Q. 147 - Intermediate Algebra

Question

In the following exercises, simplify.

(a) ${32}^{\frac{2}{5}}$

(b) ${27}^{-\frac{2}{3}}$

(c) ${\left(-25\right)}^{\frac{1}{2}}$

Step-by-Step Solution

Answer

(a) The answer is $4$

(b) The answer is $\frac{1}{9}$

(c) The answer is Not real solution.

Step 1. Part (a) Simplify the expression 32 2 5

We will use the property

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

$$\begin{gathered} {32}^{\frac{2}{5}} \\ =2^{5\times \frac{2}{5}} \\ =2^2 \\ =4 \end{gathered}$$

Step 2. Part (b) Simplify the expression 27 - 2 3

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

Step 3. Part (c) Simplify the expression - 25 1 2

$$\begin{gathered} {\left(-25\right)}^{\frac{1}{2}} \\ =\sqrt{-25} \end{gathered}$$

This is not real solution.