Chapter 11: Q. 8.61 (page 1069)

Simplify:
(a) $27^{\frac{2}{3}}$
(b) $81^{- \frac{3}{2}}$
(c) $16^{- \frac{3}{4}}$

Short Answer

Expert verified

The simplification of $27^{\frac{2}{3}}$ is $9$

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

The simplification of $16^{- \frac{3}{4}}$ is $\frac{1}{8}$

Step by step solution

Part (a) Step 1. Given

An expression $27^{\frac{2}{3}}$

To simplify the given expression

Part (a) Step 2. Change to radical form

The power of the radical is the numerator of the exponent, $2$ .
The index of the radical is the denominator of the exponent, $3$ .

$27^{\frac{2}{3}} = {(\sqrt[3]{27})}^{2}$

$= {(\sqrt[3]{3^{3}})}^{2}$

$= 3^{2}$

$= 9$

Part (b) Step 1. Given

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

To simplify the given expression

Part (b) Step 2. Change to radical form

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

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

The power of the radical is the numerator of the exponent, $3$ .
The index of the radical is the denominator of the exponent, $2$ .

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

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

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

$= \frac{1}{729}$

Part (c) Step 1. Given

An expression $16^{- \frac{3}{4}}$

To simplify the given expression

Part (c) Step 2. Change to radical form

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

$16^{- \frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}}$

The power of the radical is the numerator of the exponent, $3$ .
The index of the radical is the denominator of the exponent, $4$ .

$16^{- \frac{3}{4}} = \frac{1}{{(\sqrt[4]{16})}^{3}}$

$= \frac{1}{{(\sqrt[4]{2^{4}})}^{3}}$

$= \frac{1}{2^{3}}$

$= \frac{1}{8}$

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Simplify:
(a) $27^{\frac{2}{3}}$
(b) $81^{- \frac{3}{2}}$
(c) $16^{- \frac{3}{4}}$

Short Answer

Step by step solution

Part (a) Step 1. Given

Part (a) Step 2. Change to radical form

Part (b) Step 1. Given

Part (b) Step 2. Change to radical form

Part (c) Step 1. Given

Part (c) Step 2. Change to radical form

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Simplify:(a)2723(b)81-32(c)16-34

Short Answer

Step by step solution

Part (a) Step 1. Given

Part (a) Step 2. Change to radical form

Part (b) Step 1. Given

Part (b) Step 2. Change to radical form

Part (c) Step 1. Given

Part (c) Step 2. Change to radical form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Geometry

Theoretical and Mathematical Physics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Simplify:
(a) $27^{\frac{2}{3}}$
(b) $81^{- \frac{3}{2}}$
(c) $16^{- \frac{3}{4}}$