8.4E: Exercises
Practice Makes Perfect
In the following exercises, write as a radical expression.
- a. \(x^{\frac{1}{2}}\) b. \(y^{\frac{1}{3}}\) c. \(z^{\frac{1}{4}}\)
- a. \(r^{\frac{1}{2}}\) b. \(s^{\frac{1}{3}}\) c. \(t^{\frac{1}{4}}\)
- a. \(u^{\frac{1}{5}}\) b. \(v^{\frac{1}{9}}\) c. \(w^{\frac{1}{20}}\)
- 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. \(\sqrt[20]{w}\)
In the following exercises, write with a rational exponent.
- a. \(\sqrt[7]{x}\) b. \(\sqrt[9]{y}\) c. \(\sqrt[5]{f}\)
- a. \(\sqrt[8]{4}\) b. \(\sqrt[10]{s}\) c. \(\sqrt[4]{t}\)
- a. \(\sqrt[3]{7c}\) b. \(\sqrt[7]{12d}\) c. \(2\sqrt[4]{6b}\)
- a. \(\sqrt[4]{5x}\) b. \(\sqrt[8]{9y}\) c. \(7\sqrt[5]{3z}\)
- a. \(\sqrt{21p}\) b. \(\sqrt[4]{8q}\) c. \(4\sqrt[6]{36r}\)
- 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. \(4(36 r)^{\frac{1}{6}}\)
In the following exercises, simplify.
- a. \(81^{\frac{1}{2}}\) b. \(125^{\frac{1}{3}}\) c. \(64^{\frac{1}{2}}\)
- a. \(625^{\frac{1}{4}}\) b. \(243^{\frac{1}{5}}\) c. \(32^{\frac{1}{5}}\)
- a. \(16^{\frac{1}{4}}\) b. \(16^{\frac{1}{2}}\) c. \(625^{\frac{1}{4}}\)
- a. \(64^{\frac{1}{3}}\) b. \(32^{\frac{1}{5}}\) c. \(81^{\frac{1}{4}}\)
- a. \((-216)^{\frac{1}{3}}\) b. \(-216^{\frac{1}{3}}\) c. \((216)^{-\frac{1}{3}}\)
- a. \((-1000)^{\frac{1}{3}}\) b. \(-1000^{\frac{1}{3}}\) c. \((1000)^{-\frac{1}{3}}\)
- a. \((-81)^{\frac{1}{4}}\) b. \(-81^{\frac{1}{4}}\) c. \((81)^{-\frac{1}{4}}\)
- a. \((-49)^{\frac{1}{2}}\) b. \(-49^{\frac{1}{2}}\) c. \((49)^{-\frac{1}{2}}\)
- a. \((-36)^{\frac{1}{2}}\) b. \(-36^{\frac{1}{2}}\) c. \((36)^{-\frac{1}{2}}\)
- a. \((-16)^{\frac{1}{4}}\) b. \(-16^{\frac{1}{4}}\) c. \(16^{-\frac{1}{4}}\)
- a. \((-100)^{\frac{1}{2}}\) b. \(-100^{\frac{1}{2}}\) c. \((100)^{-\frac{1}{2}}\)
- 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. \(\frac{1}{10}\)
In the following exercises, write with a rational exponent.
- a. \(\sqrt{m^{5}}\) b. \((\sqrt[3]{3 y})^{7}\) c. \(\sqrt[5]{\left(\dfrac{4 x}{5 y}\right)^{3}}\)
- a. \(\sqrt[4]{r^{7}}\) b. \((\sqrt[5]{2 p q})^{3}\) c. \(\sqrt[4]{\left(\dfrac{12 m}{7 n}\right)^{3}}\)
- a. \(\sqrt[5]{u^{2}}\) b. \((\sqrt[3]{6 x})^{5}\) c. \(\sqrt[4]{\left(\dfrac{18 a}{5 b}\right)^{7}}\)
- 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. \(\left(\dfrac{18 a}{5 b}\right)^{\frac{7}{4}}\)
In the following exercises, simplify.
- a. \(64^{\frac{5}{2}}\) b. \(81^{\frac{-3}{2}}\) c. \((-27)^{\frac{2}{3}}\)
- a. \(25^{\frac{3}{2}}\) b. \(9^{-\frac{3}{2}}\) c. \((-64)^{\frac{2}{3}}\)
- a. \(32^{\frac{2}{5}}\) b. \(27^{-\frac{2}{3}}\) c. \((-25)^{\frac{1}{2}}\)
- a. \(100^{\frac{3}{2}}\) b. \(49^{-\frac{5}{2}}\) c. \((-100)^{\frac{3}{2}}\)
- a. \(-9^{\frac{3}{2}}\) b. \(-9^{-\frac{3}{2}}\) c. \((-9)^{\frac{3}{2}}\)
- 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
In the following exercises, simplify. Assume all variables are positive.
- a. \(c^{\frac{1}{4}} \cdot c^{\frac{5}{8}}\) b. \(\left(p^{12}\right)^{\frac{3}{4}}\) c. \(\dfrac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}\)
- a. \(6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}\) b. \(\left(b^{15}\right)^{\frac{3}{5}}\) c. \(\dfrac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}\)
- a. \(y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}\) b. \(\left(x^{12}\right)^{\frac{2}{3}}\) c. \(\dfrac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}\)
- a. \(q^{\frac{2}{3}} \cdot q^{\frac{5}{6}}\) b. \(\left(h^{6}\right)^{\frac{4}{3}}\) c. \(\dfrac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}\)
- a. \(\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}\) b. \(\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}\)
- a. \(\left(64 s^{\frac{3}{7}}\right)^{\frac{1}{6}}\) b. \(\left(m^{\frac{4}{3}} n^{\frac{1}{2}}\right)^{\frac{3}{4}}\)
- a. \(\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}\) b. \(\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}\)
- a. \(\left(625 n^{\frac{8}{3}}\right)^{\frac{3}{4}}\) b. \(\left(9 x^{\frac{2}{5}} y^{\frac{3}{5}}\right)^{\frac{5}{2}}\)
- a. \(\dfrac{r^{\frac{5}{2}} \cdot r^{-\frac{1}{2}}}{r^{-\frac{3}{2}}}\) b. \(\left(\dfrac{36 s^{\frac{1}{5}} t^{-\frac{3}{2}}}{s^{-\frac{9}{5}} t^{\frac{1}{2}}}\right)^{\frac{1}{2}}\)
- a. \(\dfrac{a^{\frac{3}{4}} \cdot a^{-\frac{1}{4}}}{a^{-\frac{10}{4}}}\) b. \(\left(\dfrac{27 b^{\frac{2}{3}} c^{-\frac{5}{2}}}{b^{-\frac{7}{3}} c^{\frac{1}{2}}}\right)^{\frac{1}{3}}\)
- a. \(\dfrac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}\) b. \(\left(\dfrac{8 x^{\frac{5}{3}} y^{-\frac{1}{2}}}{27 x^{-\frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}\)
- 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. \(\dfrac{2x}{3y}\)
- Show two different algebraic methods to simplify \(4^{\frac{3}{2}}\). Explain all your steps.
- 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.
b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?