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5.1E: Exercises

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    Practice Makes Perfect

    Rationalize the denominator

    Rationalize the denominator

    1. \(\frac{1}{\sqrt{7}}\)
    2. \(\frac{1}{\sqrt{8}}\)
    3. \(\frac{4}{\sqrt{11}}\)
    4. \(\frac{2}{\sqrt{8}}\)
    5. \(\frac{1}{1+\sqrt{2}}\)
    6. \(\frac{1}{3-\sqrt{3}}\)
    7. \(\frac{1}{2-\sqrt{8}}\)
    Answer
    1. \(\frac{\sqrt{7}}{7}\)
    2. \(\frac{\sqrt{2}}{4}\)
    3. \(\frac{4\sqrt{11}}{11}\)
    4. \(\frac{2\sqrt{2}}{\sqrt{2}}\)
    5. \(-1+\sqrt{2}\)
    6. \(\frac{3+\sqrt{3}}{6}\)
    7. \(\frac{-1-\sqrt{2}}{2}\)
    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. \(\sqrt[5]{u^{2}}\) b. \((\sqrt[3]{6 x})^{5}\) c. \(\sqrt[4]{\left(\dfrac{18 a}{5 b}\right)^{7}}\)
    Answer
    1. a. \(m^{\frac{5}{2}}\) b. \((3 y)^{\frac{7}{3}}\) c. \(\left(\dfrac{4 x}{5 y}\right)^{\frac{3}{5}}\)
    2. a. \(u^{\frac{2}{5}}\) b. \((6 x)^{\frac{5}{3}}\) c. \(\left(\dfrac{18 a}{5 b}\right)^{\frac{7}{4}}\)
    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. \(32^{\frac{2}{5}}\) b. \(27^{-\frac{2}{3}}\) c. \((-25)^{\frac{1}{2}}\)
    3. a. \(9^{\frac{3}{2}}\) b. \(-9^{-\frac{3}{2}}\) c. \(-9^{\frac{3}{2}}\)
    Answer
    1. a. \(32,768\) b. \(\frac{1}{729}\) c. \(9\)
    2. a. \(4\) b. \(\frac{1}{9}\) c. 5i
    3. a. \(27\) b. \(-\frac{1}{27}\) c. \(-27\)
    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. \(y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}\) b. \(\left(x^{12}\right)^{\frac{2}{3}}\) c. \(\dfrac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}\)
    3. a. \(\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}\) b. \(\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}\)
    4. a. \(\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}\) b. \(\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}\)
    5. a. \(\dfrac{r^{\frac{5}{2}} \cdot r^{-\frac{1}{2}}}{r^{-\frac{3}{2}}}\) b. \(\left(\dfrac{36 s^{\frac{1}{5}} t^{-\frac{3}{2}}}{s^{-\frac{9}{5}} t^{\frac{1}{2}}}\right)^{\frac{1}{2}}\)
    6. a. \(\dfrac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}\) b. \(\left(\dfrac{8 x^{\frac{5}{3}} y^{-\frac{1}{2}}}{27 x^{-\frac{4}{3}} y^{\frac{5}{2}}}\right)^{\frac{1}{3}}\)
    Answer
    1. a. \(c^{\frac{7}{8}}\) b. \(p^{9}\) c. \(\frac{1}{r}\)
    2. a. \(y^{\frac{5}{4}}\) b. \(x^{8}\) c. \(\dfrac{1}{m}\)
    3. a. \(81 q^{2}\) b. \(a^{\frac{1}{2}} b\)
    4. a. \(8 u^{\frac{1}{4}}\) b. \(8 p^{\frac{1}{2}} q^{\frac{3}{4}}\)
    5. a. \(r^{\frac{7}{2}}\) b. \(\dfrac{6 s}{t}\)
    6. a. \(c^{2}\) b. \(\dfrac{2x}{3y}\)

    This page titled 5.1E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale via source content that was edited to the style and standards of the LibreTexts platform.