Q 138 - Intermediate Algebra

Question

In the following exercises, simplify.

Part (a): ${\left(-16\right)}^{\frac{1}{4}}$

Part (b): $-{16}^{\frac{1}{4}}$

Part (c): ${16}^{-\frac{1}{4}}$

Step-by-Step Solution

Answer

Part (a): On simplifying, we know the solution is not real.

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

Part (c): On simplifying, we get $\frac{1}{2}$.

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

Consider the given question,

${\left(-16\right)}^{\frac{1}{4}}$

Rewrite as a fourth root,

$=\sqrt[4]{\left(-16\right)}$

On simplifying the expression,

$=\sqrt[4]{{\left(-2\right)}^4}$

The solution is not real.

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

Consider the given question,

$-{16}^{\frac{1}{4}}$

The exponent only applies to the $16$.

Rewrite as a fourth root,

$=-\sqrt[4]{16}$

On simplifying the expression,

$$\begin{gathered} =-\sqrt[4]{2^4} \\ =-2 \end{gathered}$$

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

Consider the given question,

${16}^{-\frac{1}{4}}$

Rewrite using the property $a^{-n}=\frac{1}{a^n}$,

$=\frac{1}{{16}^{{\displaystyle \frac{1}{4}}}}$

Rewrite as a fourth root,

$=\frac{1}{\sqrt[4]{16}}$

On simplifying the expression,

$$\begin{gathered} =\frac{1}{\sqrt[4]{2^4}} \\ =\frac{1}{2} \end{gathered}$$