8.2E: Exercises

Practice Makes Perfect

Exercise SET A: simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

1. a. $x^{\frac{1}{2}}$ b. $y^{\frac{1}{3}}$ c. $z^{\frac{1}{4}}$
2. a. $r^{\frac{1}{2}}$ b. $s^{\frac{1}{3}}$ c. $t^{\frac{1}{4}}$
3. a. $u^{\frac{1}{5}}$ b. $v^{\frac{1}{9}}$ c. $w^{\frac{1}{20}}$
4. a. $g^{\frac{1}{7}}$ b. $h^{\frac{1}{5}}$ c. $j^{\frac{1}{25}}$

Answer

1. a. $\sqrt{x}$ b. $\sqrt[3]{y}$ c. $\sqrt[4]{z}$

3. a. $\sqrt[5]{u}$ b. $\sqrt[9]{v}$ c. $\sqrt[20]{w}$

Exercise Set B: simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, write with a rational exponent.

1. a. $\sqrt[7]{x}$ b. $\sqrt[9]{y}$ c. $\sqrt[5]{f}$
2. a. $\sqrt[8]{4}$ b. $\sqrt[10]{s}$ c. $\sqrt[4]{t}$
3. a. $\sqrt[3]{7c}$ b. $\sqrt[7]{12d}$ c. $2\sqrt[4]{6b}$
4. a. $\sqrt[4]{5x}$ b. $\sqrt[8]{9y}$ c. $7\sqrt[5]{3z}$
5. a. $\sqrt{21p}$ b. $\sqrt[4]{8q}$ c. $4\sqrt[6]{36r}$
6. a. $\sqrt[3]{25a}$ b. $\sqrt{3b}$ c. $\sqrt[8]{40c}$

Answer

1. a. $x^{\frac{1}{7}}$ b. $y^{\frac{1}{9}}$ c. $f^{\frac{1}{5}}$

3. a. $(7 c)^{\frac{1}{4}}$ b. $(12 d)^{\frac{1}{7}}$ c. $2(6 b)^{\frac{1}{4}}$

5. a. $(21 p)^{\frac{1}{2}}$ b. $(8 q)^{\frac{1}{4}}$ c. $4(36 r)^{\frac{1}{6}}$

Exercise SET C: simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, simplify.

1. a. $81^{\frac{1}{2}}$ b. $125^{\frac{1}{3}}$ c. $64^{\frac{1}{2}}$
2. a. $625^{\frac{1}{4}}$ b. $243^{\frac{1}{5}}$ c. $32^{\frac{1}{5}}$
3. a. $16^{\frac{1}{4}}$ b. $16^{\frac{1}{2}}$ c. $625^{\frac{1}{4}}$
4. a. $64^{\frac{1}{3}}$ b. $32^{\frac{1}{5}}$ c. $81^{\frac{1}{4}}$
5. a. $(-216)^{\frac{1}{3}}$ b. $-216^{\frac{1}{3}}$ c. $(216)^{-\frac{1}{3}}$
6. a. $(-1000)^{\frac{1}{3}}$ b. $-1000^{\frac{1}{3}}$ c. $(1000)^{-\frac{1}{3}}$
7. a. $(-81)^{\frac{1}{4}}$ b. $-81^{\frac{1}{4}}$ c. $(81)^{-\frac{1}{4}}$
8. a. $(-49)^{\frac{1}{2}}$ b. $-49^{\frac{1}{2}}$ c. $(49)^{-\frac{1}{2}}$
9. a. $(-36)^{\frac{1}{2}}$ b. $-36^{\frac{1}{2}}$ c. $(36)^{-\frac{1}{2}}$
10. a. $(-16)^{\frac{1}{4}}$ b. $-16^{\frac{1}{4}}$ c. $16^{-\frac{1}{4}}$
11. a. $(-100)^{\frac{1}{2}}$ b. $-100^{\frac{1}{2}}$ c. $(100)^{-\frac{1}{2}}$
12. a. $(-32)^{\frac{1}{5}}$ b. $(243)^{-\frac{1}{5}}$ c. $-125^{\frac{1}{3}}$

Answer

1. a. $9$ b. $5$ c. $8$

3. a. $2$ b. $4$ c. $5$

5. a. $-6$ b. $-6$ c. $\frac{1}{6}$

7. a. not real b. $-3$ c. $\frac{1}{3}$

9. a. not real b. $-6$ c. $\frac{1}{6}$

11. a. not real b. $-10$ c. $\frac{1}{10}$

Exercise SET D: simplify expressions with $a^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

1. a. $\sqrt{m^{5}}$ b. $(\sqrt[3]{3 y})^{7}$ c. $\sqrt[5]{\left(\dfrac{4 x}{5 y}\right)^{3}}$
2. a. $\sqrt[4]{r^{7}}$ b. $(\sqrt[5]{2 p q})^{3}$ c. $\sqrt[4]{\left(\dfrac{12 m}{7 n}\right)^{3}}$
3. a. $\sqrt[5]{u^{2}}$ b. $(\sqrt[3]{6 x})^{5}$ c. $\sqrt[4]{\left(\dfrac{18 a}{5 b}\right)^{7}}$
4. a. $\sqrt[3]{a}$ b. $(\sqrt[4]{21 v})^{3}$ c. $\sqrt[4]{\left(\dfrac{2 x y}{5 z}\right)^{2}}$

Answer

1. a. $m^{\frac{5}{2}}$ b. $(3 y)^{\frac{7}{3}}$ c. $\left(\dfrac{4 x}{5 y}\right)^{\frac{3}{5}}$

3. a. $u^{\frac{2}{5}}$ b. $(6 x)^{\frac{5}{3}}$ c. $\left(\dfrac{18 a}{5 b}\right)^{\frac{7}{4}}$

Exercise SET E: simplify expressions with $a^{\frac{m}{n}}$

In the following exercises, simplify.

1. a. $64^{\frac{5}{2}}$ b. $81^{\frac{-3}{2}}$ c. $(-27)^{\frac{2}{3}}$
2. a. $25^{\frac{3}{2}}$ b. $9^{-\frac{3}{2}}$ c. $(-64)^{\frac{2}{3}}$
3. a. $32^{\frac{2}{5}}$ b. $27^{-\frac{2}{3}}$ c. $(-25)^{\frac{1}{2}}$
4. a. $100^{\frac{3}{2}}$ b. $49^{-\frac{5}{2}}$ c. $(-100)^{\frac{3}{2}}$
5. a. $-9^{\frac{3}{2}}$ b. $-9^{-\frac{3}{2}}$ c. $(-9)^{\frac{3}{2}}$
6. a. $-64^{\frac{3}{2}}$ b. $-64^{-\frac{3}{2}}$ c. $(-64)^{\frac{3}{2}}$

Answer

1. a. $32,768$ b. $\frac{1}{729}$ c. $9$

3. a. $4$ b. $\frac{1}{9}$ c. not real

5. a. $-27$ b. $-\frac{1}{27}$ c. not real

Exercise SET F: use the laws of exponents to simplify expressions with rational exponents

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

1. a. $c^{\frac{1}{4}} \cdot c^{\frac{5}{8}}$ b. $\left(p^{12}\right)^{\frac{3}{4}}$ c. $\dfrac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}$
2. a. $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$ b. $\left(b^{15}\right)^{\frac{3}{5}}$ c. $\dfrac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$
3. a. $y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}$ b. $\left(x^{12}\right)^{\frac{2}{3}}$ c. $\dfrac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}$
4. a. $q^{\frac{2}{3}} \cdot q^{\frac{5}{6}}$ b. $\left(h^{6}\right)^{\frac{4}{3}}$ c. $\dfrac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}$
5. a. $\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}$ b. $\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}$
6. a. $\left(64 s^{\frac{3}{7}}\right)^{\frac{1}{6}}$ b. $\left(m^{\frac{4}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{4}}$
7. a. $\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}$ b. $\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}$
8. a. $\left(625 n^{\frac{8}{3}}\right)^{\frac{3}{4}}$ b. $\left(9 x^{\frac{2}{5}} y^{\frac{3}{5}}\right)^{\frac{5}{2}}$
9. a. $\dfrac{r^{\frac{5}{2}} \cdot r^{-\frac{1}{2}}}{r^{-\frac{3}{2}}}$ b. $\left(\dfrac{36 s^{\frac{1}{5}} t^{-\frac{3}{2}}}{s^{-\frac{9}{5}} t^{\frac{1}{2}}}\right)^{\frac{1}{2}}$
10. a. $\dfrac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}$ b. $\left(\dfrac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}$
11. a. $\dfrac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}$ b. $\left(\dfrac{8 x^{\frac{5}{3}} y^{-\frac{1}{2}}}{27 x^{-\frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}$
12. a. $\dfrac{m^{\frac{7}{4}} \cdot m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}}$ b. $\left(\dfrac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{-\frac{1}{2}}}\right)^{\frac{1}{4}}$

Answer

1. a. $c^{\frac{7}{8}}$ b. $p^{9}$ c. $\frac{1}{r}$

3. a. $y^{\frac{5}{4}}$ b. $x^{8}$ c. $\dfrac{1}{m}$

5. a. $81 q^{2}$ b. $a^{\frac{1}{2}} b$

7. a. $8 u^{\frac{1}{4}}$ b. $8 p^{\frac{1}{2}} q^{\frac{3}{4}}$

9. a. $r^{\frac{7}{2}}$ b. $\dfrac{6 s}{t}$

11. a. $c^{2}$ b. $\dfrac{2x}{3y}$

Exercise SET G: writing exercises

1. Show two different algebraic methods to simplify $4^{\frac{3}{2}}$. Explain all your steps.
2. Explain why the expression $(-16)^{\frac{3}{2}}$ cannot be evaluated.

Answer

1. Answers will vary.

Self Check

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

Figure 8.3.4

b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?