How do you simplify (3rd root of 5) divided by (sqrt of 5)?

Sep 21, 2015

$\frac{\sqrt[6]{3125}}{5}$

Explanation:

You won't be able to multiply or divide them until they have the same index. So to simplify this, we will make them have the same index.
$\frac{\sqrt[3]{5}}{\sqrt{5}}$

Using the Law of Indices, we can rewrite this as:
$= {5}^{\frac{1}{3}} / {5}^{\frac{1}{2}}$

As you can see, they both have fraction exponents now. We must now make those fraction exponents have the same denominator. You can use any multiple of 2 and 3, but the easiest to use is 6 (since that is the least common denominator).
$= {5}^{\left(\frac{1}{3}\right) \left(\frac{2}{2}\right)} / {5}^{\left(\frac{1}{2}\right) \left(\frac{3}{3}\right)}$
$= {5}^{\frac{2}{6}} / {5}^{\frac{3}{6}}$

Using the Law of Indices again, you can rewrite this as:
$= \sqrt[6]{{5}^{2} / {5}^{3}}$
$= \textcolor{red}{\sqrt[6]{\frac{1}{5}}}$

You can have that as your final answer. However, you can still rationalize this if you want. Simply multiply something that will remove the radical from the denominator.
$= \sqrt[6]{\frac{1}{5}}$
$= \frac{\sqrt[6]{1}}{\sqrt[6]{5}}$
$= \frac{\sqrt[6]{1}}{\sqrt[6]{5}} \cdot \frac{\sqrt[6]{{5}^{5}}}{\sqrt[6]{{5}^{5}}}$
$= \frac{\sqrt[6]{1 \cdot {5}^{5}}}{\sqrt[6]{{5}^{6}}}$
$= \frac{\sqrt[6]{{5}^{5}}}{5}$
$= \textcolor{red}{\frac{\sqrt[6]{3125}}{5}}$