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. $\sqrt{675}$
9. 1. $\sqrt[4]{32}$
   2. $\sqrt[5]{64}$
10. 1. $\sqrt[3]{625}$
    2. $\sqrt[6]{128}$
11. 1. $\sqrt[5]{64}$
    2. $\sqrt[3]{256}$
12. 1. $\sqrt[4]{3125}$
    2. $\sqrt[3]{81}$

Answer

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. 1. $\sqrt{y^{11}}$
   2. $\sqrt[3]{r^{5}}$
   3. $\sqrt[4]{s^{10}}$
2. 1. $\sqrt{m^{13}}$
   2. $\sqrt[5]{u^{7}}$
   3. $\sqrt[6]{v^{11}}$
3. 1. $\sqrt{n^{21}}$
   2. $\sqrt[3]{q^{8}}$
   3. $\sqrt[8]{n^{10}}$
4. 1. $\sqrt{r^{25}}$
   2. $\sqrt[5]{p^{8}}$
   3. $\sqrt[4]{m^{5}}$
5. 1. $\sqrt{125 r^{13}}$
   2. $\sqrt[3]{108 x^{5}}$
   3. $\sqrt[4]{48 y^{6}}$
6. 1. $\sqrt{80 s^{15}}$
   2. $\sqrt[5]{96 a^{7}}$
   3. $\sqrt[6]{128 b^{7}}$
7. 1. $\sqrt{242 m^{23}}$
   2. $\sqrt[4]{405 m 10}$
   3. $\sqrt[5]{160 n^{8}}$
8. 1. $\sqrt{175 n^{13}}$
   2. $\sqrt[5]{512 p^{5}}$
   3. $\sqrt[4]{324 q^{7}}$
9. 1. $\sqrt{147 m^{7} n^{11}}$
   2. $\sqrt[3]{48 x^{6} y^{7}}$
   3. $\sqrt[4]{32 x^{5} y^{4}}$
10. 1. $\sqrt{96 r^{3} s^{3}}$
    2. $\sqrt[3]{80 x^{7} y^{6}}$
    3. $\sqrt[4]{80 x^{8} y^{9}}$
11. 1. $\sqrt{192 q^{3} r^{7}}$
    2. $\sqrt[3]{54 m^{9} n^{10}}$
    3. $\sqrt[4]{81 a^{9} b^{8}}$
12. 1. $\sqrt{150 m^{9} n^{3}}$
    2. $\sqrt[3]{81 p^{7} q^{8}}$
    3. $\sqrt[4]{162 c^{11} d^{12}}$
13. 1. $\sqrt[3]{-864}$
    2. $\sqrt[4]{-256}$
14. 1. $\sqrt[5]{-486}$
    2. $\sqrt[6]{-64}$
15. 1. $\sqrt[5]{-32}$
    2. $\sqrt[8]{-1}$
16. 1. $\sqrt[3]{-8}$
    2. $\sqrt[4]{-16}$
17. 1. $5+\sqrt{12}$
    2. $\dfrac{10-\sqrt{24}}{2}$
18. 1. $8+\sqrt{96}$
    2. $\dfrac{8-\sqrt{80}}{4}$
19. 1. $1+\sqrt{45}$
    2. $\dfrac{3+\sqrt{90}}{3}$
20. 1. $3+\sqrt{125}$
    2. $\dfrac{15+\sqrt{75}}{5}$

Answer

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

Answer

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.

Answer

1. Answers may vary

3. Answers may vary

Self Check

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

Figure 8.2.1

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