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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 11: Q. 5.31 (page 1069)

Simplify each expression:
$鈸� {(\frac{2}{3})}^{鈭� 4} 鈸� {(- \frac{m}{n})}^{鈭� 2}$

Short Answer

Expert verified

The simplified expression of ${(\frac{2}{3})}^{鈭� 4} = \frac{81}{16}$

The simplified expression oflocalid="1661181245341" ${(- \frac{m}{n})}^{鈭� 2} = \frac{n^{2}}{m^{2}}$

Step by step solution

01

Step 1. Given Information

In the given question we have to simplify each expression:

$鈸� {(\frac{2}{3})}^{鈭� 4} 鈸� {(- \frac{m}{n})}^{鈭� 2}$

02

Part (a) Step 1. The given expression is 23−4

Quotient to a Negative Power Property

If $a$ and $b$ are real numbers, $a 鈮� 0, b 鈮� 0$ and n is an integer, then ${(\frac{a}{b})}^{鈭� n} = {(\frac{b}{a})}^{n}$

03

Part (a) Step 2. The expression is 23−4

Using the Quotient to a Negative Exponent Property,

Take the reciprocal of the fraction and change the sign of the exponent.

${(\frac{2}{3})}^{鈭� 4} = {(\frac{3}{2})}^{4}$

Simplifying

${(\frac{2}{3})}^{鈭� 4} = \frac{3^{4}}{2^{4}}$

${(\frac{2}{3})}^{鈭� 4} = \frac{81}{16}$

04

Part (b) Step 1. The given expression is -mn−2

Quotient to a Negative Power Property

If $a$ and $b$ are real numbers, $a 鈮� 0, b 鈮� 0$ and n is an integer, then ${(\frac{a}{b})}^{鈭� n} = {(\frac{b}{a})}^{n}$

05

Part (b) Step 2. The expression is-mn−2

Using the Quotient to a Negative Exponent Property,

Take the reciprocal of the fraction and change the sign of the exponent.

${(- \frac{m}{n})}^{鈭� 2} = {(- \frac{n}{m})}^{2}$

Simplifying

${(- \frac{m}{n})}^{鈭� 2} = \frac{n^{2}}{m^{2}}$

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