Question

Simplify:

(a)$-{81}^{\frac{3}{2}}$ -

(b) $-{81}^{-\frac{3}{2}}$

(c) ${(-81)}^{-\frac{3}{2}}$

Short answer

The simplification of $-{81}^{\frac{3}{2}}$is $-729$

The simplification of $-{81}^{-\frac{3}{2}}$is $-\frac{1}{729}$

The simplification of ${(-81)}^{-\frac{3}{2}}$ has no real root.

Step-by-step solution

Part (a) Step 1. Given

An expression$-{81}^{\frac{3}{2}}$

To simplify the given expression

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

Rewrite in radical form and simplify.

$-{81}^{\frac{3}{2}}=-{\left(\sqrt{81}\right)}^3$

$=-{\left(\sqrt{9^2}\right)}^3$

$=-9^3$

$=-729$

Part (b) Step 1. Given

An expression$-{81}^{-\frac{3}{2}}$

To simplify the given expression

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

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

$-{81}^{-\frac{3}{2}}=-\left(\frac{1}{{\left(81\right)}^{{\displaystyle \frac{3}{2}}}}\right)$

$=-\left(\frac{1}{{\left(\sqrt{81}\right)}^3}\right)$

$=-\left(\frac{1}{{\left(\sqrt{9^2}\right)}^3}\right)$

$=-\frac{1}{9^3}$

$=-\frac{1}{729}$

Part (c) Step 1. Given

An expression${(-81)}^{-\frac{3}{2}}$

To simplify the given expression

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

Rewrite in radical form and simplify

${(-81)}^{-\frac{3}{2}}=\frac{1}{(-81)\frac{3}{3}}$

$=\frac{1}{{\left(\sqrt{-81}\right)}^3}$

There is no real number whose square root is $-81$.

So it has no real root.