
# 8.4E: 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. $$j^{\frac{1}{25}}$$

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. $$\sqrt[8]{40c}$$

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. $$-125^{\frac{1}{3}}$$

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. $$\sqrt[4]{\left(\dfrac{2 x y}{5 z}\right)^{2}}$$

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. $$(-64)^{\frac{3}{2}}$$

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. $$\left(\dfrac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{-\frac{1}{2}}}\right)^{\frac{1}{4}}$$

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.