Exercises
- Rewrite each of the following radical expressions in terms of exponents, and simplify if possible.
$ $
$ $
$ $
$ $
$ ()^{3} $
$ (){4}(\sqrt[3]{a{2}})^{6} $
$ $
- Rewrite each of the following in terms of radicals.
- $ x^{1/5} $ b) $ y^{2/3} $ c) $ 3^{-1/4} $ d) $ (a + bc)^{3/8} $ e) $ 2^{0.5} $ f) $ w^{-1.5} $
Explain why $ = $ . [Hint: Rewrite this equation in terms of exponents.]
We use the previous exercise’s identity to pull factors out of radicals. (Ex: $ = = = 2 $ .) Rewrite the following expressions, making the number under the radical as small as possible:
- $ $ b) $ $ c) $ $ d) $ $ e) $ $
Explain why $ = $ .
- Compute $ $ , then compute $ + $ . Can radicals be distributed over addition?
- True or false: $ = a + 5 $ .
- Compute $ $ , then compute $ - $ . Can radicals be distributed over subtraction?
- True or false: $ =x-y $ .
- Simplify the following as much as possible:
\[ \sqrt[3]{\frac{64}{125}} \]
\[ \sqrt[3]{-1000} \]
\[ \mathsf{d})\sqrt{2}+\sqrt{8} \]
\[ \sqrt{\frac{200}{144}} \]
\[ )\sqrt{3-\left(\frac{\sqrt{3}}{2}\right)^{2}} \]
\[ 36^{3/2} \]
\[ 32^{4/5} \]
- 216^{2/3}
\[ 100^{5/2} \]
\[ \mathrm{j)}\;7^{2/5}\cdot7^{8/5} \]
\[ (49a^{8}b^{-4})^{1/2} \]
\[ \sqrt[5]{x^{2}y}\cdot\sqrt[5]{x^{3}y^{4}} \]
\[ \mathsf{m})\left(x^{1/2}+y^{1/2}\right)\left(x^{1/2}-y^{1/2}\right) \]
\[ \mathsf{n})a^{-1/6}\left[a^{2/3}\left(\frac{a^{2/3}}{a^{1/4}}\right)^{6}\right]^{1/3} \]
\[ 0)\sqrt{y}\left(\frac{x^{2}y^{-3}}{y^{3}}\right)\left(\frac{y^{13/2}}{x}\right) \]
- A simple trick for removing square roots from a fraction’s denominator is called rationalizing the denominator. It comes in two basic versions. Both rely on the old “multiply by one” trick. I’ll explain each with an example.
Example
\[ \frac{5}{\sqrt{7}}=\left(\frac{5}{\sqrt{7}}\right)\left(\frac{\sqrt{7}}{\sqrt{7}}\right)=\frac{5\sqrt{7}}{7}. \]
The second version is a bit more elaborate, and requires the difference of squares identity.
\[ \frac{3}{1-\sqrt{5}}=\left(\frac{3}{1-\sqrt{5}}\right)\left(\frac{1+\sqrt{5}}{1+\sqrt{5}}\right)=\frac{3+3\sqrt{5}}{1^{2}-\left(\sqrt{5}\right)^{2}}=\frac{3+3\sqrt{5}}{-4} \]
In the second version, we always multiply the top and bottom by the so-called conjugate of the bottom. Rationalize the denominator in the following expressions:
\[ \begin{aligned}a)\frac{2}{\sqrt{2}}\qquad&b)\frac{30}{\sqrt{6}}\qquad c)\frac{14}{\sqrt{7}}\qquad d)\frac{3x}{3+\sqrt{x}}\qquad e)\frac{2}{\sqrt{7}+\sqrt{5}}\qquad f)\frac{\sqrt{6}+2}{\sqrt{6}-2}\end{aligned} \]
Rationalize the numerator in the following expressions.
\[ g)\frac{5\sqrt{3}}{9}\qquad h)\frac{2-\sqrt{x}}{5\sqrt{x}} \]