Q 130E - Intermediate Algebra

Question

In the following exercises, simplify.

Part (a): ${625}^{\frac{1}{4}}$

Part (b): ${243}^{\frac{1}{5}}$

Part (c): ${32}^{\frac{1}{5}}$

Step-by-Step Solution

Answer

Part (a): On simplifying, we get $5$.

Part (b): On simplifying, we get $3$.

Part (c): On simplifying, we get $2$.

Part (a) Step 1. Rewrite as a root.

Consider the given question,

${625}^{\frac{1}{4}}$

Rewrite as a fourth root,

$=\sqrt[4]{625}$

Recognize $625$ is a perfect fourth power. Then on simplifying the expression,

$$\begin{gathered} =\sqrt[4]{5^4} \\ =5 \end{gathered}$$

Part (b) Step 1. Rewrite as a root.

Consider the given question,

${243}^{\frac{1}{5}}$

Rewrite as a fifth root,

$=\sqrt[5]{243}$

Recognize $243$is a perfect fifth power. Then on simplifying the expression,

$$\begin{gathered} =\sqrt[5]{3^5} \\ =3 \end{gathered}$$

Part (c) Step 1. Rewrite as a root.

Consider the given question,

${32}^{\frac{1}{5}}$

Rewrite as a fifth root,

$=\sqrt[5]{32}$

Recognize $32$ is a perfect fifth power. Then on simplifying the expression,

$$\begin{gathered} =\sqrt[5]{2^5} \\ =2 \end{gathered}$$