Rational Exponents: example 9.110

We will rewrite each expression as a radical first using the property, $a^{\frac{m}{n}}={\left(\sqrt[n]{a}\right)}^m$. This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

1. ⓐ
   $\begin{array}{cccc}& & & 9^{\frac{3}{2}}\hfill \\ \begin{array}{c}\text{The power of the radical is the numerator}\hfill \\ \text{of the exponent, 3. Since the denominator}\hfill \\ \text{of the exponent is 2, this is a square root.}\hfill \end{array}\hfill & & & {\left(\sqrt{9}\right)}^3\hfill \\ \\ \\ \text{Simplify.}\hfill & & & {\left(3\right)}^3\hfill \\ & & & 27\hfill \end{array}$
   
   
2. ⓑ
   $\begin{array}{cccc}& & & {125}^{\frac{2}{3}}\hfill \\ \begin{array}{c}\text{The power of the radical is the numerator}\hfill \\ \text{of the exponent, 2. The index of the radical}\hfill \\ \text{is the denominator of the exponent, 3.}\hfill \end{array}\hfill & & & {\left(\sqrt[3]{125}\right)}^2\hfill \\ \\ \\ \text{Simplify.}\hfill & & & {\left(5\right)}^2\hfill \\ & & & 25\hfill \end{array}$
   
   
3. ⓒ
   $\begin{array}{cccc}& & & {81}^{\frac{3}{4}}\hfill \\ \begin{array}{c}\text{The power of the radical is the numerator}\hfill \\ \text{of the exponent, 3. The index of the radical}\hfill \\ \text{is the denominator of the exponent, 4.}\hfill \end{array}\hfill & & & {\left(\sqrt[4]{81}\right)}^3\hfill \\ \\ \\ \text{Simplify.}\hfill & & & {\left(3\right)}^3\hfill \\ & & & 27\hfill \end{array}$