Question

How to evaluate a number with a negative fraction exponent?

Answer 1

To evaluate a number with a negative fraction exponent, you need to take care of two things.
1) The negative exponent.
2) The fraction as an exponent.

1) Negative exponent

$a^{-n} = \dfrac{1}{a^n}$

2) Fractional exponent

$a^{\frac{m}{n}} = \sqrt[n] {a^m} = (\sqrt[n]{a})^m$

Example:

Evaluate

$8^{- \frac{2}{3}}$

First, take care of the negative exponent.

$8^{-\frac{2}{3}} = \dfrac{1}{8^{\frac{2}{3}}}=$

Now we take care of the fractional exponent by using a root.

$= \dfrac{1}{\sqrt[3] {8^2}} = \dfrac{1}{(\sqrt[3] {8})^2} =$

$= \dfrac{1}{2^2} = \dfrac{1}{4}$