Question

In the following exercises, simplify.

Part (a): ${64}^{\frac{1}{3}}$

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

Part (c):${81}^{\frac{1}{4}}$

Short answer

Part (a): On simplifying, we get $4$.

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

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

Step-by-step solution

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

Consider the given question,

${64}^{\frac{1}{3}}$

Rewrite as a cube root,

$=\sqrt[3]{64}$

On simplifying the expression,

$=4$

Part (b) 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,

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

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

Consider the given question,

${81}^{\frac{1}{4}}$

Rewrite as a fourth root,

$=\sqrt[4]{81}$

Recognize $81$is a perfect fourth power. Then on simplifying the expression,

$$\begin{gathered} =\sqrt[4]{3^4} \\ =3 \end{gathered}$$