Question

In the following exercises, simplify.

Part (a):${\left(-1000\right)}^{\frac{1}{3}}$

Part (b):$-{1000}^{\frac{1}{3}}$

Part (c): ${\left(1000\right)}^{-\frac{1}{3}}$

Short answer

Part (a): On simplifying, we get $-10$.

Part (b): On simplifying, we get $-10$.

Part (c): On simplifying, we get $\frac{1}{10}$.

Step-by-step solution

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

Consider the given question,

${\left(-1000\right)}^{\frac{1}{3}}$

Rewrite as a cube root,

localid="1644767968693" $=\sqrt[3]{\left(-1000\right)}$

On simplifying the expression,

localid="1644770332482" $$\begin{gathered} =\sqrt[3]{{\left(-10\right)}^3} \\ =-10 \end{gathered}$$

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

Consider the given question,

$-{1000}^{\frac{1}{3}}$

The exponent only applies to the $1000$.

Rewrite as a cube root,

role="math" localid="1644768108079" $=-\sqrt[3]{1000}$

On simplifying the expression,

$$\begin{gathered} =-\sqrt[3]{{\left(10\right)}^3} \\ =-10 \end{gathered}$$

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

Consider the given question,

${\left(1000\right)}^{-\frac{1}{3}}$

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

$=\frac{1}{{\left(1000\right)}^{{\displaystyle \frac{1}{3}}}}$

Rewrite as a cube root,

$=\frac{1}{\sqrt[3]{1000}}$

On simplifying the expression,

$$\begin{gathered} =\frac{1}{\sqrt[3]{{\left(10\right)}^3}} \\ =\frac{1}{10} \end{gathered}$$