5.1E: Exercises
- Page ID
- 104849
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Practice Makes Perfect
Rationalize the denominator
- \(\frac{1}{\sqrt{7}}\)
- \(\frac{1}{\sqrt{8}}\)
- \(\frac{4}{\sqrt{11}}\)
- \(\frac{2}{\sqrt{8}}\)
- \(\frac{1}{1+\sqrt{2}}\)
- \(\frac{1}{3-\sqrt{3}}\)
- \(\frac{1}{2-\sqrt{8}}\)
- Answer
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- \(\frac{\sqrt{7}}{7}\)
- \(\frac{\sqrt{2}}{4}\)
- \(\frac{4\sqrt{11}}{11}\)
- \(\frac{2\sqrt{2}}{\sqrt{2}}\)
- \(-1+\sqrt{2}\)
- \(\frac{3+\sqrt{3}}{6}\)
- \(\frac{-1-\sqrt{2}}{2}\)
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. \(\sqrt[5]{u^{2}}\) b. \((\sqrt[3]{6 x})^{5}\) c. \(\sqrt[4]{\left(\dfrac{18 a}{5 b}\right)^{7}}\)
- Answer
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- a. \(m^{\frac{5}{2}}\) b. \((3 y)^{\frac{7}{3}}\) c. \(\left(\dfrac{4 x}{5 y}\right)^{\frac{3}{5}}\)
- 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. \(32^{\frac{2}{5}}\) b. \(27^{-\frac{2}{3}}\) c. \((-25)^{\frac{1}{2}}\)
- a. \(9^{\frac{3}{2}}\) b. \(-9^{-\frac{3}{2}}\) c. \(-9^{\frac{3}{2}}\)
- Answer
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- a. \(32,768\) b. \(\frac{1}{729}\) c. \(9\)
- a. \(4\) b. \(\frac{1}{9}\) c. 5i
- a. \(27\) b. \(-\frac{1}{27}\) c. \(-27\)
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. \(y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}\) b. \(\left(x^{12}\right)^{\frac{2}{3}}\) c. \(\dfrac{m^{\frac{5}{8}}}{m^{\frac{13}{8}}}\)
- a. \(\left(27 q^{\frac{3}{2}}\right)^{\frac{4}{3}}\) b. \(\left(a^{\frac{1}{3}} b^{\frac{2}{3}}\right)^{\frac{3}{2}}\)
- a. \(\left(16 u^{\frac{1}{3}}\right)^{\frac{3}{4}}\) b. \(\left(4 p^{\frac{1}{3}} q^{\frac{1}{2}}\right)^{\frac{3}{2}}\)
- a. \(\dfrac{r^{\frac{5}{2}} \cdot r^{-\frac{1}{2}}}{r^{-\frac{3}{2}}}\) 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. \(\dfrac{c^{\frac{5}{3}} \cdot c^{-\frac{1}{3}}}{c^{-\frac{2}{3}}}\) 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
-
- a. \(c^{\frac{7}{8}}\) b. \(p^{9}\) c. \(\frac{1}{r}\)
- a. \(y^{\frac{5}{4}}\) b. \(x^{8}\) c. \(\dfrac{1}{m}\)
- a. \(81 q^{2}\) b. \(a^{\frac{1}{2}} b\)
- a. \(8 u^{\frac{1}{4}}\) b. \(8 p^{\frac{1}{2}} q^{\frac{3}{4}}\)
- a. \(r^{\frac{7}{2}}\) b. \(\dfrac{6 s}{t}\)
- a. \(c^{2}\) b. \(\dfrac{2x}{3y}\)