How to transform an expression to radical form (125)^(-1/3)?

Jul 25, 2015

You use the fact that ${x}^{\frac{1}{n}}$ is equivalent to $\sqrt[n]{x}$.

Explanation:

In order to convert your expression to radical form, you need to use two proprties of fractional exponents. The first one tells you that

${x}^{- y} = \frac{1}{x} ^ y$

In your case, the exponent is equal to $- \frac{1}{3}$, which means that the expression is equivalent to

${\left(125\right)}^{- \frac{1}{3}} = \frac{1}{125} ^ \left(\frac{1}{3}\right)$

Next, use the fact that raising a number to a fractional exponent that has the form $\frac{1}{n}$ is equivalent to extracting the ${n}^{\text{th}}$ root of that number.

In your case, the fractional exponent is $\frac{1}{3}$, which means that you have

$\frac{1}{125} ^ \left(\frac{1}{3}\right) = \frac{1}{\sqrt[3]{125}}$

As it turns out, $125$ is a perfect cube, which means that you can write it as

$125 = 25 \cdot 5 = 5 \cdot 5 \cdot 5 = {5}^{3}$

The expression becomes

$\frac{1}{\sqrt[3]{125}} = \frac{1}{\sqrt[3]{{5}^{3}}} = \textcolor{g r e e n}{\frac{1}{5}}$