Rational Exponents: example 9.108

ⓐ
$\begin{array}{cccc}& & & {\left(\mathrm{-16}\right)}^{\frac{1}{4}}\hfill \\ \text{Rewrite as a fourth root.}\hfill & & & \sqrt[4]{\mathrm{-16}}\hfill \\ \text{There is no real number whose fourth power is}\mathrm{-16}.\hfill & & & \end{array}$

ⓑ
$\begin{array}{cccc}& & & \text{−}{16}^{\frac{1}{4}}\hfill \\ \begin{array}{c}\text{The exponent only applies to the 16.}\hfill \\ \text{Rewrite as a fourth root.}\hfill \end{array}\hfill & & & \text{−}\sqrt[4]{16}\hfill \\ \text{Rewrite}16\text{as}{2}^4.\hfill & & & \text{−}\sqrt[4]{2^4}\hfill \\ \text{Simplify.}\hfill & & & \mathrm{-2}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & {\left(16\right)}^{-\frac{1}{4}}\hfill \\ \text{Rewrite using the property}{a}^{\text{−}n}=\frac{1}{a^n}.\hfill & & & \frac{1}{{\left(16\right)}^{\frac{1}{4}}}\hfill \\ \text{Rewrite as a fourth root.}\hfill & & & \frac{1}{\sqrt[4]{16}}\hfill \\ \text{Rewrite}16\text{as}{2}^4.\hfill & & & \frac{1}{\sqrt[4]{2^4}}\hfill \\ \text{Simplify.}\hfill & & & \frac{1}{2}\hfill \end{array}$