Question

In the following exercises, simplify.

(a) ${64}^{\frac{5}{2}}$

(b) ${81}^{\frac{-3}{2}}$

(c)${\left(-27\right)}^{\frac{2}{3}}$

Short answer

(a) The answer is$32,768$

(b) The answer is$\frac{1}{729}$

(c) The answer is$9$

Step-by-step solution

Step 1. Part (a) Simplify the expression 6452

We will use the property

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

$$\begin{gathered} {64}^{\frac{5}{2}} \\ ={\left(8\right)}^{2\times \frac{5}{2}} \\ =8^5 \\ =32,768 \end{gathered}$$

Step 2. Part (b) Simplify the expression 81-32

$$\begin{gathered} {\left(81\right)}^{\frac{-3}{2}} \\ =9^{2\times \frac{-3}{2}} \\ =9^{-3} \\ =\frac{1}{9^3} \\ =\frac{1}{729} \end{gathered}$$

Step 3. Part (c) Simplify the expression -2723

$$\begin{gathered} {\left(-27\right)}^{\frac{2}{3}} \\ ={\left(-3\right)}^{3\times \frac{2}{3}} \\ ={\left(-3\right)}^2 \\ =9 \end{gathered}$$