Q 133E - Intermediate Algebra

Question

In the following exercises, simplify.

Part (a): ${\left(-216\right)}^{\frac{1}{3}}$

Part (b): $-{216}^{\frac{1}{3}}$

Part (c): ${\left(216\right)}^{-\frac{1}{3}}$

Step-by-Step Solution

Answer

Part (a): On simplifying, we get $-6$.

Part (b): On simplifying, we get $-6$.

Part (c): On simplifying, we get $\frac{1}{6}$.

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

Consider the given question,

${\left(-216\right)}^{\frac{1}{3}}$

Rewrite as a cube root,

$=\sqrt[3]{\left(-216\right)}$

On simplifying the expression,

$$\begin{gathered} =\sqrt[3]{{\left(-6\right)}^3} \\ =-6 \end{gathered}$$

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

Consider the given question,

$-{216}^{\frac{1}{3}}$

The exponent only applies to the $216$.

Rewrite as a cube root,

$=-\sqrt[3]{216}$

On simplifying the expression,

$$\begin{gathered} =-\sqrt[3]{6^3} \\ =-6 \end{gathered}$$

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

Consider the given question,

${\left(216\right)}^{-\frac{1}{3}}$

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

$=\frac{1}{{\left(216\right)}^{{\displaystyle \frac{1}{3}}}}$

Rewrite as a cube root,

$=\frac{1}{\sqrt[3]{216}}$

On simplifying the expression,

$$\begin{gathered} =\frac{1}{\sqrt[3]{6^3}} \\ =\frac{1}{6} \end{gathered}$$