §9.7 source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[4]{3125}$ ⓑ $\sqrt[3]{81}$

Answer

ⓐ

5 \sqrt[4]{5}

ⓑ

3 \sqrt[3]{3}

9.7-471 §9.7 source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[5]{96 a^{7}}$ ⓑ $\sqrt[3]{375 b^{4}}$

Answer

ⓐ

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

ⓑ

5 b \sqrt[3]{3 b}

9.7-473 §9.7 source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{512 p^{5}}$ ⓑ $\sqrt[4]{324 q^{7}}$

Answer

ⓐ

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

ⓑ

3 q \sqrt[4]{4 q^{3}}

9.7-475 §9.7 source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[5]{−486}$ ⓑ $\sqrt[6]{−64}$

Answer

ⓐ

−3 \sqrt[5]{2}

ⓑ

not real

9.7-477 §9.7 source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{−8}$ ⓑ $\sqrt[4]{−16}$

Answer

ⓐ

−2

ⓑ

not real

9.7-479 §9.7 source ↗

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[5]{\frac{d^{12}}{d^{7}}}$ ⓑ $\sqrt[8]{\frac{m^{12}}{m^{4}}}$

Answer

ⓐ

d

ⓑ

| m |

9.7-481 §9.7 source ↗

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{\frac{r^{14}}{r^{5}}}$ ⓑ $\sqrt[4]{\frac{c^{21}}{c^{9}}}$

Answer

ⓐ

r^{3}

ⓑ

| c^{3} |

9.7-483 §9.7 source ↗

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt[3]{−625}}{\sqrt[3]{5}}$ ⓑ $\frac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$

Answer

ⓐ

−5

ⓑ

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

9.7-485 §9.7 source ↗

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{\frac{162}{6}}$ ⓑ $\sqrt[4]{\frac{160 r^{10}}{5 r^{3}}}$

Answer

ⓐ

3 \sqrt[3]{6}

ⓑ

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

9.7-487 §9.7 source ↗

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[5]{\frac{96 r^{11}}{s^{3}}}$ ⓑ $\sqrt[6]{\frac{128 u^{7}}{v^{3}}}$

Answer

ⓐ

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

ⓑ

2 u \sqrt[6]{\frac{2 u}{v^{3}}}

9.7-489 §9.7 source ↗

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{\frac{625 u^{10}}{v^{3}}}$ ⓑ $\sqrt[4]{\frac{729 c^{21}}{d^{8}}}$

Answer

ⓐ

\frac{5 u^{3} \sqrt[3]{5 u}}{v}

ⓑ

\frac{3 c^{5} \sqrt[4]{9 c}}{d^{2}}

9.7-491 §9.7 source ↗

Add and Subtract Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{15 q} + \sqrt[3]{15 q}$ ⓑ $2 \sqrt[4]{27} - 6 \sqrt[4]{27}$

Answer

ⓐ

2 \sqrt[3]{15 q}

ⓑ

−4 \sqrt[4]{27}

9.7-493 §9.7 source ↗

Add and Subtract Higher Roots

In the following exercises, simplify.

ⓐ

ⓑ

Answer

ⓐ

ⓑ

9.7-495 §9.7 source ↗

Add and Subtract Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{250} - \sqrt[3]{54}$ ⓑ $\sqrt[4]{243} - \sqrt[4]{1875}$

Answer

ⓐ

2 \sqrt[3]{2}

ⓑ

−2 \sqrt[4]{3}

9.7-497 §9.7 source ↗

Add and Subtract Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[4]{243} + \sqrt[4]{1250}$ ⓑ $\sqrt[3]{2000} + \sqrt[3]{54}$

Answer

ⓐ

3 \sqrt[4]{3} + 5 \sqrt[4]{2}

ⓑ

13 \sqrt[3]{2}

9.7-499 §9.7 source ↗

Add and Subtract Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{80 b^{5}} -$ $\sqrt[3]{−270 b^{3}}$ ⓑ $\sqrt[4]{160 v^{10}} -$ $\sqrt[4]{1280 v^{3}}$

Answer

ⓐ

2 b \sqrt[3]{10 b^{2}} +

3 b \sqrt[3]{10}

ⓑ

2 v^{2} \sqrt[4]{10 v^{2}} -

4 \sqrt[4]{5 v^{3}}

9.7-501 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\sqrt[6]{64}$

Answer

2

9.7-503 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

Answer

| b |

9.7-505 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\sqrt[3]{125 d^{15}}$

Answer

5 d^{5}

9.7-507 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\sqrt[4]{s^{10}}$

Answer

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

9.7-509 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\sqrt[4]{48 y^{6}}$

Answer

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

9.7-511 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\sqrt[6]{−64}$

Answer

not real

9.7-513 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\frac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$

Answer

2 x \sqrt[5]{2 x}

9.7-515 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\sqrt[6]{\frac{128 u^{7}}{v^{3}}}$

Answer

2 u \sqrt[6]{\frac{2 u}{v^{3}}}

9.7-517 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\sqrt[4]{512} - \sqrt[4]{32}$

Answer

2 \sqrt[4]{2}

9.7-519 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

$\sqrt[4]{486 u^{7}} +$ $\sqrt[4]{768 u^{3}}$

Answer

3 │ u │ \sqrt[4]{6 u^{3}} +

4 \sqrt[4]{3 u^{3}}

9.7-521 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

Spread of a virus The expression $3 \cdot x^{n}$ models the spread of a virus after $n$ cycles. There were three people originally infected with the virus, and each of them infected $x$ people. So $3 \cdot x^{4}$ is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression $\sqrt[4]{\frac{1875}{3}}$ to determine the number of people each person infected.

Answer

5

9.7-523 §9.7 source ↗

Mixed Practice

In the following exercises, simplify.

Explain why $\sqrt[4]{−64}$ is not a real number but $\sqrt[3]{−64}$ is.

Answer

Answers may vary.

Be Prepared 9.21 §9.8 source ↗

Add:

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

.
If you missed this problem, review Example 1.81.

Example 1.81 in the book ↗

Answer

\frac{53}{60}

Be Prepared 9.22 §9.8 source ↗

Simplify:

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

.
If you missed this problem, review Example 6.24.

Example 6.24 in the book ↗

Answer

64 x^{6} y^{15}

Be Prepared 9.23 §9.8 source ↗

Simplify:

5^{−3}

.
If you missed this problem, review Example 6.89.

Example 6.89 in the book ↗

Answer

\frac{1}{125}

Try It 9.205 §9.8 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 9.206 §9.8 source ↗

Write as a radial expression: ⓐ

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

ⓑ

z^{\frac{1}{3}}

ⓒ

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

Answer

ⓐ

\sqrt{b}

ⓑ

\sqrt[3]{z}

ⓒ

\sqrt[4]{p}

Try It 9.207 §9.8 source ↗

Write with a rational exponent: ⓐ

\sqrt{s}

ⓑ

\sqrt[3]{x}

ⓒ

\sqrt[4]{b}

Answer

ⓐ

s^{\frac{1}{2}}

ⓑ

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

ⓒ

b^{\frac{1}{4}}

Try It 9.208 §9.8 source ↗

Write with a rational exponent: ⓐ

\sqrt{v}

ⓑ

\sqrt[3]{p}

ⓒ

\sqrt[4]{p}

Answer

ⓐ

v^{\frac{1}{2}}

ⓑ

p^{\frac{1}{3}}

ⓒ

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

Try It 9.209 §9.8 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}}

ⓒ

{(486 y)}^{\frac{1}{4}}

Try It 9.210 §9.8 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}}

ⓒ

{(1024 a)}^{\frac{1}{3}}

Try It 9.211 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

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

ⓒ

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

Answer

ⓐ 6 ⓑ 2 ⓒ 2

Try It 9.212 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

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

ⓒ

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

Answer

ⓐ 10 ⓑ 3 ⓒ 3

Try It 9.213 §9.8 source ↗

Simplify: ⓐ

{(−125)}^{\frac{1}{3}}

ⓑ

− 125^{\frac{1}{3}}

ⓒ

{(125)}^{- \frac{1}{3}}

Answer

ⓐ

−5

ⓑ

−5

ⓒ

\frac{1}{5}

Try It 9.214 §9.8 source ↗

Simplify: ⓐ

{(−32)}^{\frac{1}{5}}

ⓑ

− 32^{\frac{1}{5}}

ⓒ

{(32)}^{- \frac{1}{5}}

Answer

ⓐ

−2

ⓑ

−2

ⓒ

\frac{1}{2}

Try It 9.215 §9.8 source ↗

Simplify: ⓐ

{(−64)}^{\frac{1}{2}}

ⓑ

− 64^{\frac{1}{2}}

ⓒ

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

Answer

ⓐ not real ⓑ

−8

ⓒ

\frac{1}{8}

Try It 9.216 §9.8 source ↗

Simplify: ⓐ

{(−256)}^{\frac{1}{4}}

ⓑ

− 256^{\frac{1}{4}}

ⓒ

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

Answer

ⓐ not real ⓑ

−4

ⓒ

\frac{1}{4}

Try It 9.217 §9.8 source ↗

Write with a rational exponent: ⓐ

\sqrt{x^{5}}

ⓑ

\sqrt[4]{z^{3}}

ⓒ

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

Answer

ⓐ

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

ⓑ

z^{\frac{3}{4}}

ⓒ

y^{\frac{2}{5}}

Try It 9.218 §9.8 source ↗

Write with a rational exponent: ⓐ

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

ⓑ

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

ⓒ

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

Answer

ⓐ

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

ⓑ

b^{\frac{7}{3}}

ⓒ

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

Try It 9.219 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

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

ⓒ

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

Answer

ⓐ 8 ⓑ 9 ⓒ 125

Try It 9.220 §9.8 source ↗

Simplify: ⓐ

8^{\frac{5}{3}}

ⓑ

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

ⓒ

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

Answer

ⓐ 32 ⓑ 729 ⓒ 8

Try It 9.221 §9.8 source ↗

Simplify: ⓐ

8^{- \frac{5}{3}}

ⓑ

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

ⓒ

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

Answer

ⓐ

\frac{1}{32}

ⓑ

\frac{1}{729}

ⓒ

\frac{1}{8}

Try It 9.222 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

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

ⓒ

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

Answer

ⓐ

\frac{1}{8}

ⓑ

\frac{1}{9}

ⓒ

\frac{1}{125}

Try It 9.223 §9.8 source ↗

Simplify: ⓐ

{−16}^{\frac{3}{2}}

ⓑ

{−16}^{- \frac{3}{2}}

ⓒ

{(−16)}^{- \frac{3}{2}}

Answer

ⓐ

−64

ⓑ

- \frac{1}{64}

Try It 9.224 §9.8 source ↗

Simplify: ⓐ

{−81}^{\frac{3}{2}}

ⓑ

{−81}^{- \frac{3}{2}}

ⓒ

{(−81)}^{- \frac{3}{2}}

Answer

ⓐ

−729

ⓑ

- \frac{1}{729}

Try It 9.225 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

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

ⓒ

m^{\frac{1}{4}} \cdot m^{\frac{3}{4}}

Answer

ⓐ 9 ⓑ

y^{3}

ⓒ m

Try It 9.226 §9.8 source ↗

Simplify: ⓐ

5^{\frac{3}{5}} \cdot 5^{\frac{7}{5}}

ⓑ

z^{\frac{1}{8}} \cdot z^{\frac{7}{8}}

ⓒ

n^{\frac{2}{7}} \cdot n^{\frac{5}{7}}

Answer

ⓐ 25 ⓑ z ⓒ n

Try It 9.227 §9.8 source ↗

Simplify: ⓐ

{(p^{10})}^{\frac{1}{5}}

ⓑ

{(q^{8})}^{\frac{3}{4}}

ⓒ

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

Answer

ⓐ

p^{2}

ⓑ

q^{6}

ⓒ

x^{8}

Try It 9.228 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

{(s^{12})}^{\frac{3}{4}}

ⓒ

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

Answer

ⓐ

r^{10}

ⓑ

s^{9}

ⓒ

m^{2}

Try It 9.229 §9.8 source ↗

Simplify: ⓐ

\frac{u^{\frac{5}{4}}}{u^{\frac{1}{4}}}

ⓑ

\frac{v^{\frac{3}{5}}}{v^{\frac{2}{5}}}

ⓒ

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

Answer

ⓐ u ⓑ

v^{\frac{1}{5}}

ⓒ

\frac{1}{x}

Try It 9.230 §9.8 source ↗

Simplify: ⓐ

\frac{c^{\frac{12}{5}}}{c^{\frac{2}{5}}}

ⓑ

\frac{m^{\frac{5}{4}}}{m^{\frac{9}{4}}}

ⓒ

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

Answer

ⓐ

c^{2}

ⓑ

\frac{1}{m}

ⓒ

\frac{1}{d}

Try It 9.231 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

{(64 y^{\frac{2}{3}})}^{\frac{1}{3}}

Answer

ⓐ

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

ⓑ

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

Try It 9.232 §9.8 source ↗

Simplify: ⓐ

{(16 m^{\frac{1}{3}})}^{\frac{3}{2}}

ⓑ

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

Answer

ⓐ

64 m^{\frac{1}{2}}

ⓑ

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

Try It 9.233 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

\frac{n^{\frac{1}{6}} \cdot n}{n^{- \frac{11}{6}}}

Answer

ⓐ

m^{2}

ⓑ

n^{3}

Try It 9.234 §9.8 source ↗

Simplify: ⓐ

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

ⓑ

\frac{v^{\frac{1}{2}} \cdot v}{v^{- \frac{7}{2}}}

Answer

ⓐ

u^{3}

ⓑ

v^{5}

9.8-525 §9.8 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}

9.8-527 §9.8 source ↗

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

In the following exercises, write as a radical expression.

Answer

ⓐ

\sqrt[7]{g}

ⓑ

\sqrt[5]{h}

ⓒ

\sqrt[25]{j}

9.8-529 §9.8 source ↗

In the following exercises, write with a rational exponent.

ⓐ $\sqrt[8]{r}$ ⓑ

Answer

ⓐ

r^{\frac{1}{8}}

ⓑ

s^{\frac{1}{10}}

ⓒ

t^{\frac{1}{4}}

9.8-531 §9.8 source ↗

In the following exercises, write with a rational exponent.

Answer

ⓐ

u^{\frac{1}{5}}

ⓑ

v^{\frac{1}{2}}

ⓒ

w^{\frac{1}{16}}

9.8-533 §9.8 source ↗

In the following exercises, write with a rational exponent.

Answer

ⓐ

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

ⓑ

{(9 y)}^{\frac{1}{8}}

ⓒ

7 {(3 z)}^{\frac{1}{5}}

9.8-535 §9.8 source ↗

In the following exercises, write with a rational exponent.

Answer

ⓐ

{(25 a)}^{\frac{1}{3}}

ⓑ

{(3 b)}^{\frac{1}{2}}

ⓒ

{(40 c)}^{\frac{1}{10}}

9.8-537 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ 5 ⓑ 3 ⓒ 2

9.8-539 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ 6 ⓑ 2 ⓒ 3

9.8-541 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ

−3

ⓑ

−3

ⓒ

\frac{1}{3}

9.8-543 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ

−10

ⓑ

−10

ⓒ

\frac{1}{10}

9.8-545 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ not a real number ⓑ

−7

ⓒ

\frac{1}{7}

9.8-547 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ not a real number ⓑ

1

ⓒ

−1

9.8-549 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ

−2

ⓑ

\frac{1}{3}

ⓒ

−5

9.8-551 §9.8 source ↗

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

In the following exercises, write with a rational exponent.

Answer

ⓐ

r^{\frac{7}{4}}

ⓑ

s^{\frac{3}{5}}

ⓒ

t^{\frac{7}{3}}

9.8-553 §9.8 source ↗

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

In the following exercises, write with a rational exponent.

Answer

ⓐ

a^{\frac{1}{3}}

ⓑ

b^{\frac{5}{2}}

ⓒ

c^{\frac{5}{3}}

9.8-555 §9.8 source ↗

In the following exercises, simplify.

Answer

9.8-557 §9.8 source ↗

In the following exercises, simplify.

Answer

9.8-559 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ 32,768 ⓑ

\frac{1}{729}

ⓒ

\frac{1}{81}

9.8-561 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ 1000 ⓑ

\frac{1}{16,807}

9.8-563 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ

−512

ⓑ

- \frac{1}{512}

9.8-565 §9.8 source ↗

In the following exercises, simplify.

Answer

ⓐ

−343

ⓑ

- \frac{1}{343}

9.8-567 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$ ⓑ $n^{\frac{2}{10}} \cdot n^{\frac{8}{10}}$ ⓒ $q^{\frac{2}{5}} \cdot q^{\frac{13}{5}}$

Answer

ⓐ 216 ⓑ

n

ⓒ

q^{3}

9.8-569 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $10^{\frac{1}{3}} \cdot 10^{\frac{5}{3}}$ ⓑ $x^{\frac{5}{6}} \cdot x^{\frac{7}{6}}$ ⓒ $y^{\frac{11}{8}} \cdot y^{\frac{21}{8}}$

Answer

ⓐ 100 ⓑ

x^{2}

ⓒ

y^{4}

9.8-571 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Answer

ⓐ

a^{2}

ⓑ

b^{9}

ⓒ

c

9.8-573 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Answer

ⓐ

h^{8}

ⓑ

k^{9}

ⓒ

j^{14}

9.8-575 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $\frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}}$ ⓑ $\frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}}$ ⓒ $\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

Answer

ⓐ

s

ⓑ

z^{2}

ⓒ

\frac{1}{w}

9.8-577 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $\frac{u^{\frac{13}{9}}}{u^{\frac{4}{9}}}$ ⓑ $\frac{r^{\frac{15}{7}}}{r^{\frac{8}{7}}}$ ⓒ $\frac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}$

Answer

ⓐ

u

ⓑ

r

ⓒ

\frac{1}{n}

9.8-579 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ ${(81 r^{\frac{4}{5}})}^{\frac{1}{4}}$ ⓑ ${(64 s^{\frac{3}{7}})}^{\frac{1}{6}}$

Answer

ⓐ

3 r^{\frac{1}{5}}

ⓑ

2 s^{\frac{1}{14}}

9.8-581 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ ${(27 m^{\frac{3}{4}})}^{\frac{2}{3}}$ ⓑ ${(625 n^{\frac{8}{3}})}^{\frac{3}{4}}$

Answer

ⓐ

9 m^{\frac{1}{2}}

ⓑ

125 n^{2}

9.8-583 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ ${(r^{8} s^{4})}^{\frac{1}{4}}$ ⓑ ${(u^{15} v^{20})}^{\frac{1}{5}}$

Answer

ⓐ

r^{2} s

ⓑ

u^{3} v^{4}

9.8-585 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ ${(r^{16} s^{10})}^{\frac{1}{2}}$ ⓑ ${(u^{10} v^{5})}^{\frac{4}{5}}$

Answer

ⓐ

r^{8} s^{5}

ⓑ

u^{8} v^{4}

9.8-587 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $\frac{a^{\frac{3}{4}} \cdot a^{- \frac{1}{4}}}{a^{- \frac{10}{4}}}$ ⓑ $\frac{b^{\frac{2}{3}} \cdot b}{b^{- \frac{7}{3}}}$

Answer

ⓐ

a^{3}

ⓑ

b^{4}

9.8-589 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $\frac{m^{\frac{7}{4}} \cdot m^{- \frac{5}{4}}}{m^{- \frac{2}{4}}}$ ⓑ $\frac{n^{\frac{3}{7}} \cdot n}{n^{- \frac{4}{7}}}$

Answer

ⓐ

m

ⓑ

n^{2}

9.8-591 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

$n^{\frac{2}{6}} \cdot n^{\frac{4}{6}}$

Answer

n

9.8-593 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${(b^{10})}^{\frac{3}{5}}$

Answer

b^{6}

9.8-595 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

$\frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}}$

Answer

\frac{1}{z^{2}}

9.8-597 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${(64 s^{\frac{3}{5}})}^{\frac{1}{6}}$

Answer

2 s^{\frac{1}{10}}

9.8-599 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${(u^{12} v^{18})}^{\frac{1}{6}}$

Answer

u^{2} v^{3}

9.8-601 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify $242^{\frac{1}{2}}$ to find the length of each side of his patio.Round to the nearest tenth of a foot.

Answer

15.6 feet

9.8-603 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify $\frac{1024^{\frac{1}{2}}}{16^{\frac{1}{2}}}$ to find how many seconds it took for the flare to reach the water.

Answer

8 seconds

9.8-605 §9.8 source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Explain why the expression ${(−16)}^{\frac{3}{2}}$ cannot be evaluated.

Answer

Answers will vary.

9.R-607 §9.R source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{144}$

Answer

9.R-609 §9.R source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$- \sqrt{81}$

Answer

−9

9.R-611 §9.R source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{−36}$

Answer

not a real number

9.R-613 §9.R source ↗

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{64 + 225}$

Answer

9.R-615 §9.R source ↗

Estimate Square Roots

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

$\sqrt{155}$

Answer

12 <

\sqrt{155} <

13

9.R-617 §9.R source ↗

Approximate Square Roots

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

$\sqrt{57}$

Answer

7.55

9.R-619 §9.R source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{64 b^{2}}$

Answer

8 b

9.R-621 §9.R source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

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

Answer

15 m n

9.R-623 §9.R source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{49 y^{2}}$

Answer

7 y

9.R-625 §9.R source ↗

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{121 c^{2} d^{2}}$

Answer

11 c d

9.R-627 §9.R source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{98}$

Answer

7 \sqrt{2}

9.R-629 §9.R source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{y^{19}}$

Answer

y^{9} \sqrt{y}

9.R-631 §9.R source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{36 n^{13}}$

Answer

6 n^{6} \sqrt{n}

9.R-633 §9.R source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{150 n^{7}}$

Answer

5 n^{3} \sqrt{6 n}

9.R-635 §9.R source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{108 r^{5} s^{3}}$

Answer

6 r^{2} s \sqrt{3 r s}

9.R-637 §9.R source ↗

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\frac{6 + \sqrt{72}}{6}$

Answer

1 + \sqrt{2}

9.R-639 §9.R source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{81}{36}}$

Answer

\frac{3}{2}

9.R-641 §9.R source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{y^{6}}{y^{2}}}$

Answer

y^{2}

9.R-643 §9.R source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{72 q^{8}}{2 q^{4}}}$

Answer

6 q^{2}

9.R-645 §9.R source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{26}{169}}$

Answer

\frac{\sqrt{26}}{13}

9.R-647 §9.R source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{36 r^{10}}{16 r^{5}}}$

Answer

\frac{3 r^{2} \sqrt{r}}{2}

9.R-649 §9.R source ↗

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{12 r^{5} s^{7}}{75 r^{2} s}}$

Answer

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

9.R-651 §9.R source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$5 \sqrt{5} + 7 \sqrt{5}$

Answer

12 \sqrt{5}

9.R-653 §9.R source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$6 \sqrt{m} - 2 \sqrt{m}$

Answer

4 \sqrt{m}

9.R-655 §9.R source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$8 \sqrt{13} + 2 \sqrt{3} + 3 \sqrt{13}$

Answer

11 \sqrt{13} + 2 \sqrt{3}

9.R-657 §9.R source ↗

Add and Subtract Like Square Roots

In the following exercises, simplify.

$2 \sqrt{3 r s} +$ $\sqrt{3 r s} -$ $5 \sqrt{r s}$

Answer

3 \sqrt{3 r s} - 5 \sqrt{r s}

9.R-659 §9.R source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{8} + 3 \sqrt{2}$

Answer

5 \sqrt{2}

9.R-661 §9.R source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{48} + \sqrt{75}$

Answer

9 \sqrt{3}

9.R-663 §9.R source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\frac{1}{3} \sqrt{27} - \frac{1}{8} \sqrt{192}$

Answer

0

9.R-665 §9.R source ↗

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$6 \sqrt{18 n^{4}} -$ $3 \sqrt{8 n^{4}} +$ $n^{2} \sqrt{50}$

Answer

17 n^{2} \sqrt{2}

9.R-667 §9.R source ↗

Multiply Square Roots

In the following exercises, simplify.

$2 \sqrt{2} \cdot 6 \sqrt{14}$

Answer

24 \sqrt{7}

9.R-669 §9.R source ↗

Multiply Square Roots

In the following exercises, simplify.

$(6 \sqrt{2 y}) (3 \sqrt{50 y^{3}})$

Answer

180 y^{2}

9.R-671 §9.R source ↗

Multiply Square Roots

In the following exercises, simplify.

${(\sqrt{8})}^{2}$

Answer

9.R-673 §9.R source ↗

Multiply Square Roots

In the following exercises, simplify.

$(2 \sqrt{5}) (5 \sqrt{5})$

Answer

9.R-675 §9.R source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$10 (2 - \sqrt{7})$

Answer

20 - 10 \sqrt{7}

9.R-677 §9.R source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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

Answer

13 - 2 \sqrt{2}

9.R-679 §9.R source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$(1 - 3 \sqrt{x}) (5 + 2 \sqrt{x})$

Answer

5 - 13 \sqrt{x} - 6 x

9.R-681 §9.R source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

${(1 + 6 \sqrt{p})}^{2}$

Answer

1 + 12 \sqrt{p} + 36 p

9.R-683 §9.R source ↗

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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

Answer

−19

9.R-685 §9.R source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{75}}{10}$

Answer

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

9.R-687 §9.R source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{48}}{\sqrt{27}}$

Answer

\frac{4}{3}

9.R-689 §9.R source ↗

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{20 y^{5}}}{\sqrt{2 y}}$

Answer

y^{2} \sqrt{10}

9.R-691 §9.R source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

$\frac{10}{\sqrt{15}}$

Answer

\frac{2 \sqrt{15}}{3}

9.R-693 §9.R source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

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

Answer

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

9.R-695 §9.R source ↗

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

$\sqrt{\frac{3}{28}}$

Answer

\frac{\sqrt{21}}{14}

9.R-697 §9.R source ↗

Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

$\frac{4}{4 + \sqrt{27}}$

Answer

\frac{16 - 12 \sqrt{3}}{−11}

9.R-699 §9.R source ↗

Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

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

Answer

−8 - 4 \sqrt{5}

9.R-701 §9.R source ↗

Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

$\frac{\sqrt{2}}{\sqrt{p} + \sqrt{3}}$

Answer

\frac{\sqrt{2 p} - \sqrt{6}}{p - 3}

9.R-703 §9.R source ↗

Solve Radical Equations

In the following exercises, solve the equation.

$\sqrt{7 z + 1} = 6$

Answer

9.R-705 §9.R source ↗

Solve Radical Equations

In the following exercises, solve the equation.

$\sqrt{6 m + 4} - 5 = 0$

Answer

\frac{7}{2}

9.R-707 §9.R source ↗

Solve Radical Equations

In the following exercises, solve the equation.

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

Answer

4 and 5

9.R-709 §9.R source ↗

Solve Radical Equations

In the following exercises, solve the equation.

$\sqrt{r - 4} - r = −10$

Answer

9.R-711 §9.R source ↗

Solve Radical Equations

In the following exercises, solve the equation.

$2 \sqrt{2 x - 7} - 4 = 8$

Answer

\frac{43}{2}

9.R-713 §9.R source ↗

Solve Radical Equations

In the following exercises, solve the equation.

$\sqrt{a} + 3 = \sqrt{a + 9}$

Answer

9.R-715 §9.R source ↗

Solve Radical Equations

In the following exercises, solve the equation.

$\sqrt{u} + 2 = \sqrt{u + 5}$

Answer

\frac{1}{16}

9.R-717 §9.R source ↗

Solve Radical Equations

In the following exercises, solve the equation.

$\sqrt{y + 5} + 1 = \sqrt{2 y + 3}$

Answer

9.R-719 §9.R source ↗

Use Square Roots in Applications

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

A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker.

Answer

7.5 seconds

9.R-721 §9.R source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[6]{64}$
ⓑ $\sqrt[3]{64}$

Answer

ⓐ 2 ⓑ 4

9.R-723 §9.R source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[9]{d^{9}}$
ⓑ $\sqrt[8]{v^{8}}$

Answer

ⓐ

d

ⓑ

| v |

9.R-725 §9.R source ↗

Simplify Expressions with Higher Roots

In the following exercises, simplify.

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

Answer

ⓐ

2 x^{2}

ⓑ

2 y^{2}

9.R-727 §9.R source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[9]{d^{9}}$
ⓑ

Answer

ⓐ

d

ⓑ

m \sqrt[11]{m^{6}}

9.R-729 §9.R source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[5]{64 c^{8}}$
ⓑ $\sqrt[4]{48 d^{7}}$

Answer

ⓐ

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

ⓑ

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

9.R-731 §9.R source ↗

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[3]{−500}$
ⓑ $\sqrt[4]{−16}$

Answer

ⓐ

−5 \sqrt[3]{4}

ⓑ not a real number

9.R-733 §9.R source ↗

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

$\sqrt[3]{\frac{w^{12}}{w^{2}}}$

Answer

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

9.R-735 §9.R source ↗

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

$\sqrt[3]{\frac{54 z^{9}}{2 z^{3}}}$

Answer

3 z^{2}

9.R-737 §9.R source ↗

Add and Subtract Higher Roots

In the following exercises, simplify.

$4 \sqrt[5]{20} - 2 \sqrt[5]{20}$

Answer

2 \sqrt[5]{20}

9.R-739 §9.R source ↗

Add and Subtract Higher Roots

In the following exercises, simplify.

$\sqrt[4]{1250} - \sqrt[4]{162}$

Answer

2 \sqrt[4]{2}

9.R-741 §9.R source ↗

Add and Subtract Higher Roots

In the following exercises, simplify.

$\sqrt[5]{96 t^{8}} +$ $\sqrt[5]{486 t^{4}}$

Answer

2 t \sqrt[5]{3 t^{3}} +

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

9.R-743 §9.R source ↗

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

In the following exercises, write as a radical expression.

$s^{\frac{1}{10}}$

Answer

9.R-745 §9.R source ↗

In the following exercises, write with a rational exponent.

$\sqrt[6]{v}$

Answer

v^{\frac{1}{6}}

9.R-747 §9.R source ↗

In the following exercises, write with a rational exponent.

$\sqrt[6]{10 z}$

Answer

{(10 z)}^{\frac{1}{6}}

9.R-749 §9.R source ↗

In the following exercises, simplify.

$32^{\frac{1}{5}}$

Answer

9.R-751 §9.R source ↗

In the following exercises, simplify.

${(125)}^{- \frac{1}{3}}$

Answer

\frac{1}{5}

9.R-753 §9.R source ↗

In the following exercises, simplify.

${(36)}^{- \frac{1}{2}}$

Answer

\frac{1}{6}

9.R-755 §9.R source ↗

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

In the following exercises, write with a rational exponent.

$\sqrt[5]{n^{8}}$

Answer

n^{\frac{8}{5}}

9.R-757 §9.R source ↗

In the following exercises, simplify.

$64^{\frac{5}{2}}$

Answer

32,768

9.R-759 §9.R source ↗

In the following exercises, simplify.

$81^{- \frac{5}{2}}$

Answer

\frac{1}{59,049}

9.R-761 §9.R source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

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

Answer

x^{8}

9.R-763 §9.R source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

${(16 s^{\frac{9}{4}})}^{\frac{1}{4}}$

Answer

2 s^{\frac{9}{16}}

9.R-765 §9.R source ↗

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

$\frac{z^{\frac{2}{3}} \cdot z^{- \frac{1}{3}}}{z^{- \frac{5}{3}}}$

Answer

z^{2}

9.T-767 §9.T source ↗

In the following exercises, simplify.

$\sqrt{169 m^{4} n^{2}}$

Answer

13 m^{2} | n |

9.T-769 §9.T source ↗

In the following exercises, simplify.

$3 \sqrt{13} + 5 \sqrt{2} + \sqrt{13}$

Answer

4 \sqrt{13} + 5 \sqrt{2}

9.T-771 §9.T source ↗

In the following exercises, simplify.

$(3 \sqrt{6 y}) (2 \sqrt{50 y^{3}})$

Answer

60 y^{2} \sqrt{3}

9.T-773 §9.T source ↗

In the following exercises, simplify.

${(1 - 2 \sqrt{q})}^{2}$

Answer

1 - 4 \sqrt{q} + 4 q

9.T-775 §9.T source ↗

In the following exercises, simplify.

ⓐ $\sqrt[4]{81 x^{12}}$
ⓑ $\sqrt[6]{64 y^{18}}$

Answer

ⓐ

3 x^{3}

ⓑ

2 y^{3}

9.T-777 §9.T source ↗

In the following exercises, simplify.

$\sqrt{\frac{14 y^{3}}{7 y}}$

Answer

y \sqrt{2}

9.T-779 §9.T source ↗

In the following exercises, simplify.

$\sqrt[4]{512} - 2 \sqrt[4]{32}$

Answer

0

9.T-781 §9.T source ↗

In the following exercises, simplify.

$49^{\frac{3}{2}}$

Answer

343

9.T-783 §9.T source ↗

In the following exercises, simplify.

$\frac{w^{\frac{3}{4}}}{w^{\frac{7}{4}}}$

Answer

\frac{1}{w}

9.T-785 §9.T source ↗

In the following exercises, rationalize the denominator.

$\frac{3}{2 \sqrt{6}}$

Answer

\frac{\sqrt{6}}{4}

9.T-787 §9.T source ↗

In the following exercises, solve.

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

Answer

9.T-789 §9.T source ↗

In the following exercise, solve.

A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.

Answer

6.1 seconds

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Problems and answers are adapted from Elementary Algebra 2e by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/elementary-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.