Question

In the following exercises, simplify.

Part (a):${625}^{\frac{1}{4}}$

Part (b):${243}^{\frac{1}{5}}$

Part (c):${32}^{\frac{1}{5}}$

Short answer

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

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

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

Step-by-step solution

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,

role="math" localid="1644764051727" $$\begin{gathered} =\sqrt[4]{5^4} \\ =5 \end{gathered}$$

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,

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