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# 8.3E: Exercises

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### 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.