Question

In the following exercises, simplify.

(a) ${25}^{\frac{3}{2}}$

(b) $9^{\frac{-3}{2}}$

(c)${\left(-64\right)}^{\frac{2}{3}}$

Short answer

(a) The answer is$125$

(b) The answer is$\frac{1}{27}$

(c) The answer is$16$

Step-by-step solution

Step 1. Part (a) Simplify the expression 2532

We will use the property

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

$$\begin{gathered} {25}^{\frac{3}{2}} \\ =5^{2\times \frac{3}{2}} \\ =5^3 \\ =125 \end{gathered}$$

Step 2. Part (b) Simplify the expression 9-32

$$\begin{gathered} 9^{\frac{-3}{2}} \\ =3^{2\times \frac{-3}{2}} \\ =3^{-3} \\ =\frac{1}{3^3} \\ =\frac{1}{27} \end{gathered}$$

Step 3. Part (c) Simplify the expression -6423

$$\begin{gathered} {\left(-64\right)}^{\frac{2}{3}} \\ ={\left(-4\right)}^{3\times \frac{2}{3}} \\ ={\left(-4\right)}^2 \\ =16 \end{gathered}$$