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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 8: Q. 8.64 (page 781)

Simplify:
(a) $- 81^{\frac{3}{2}}$ -
(b) $- 81^{- \frac{3}{2}}$
(c) ${(- 81)}^{- \frac{3}{2}}$

Short Answer

Expert verified

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

01

Part (a) Step 1. Given

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

To simplify the given expression

02

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

Rewrite in radical form and simplify.

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

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

$= - 9^{3}$

$= - 729$

03

Part (b) Step 1. Given

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

To simplify the given expression

04

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

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

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

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

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

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

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

05

Part (c) Step 1. Given

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

To simplify the given expression

06

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

Rewrite in radical form and simplify

${(- 81)}^{- \frac{3}{2}} = \frac{1}{(- 81)}$ ³³

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

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

So it has no real root.

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