Question

Simplify:

(a) $-{16}^{\frac{3}{2}}$

(b) $-{16}^{-\frac{3}{2}}$

(c)${(-16)}^{-\frac{3}{2}}$

Short answer

The simplification of $-{16}^{\frac{3}{2}}$is $-64$

The simplification of $-{16}^{-\frac{3}{2}}$is $-\frac{1}{64}$

The simplification of ${(-16)}^{-\frac{3}{2}}$ has no real root.

Step-by-step solution

Part (a) Step 1. Given

An expression $-{16}^{\frac{3}{2}}$

To simplify the given expression

Part (a) Step 2. Rewrite in radical form and simplify

Rewrite in radical form and simplify.

$-{16}^{\frac{3}{2}}=-{\left(\sqrt{16}\right)}^3$

$=-{\left(\sqrt{4^2}\right)}^3$

$=-\left(4^3\right)$

$=-64$

Part (b) Step 1. Given

An expression $-{16}^{-\frac{3}{2}}$

To simplify the given expression

Part (b) Step 2. Rewrite in radical form and simplify

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

$-{16}^{-\frac{3}{2}}=-\left(\frac{1}{{16}^{{\displaystyle \frac{3}{2}}}}\right)$

Rewrite in radical form,

$=-\left(\frac{1}{{\left(\sqrt{16}\right)}^3}\right)$

$=-\left(\frac{1}{{\left(\sqrt{4^2}\right)}^3}\right)$

$=-\left(\frac{1}{4^3}\right)$

$=-\frac{1}{64}$

Part (c) Step 1. Given

An expression${(-16)}^{-\frac{3}{2}}$

To simplify the given expression

Part (c) Step 2. Rewrite in radical form and simplify

Rewrite in radical form and simplify
${(-16)}^{-\frac{3}{2}}=\frac{1}{{(-16)}^{{\displaystyle \frac{3}{2}}}}$

$=\frac{1}{{\left(\sqrt{-16}\right)}^3}$

There is no real number whose square root is $-16$.

So it has no real root.