
# 8.3E: Exercises


### Practice Makes Perfect

##### Exercise SET A: use the product property to simplify radical expressions

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

1. $$\sqrt{27}$$
2. $$\sqrt{80}$$
3. $$\sqrt{125}$$
4. $$\sqrt{96}$$
5. $$\sqrt{147}$$
6. $$\sqrt{450}$$
7. $$\sqrt{800}$$
8. $$\sqrt{675}$$
1. $$\sqrt[4]{32}$$
2. $$\sqrt[5]{64}$$
1. $$\sqrt[3]{625}$$
2. $$\sqrt[6]{128}$$
1. $$\sqrt[5]{64}$$
2. $$\sqrt[3]{256}$$
1. $$\sqrt[4]{3125}$$
2. $$\sqrt[3]{81}$$

1. $$3\sqrt{3}$$

3. $$5\sqrt{5}$$

5. $$7\sqrt{3}$$

7. $$20\sqrt{2}$$

9.

1. $$2 \sqrt[4]{2}$$
2. $$2 \sqrt[5]{2}$$

11.

1. $$2 \sqrt[5]{2}$$
2. $$4 \sqrt[3]{4}$$
##### Exercise SET B: use the product property to simplify radical expressions

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

1. $$\sqrt{y^{11}}$$
2. $$\sqrt[3]{r^{5}}$$
3. $$\sqrt[4]{s^{10}}$$
1. $$\sqrt{m^{13}}$$
2. $$\sqrt[5]{u^{7}}$$
3. $$\sqrt[6]{v^{11}}$$
1. $$\sqrt{n^{21}}$$
2. $$\sqrt[3]{q^{8}}$$
3. $$\sqrt[8]{n^{10}}$$
1. $$\sqrt{r^{25}}$$
2. $$\sqrt[5]{p^{8}}$$
3. $$\sqrt[4]{m^{5}}$$
1. $$\sqrt{125 r^{13}}$$
2. $$\sqrt[3]{108 x^{5}}$$
3. $$\sqrt[4]{48 y^{6}}$$
1. $$\sqrt{80 s^{15}}$$
2. $$\sqrt[5]{96 a^{7}}$$
3. $$\sqrt[6]{128 b^{7}}$$
1. $$\sqrt{242 m^{23}}$$
2. $$\sqrt[4]{405 m 10}$$
3. $$\sqrt[5]{160 n^{8}}$$
1. $$\sqrt{175 n^{13}}$$
2. $$\sqrt[5]{512 p^{5}}$$
3. $$\sqrt[4]{324 q^{7}}$$
1. $$\sqrt{147 m^{7} n^{11}}$$
2. $$\sqrt[3]{48 x^{6} y^{7}}$$
3. $$\sqrt[4]{32 x^{5} y^{4}}$$
1. $$\sqrt{96 r^{3} s^{3}}$$
2. $$\sqrt[3]{80 x^{7} y^{6}}$$
3. $$\sqrt[4]{80 x^{8} y^{9}}$$
1. $$\sqrt{192 q^{3} r^{7}}$$
2. $$\sqrt[3]{54 m^{9} n^{10}}$$
3. $$\sqrt[4]{81 a^{9} b^{8}}$$
1. $$\sqrt{150 m^{9} n^{3}}$$
2. $$\sqrt[3]{81 p^{7} q^{8}}$$
3. $$\sqrt[4]{162 c^{11} d^{12}}$$
1. $$\sqrt[3]{-864}$$
2. $$\sqrt[4]{-256}$$
1. $$\sqrt[5]{-486}$$
2. $$\sqrt[6]{-64}$$
1. $$\sqrt[5]{-32}$$
2. $$\sqrt[8]{-1}$$
1. $$\sqrt[3]{-8}$$
2. $$\sqrt[4]{-16}$$
1. $$5+\sqrt{12}$$
2. $$\dfrac{10-\sqrt{24}}{2}$$
1. $$8+\sqrt{96}$$
2. $$\dfrac{8-\sqrt{80}}{4}$$
1. $$1+\sqrt{45}$$
2. $$\dfrac{3+\sqrt{90}}{3}$$
1. $$3+\sqrt{125}$$
2. $$\dfrac{15+\sqrt{75}}{5}$$

1.

1. $$\left|y^{5}\right| \sqrt{y}$$
2. $$r \sqrt[3]{r^{2}}$$
3. $$s^{2} \sqrt[4]{s^{2}}$$

3.

1. $$n^{10} \sqrt{n}$$
2. $$q^{2} \sqrt[3]{q^{2}}$$
3. $$|n| \sqrt[8]{n^{2}}$$

5.

1. $$5 r^{6} \sqrt{5 r}$$
2. $$3 x \sqrt[3]{4 x^{2}}$$
3. $$2|y| \sqrt[4]{3 y^{2}}$$

7.

1. $$11\left|m^{11}\right| \sqrt{2 m}$$
2. $$3 m^{2} \sqrt[4]{5 m^{2}}$$
3. $$2 n \sqrt[5]{5 n^{3}}$$

9.

1. $$7\left|m^{3} n^{5}\right| \sqrt{3 m n}$$
2. $$2 x^{2} y^{2} \sqrt[3]{6 y}$$
3. $$2|x y| \sqrt[4]{2 x}$$

11.

1. $$8\left|q r^{3}\right| \sqrt{3 q r}$$
2. $$3 m^{3} n^{3} \sqrt[3]{2 n}$$
3. $$3 a^{2} b^{2} \sqrt[4]{a}$$

13.

1. $$-6 \sqrt[3]{4}$$
2. not real

15.

1. $$-2$$
2. not real

17.

1. $$5+2 \sqrt{3}$$
2. $$5-\sqrt{6}$$

19.

1. $$1+3 \sqrt{5}$$
2. $$1+\sqrt{10}$$
##### Exercise Set C: use the quotient property to simplify radical expressions

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

1. $$\sqrt{\dfrac{45}{80}}$$
2. $$\sqrt[3]{\dfrac{8}{27}}$$
3. $$\sqrt[4]{\dfrac{1}{81}}$$
1. $$\sqrt{\dfrac{72}{98}}$$
2. $$\sqrt[3]{\dfrac{24}{81}}$$
3. $$\sqrt[4]{\dfrac{6}{96}}$$
1. $$\sqrt{\dfrac{100}{36}}$$
2. $$\sqrt[3]{\dfrac{81}{375}}$$
3. $$\sqrt[4]{\dfrac{1}{256}}$$
1. $$\sqrt{\dfrac{121}{16}}$$
2. $$\sqrt[3]{\dfrac{16}{250}}$$
3. $$\sqrt[4]{\dfrac{32}{162}}$$
1. $$\sqrt{\dfrac{x^{10}}{x^{6}}}$$
2. $$\sqrt[3]{\dfrac{p^{11}}{p^{2}}}$$
3. $$\sqrt[4]{\dfrac{q^{17}}{q^{13}}}$$
1. $$\sqrt{\dfrac{p^{20}}{p^{10}}}$$
2. $$\sqrt[5]{\dfrac{d^{12}}{d^{7}}}$$
3. $$\sqrt[8]{\dfrac{m^{12}}{m^{4}}}$$
1. $$\sqrt{\dfrac{y^{4}}{y^{8}}}$$
2. $$\sqrt[5]{\dfrac{u^{21}}{u^{11}}}$$
3. $$\sqrt[6]{\dfrac{v^{30}}{v^{12}}}$$
1. $$\sqrt{\dfrac{q^{8}}{q^{14}}}$$
2. $$\sqrt[3]{\dfrac{r^{14}}{r^{5}}}$$
3. $$\sqrt[4]{\dfrac{c^{21}}{c^{9}}}$$
1. $$\sqrt{\dfrac{96 x^{7}}{121}}$$
2. $$\sqrt{\dfrac{108 y^{4}}{49}}$$
3. $$\sqrt{\dfrac{300 m^{5}}{64}}$$
4. $$\sqrt{\dfrac{125 n^{7}}{169}}$$
5. $$\sqrt{\dfrac{98 r^{5}}{100}}$$
6. $$\sqrt{\dfrac{180 s^{10}}{144}}$$
7. $$\sqrt{\dfrac{28 q^{6}}{225}}$$
8. $$\sqrt{\dfrac{150 r^{3}}{256}}$$
1. $$\sqrt{\dfrac{75 r^{9}}{s^{8}}}$$
2. $$\sqrt[3]{\dfrac{54 a^{8}}{b^{3}}}$$
3. $$\sqrt[4]{\dfrac{64 c^{5}}{d^{4}}}$$
1. $$\sqrt{\dfrac{72 x^{5}}{y^{6}}}$$
2. $$\sqrt[5]{\dfrac{96 r^{11}}{s^{5}}}$$
3. $$\sqrt[6]{\dfrac{128 u^{7}}{v^{12}}}$$
1. $$\sqrt{\dfrac{28 p^{7}}{q^{2}}}$$
2. $$\sqrt[3]{\dfrac{81 s^{8}}{t^{3}}}$$
3. $$\sqrt[4]{\dfrac{64 p^{15}}{q^{12}}}$$
1. $$\sqrt{\dfrac{45 r^{3}}{s^{10}}}$$
2. $$\sqrt[3]{\dfrac{625 u^{10}}{v^{3}}}$$
3. $$\sqrt[4]{\dfrac{729 c^{21}}{d^{8}}}$$
1. $$\sqrt{\dfrac{32 x^{5} y^{3}}{18 x^{3} y}}$$
2. $$\sqrt[3]{\dfrac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}$$
3. $$\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}$$
1. $$\sqrt{\dfrac{75 r^{6} s^{8}}{48 r s^{4}}}$$
2. $$\sqrt[3]{\dfrac{24 x^{8} y^{4}}{81 x^{2} y}}$$
3. $$\sqrt[4]{\dfrac{32 m^{9} n^{2}}{162 m n^{2}}}$$
1. $$\sqrt{\dfrac{27 p^{2} q}{108 p^{4} q^{3}}}$$
2. $$\sqrt[3]{\dfrac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}$$
3. $$\sqrt[6]{\dfrac{2 m^{9} n^{7}}{128 m^{3} n}}$$
1. $$\sqrt{\dfrac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}$$
2. $$\sqrt[3]{\dfrac{24 m^{9} n^{7}}{375 m^{4} n}}$$
3. $$\sqrt[4]{\dfrac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}$$
1. $$\dfrac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}$$
2. $$\dfrac{\sqrt[4]{64}}{\sqrt[4]{2}}$$
3. $$\dfrac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}$$
1. $$\dfrac{\sqrt{80 q^{5}}}{\sqrt{5 q}}$$
2. $$\dfrac{\sqrt[3]{-625}}{\sqrt[3]{5}}$$
3. $$\dfrac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}$$
1. $$\dfrac{\sqrt{50 m^{7}}}{\sqrt{2 m}}$$
2. $$\sqrt[3]{\dfrac{1250}{2}}$$
3. $$\sqrt[4]{\dfrac{486 y^{9}}{2 y^{3}}}$$
1. $$\dfrac{\sqrt{72 n^{11}}}{\sqrt{2 n}}$$
2. $$\sqrt[3]{\dfrac{162}{6}}$$
3. $$\sqrt[4]{\dfrac{160 r^{10}}{5 r^{3}}}$$

1.

1. $$\dfrac{3}{4}$$
2. $$\dfrac{2}{3}$$
3. $$\dfrac{1}{3}$$

3.

1. $$\dfrac{5}{3}$$
2. $$\dfrac{3}{5}$$
3. $$\dfrac{1}{4}$$

5.

1. $$x^{2}$$
2. $$p^{3}$$
3. $$|q|$$

7.

1. $$\dfrac{1}{y^{2}}$$
2. $$u^{2}$$
3. $$|v^{3}|$$

9. $$\dfrac{4\left|x^{3}\right| \sqrt{6 x}}{11}$$

11. $$\dfrac{10 m^{2} \sqrt{3 m}}{8}$$

13. $$\dfrac{7 r^{2} \sqrt{2 r}}{10}$$

15. $$\dfrac{2\left|q^{3}\right| \sqrt{7}}{15}$$

17.

1. $$\dfrac{5 r^{4} \sqrt{3 r}}{s^{4}}$$
2. $$\dfrac{3 a^{2} \sqrt[3]{2 a^{2}}}{|b|}$$
3. $$\dfrac{2|c| \sqrt[4]{4 c}}{|d|}$$

19.

1. $$\dfrac{2\left|p^{3}\right| \sqrt{7 p}}{|q|}$$
2. $$\dfrac{3 s^{2} \sqrt[3]{3 s^{2}}}{t}$$
3. $$\dfrac{2\left|p^{3}\right| \sqrt[4]{4 p^{3}}}{\left|q^{3}\right|}$$

21.

1. $$\dfrac{4|x y|}{3}$$
2. $$\dfrac{y^{2} \sqrt[3]{x}}{2}$$
3. $$\dfrac{|a b| \sqrt[4]{a}}{4}$$

23.

1. $$\dfrac{1}{2|p q|}$$
2. $$\dfrac{2 c d \sqrt[5]{2 d^{2}}}{5}$$
3. $$\dfrac{|m n| \sqrt[6]{2}}{2}$$

25.

1. $$\dfrac{3 p^{4} \sqrt{p}}{|q|}$$
2. $$2 \sqrt[4]{2}$$
3. $$2 x \sqrt[5]{2 x}$$

27.

1. $$5\left|m^{3}\right|$$
2. $$5 \sqrt[3]{5}$$
3. $$3|y| \sqrt[4]{3 y^{2}}$$
##### Exercise SET D: writing exercises
1. Explain why $$\sqrt{x^{4}}=x^{2}$$. Then explain why $$\sqrt{x^{16}}=x^{8}$$.
2. Explain why $$7+\sqrt{9}$$ is not equal to $$\sqrt{7+9}$$.
3. Explain how you know that $$\sqrt[5]{x^{10}}=x^{2}$$.
4. Explain why $$\sqrt[4]{-64}$$ is not a real number but $$\sqrt[3]{-64}$$ is.

## Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. After reviewing this checklist, what will you do to become confident for all objectives?

8.3E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.