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Mathematics LibreTexts

8.2E: Exercises

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    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.

    This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions with a to the power of 1 divided by n.”, “simplify expression with a to the power of m divided by n”, and “use the laws of exponents to simplify expression with rational exponents”. The other columns are left blank so that the learner may indicate their mastery level for each topic.
    Figure 8.3.4

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

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