Q 135E - Intermediate Algebra

Question

In the following exercises, simplify.

Part (a): ${\left(-81\right)}^{\frac{1}{4}}$

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

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

Step-by-Step Solution

Answer

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

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

Part (c): On simplifying, we get $\frac{1}{3}$.

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

Consider the given question,

${\left(-81\right)}^{\frac{1}{4}}$

Rewrite as a fourth root,

$=\sqrt[4]{\left(-81\right)}$

On simplifying the expression,

$=\sqrt[4]{{\left(-3\right)}^4}$

The solution is not real.

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

Consider the given question,

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

The exponent only applies to the $81$.

Rewrite as a fourth root,

$=-\sqrt[4]{81}$

On simplifying the expression,

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

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

Consider the given question,

${\left(81\right)}^{-\frac{1}{4}}$

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

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

Rewrite as a fourth root,

$=\frac{1}{\sqrt[4]{81}}$

On simplifying the expression,

$$\begin{gathered} =\frac{1}{\sqrt[4]{3^4}} \\ =\frac{1}{3} \end{gathered}$$