Question

Simplify each expression:

$ⓐ{\left(\frac{2}{3}\right)}^{-4}ⓑ{\left(-\frac{m}{n}\right)}^{-2}$

Short answer

The simplified expression of${\left(\frac{2}{3}\right)}^{-4}=\frac{81}{16}$

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

Step-by-step solution

Step 1. Given Information 

In the given question we have to simplify each expression:

$ⓐ{\left(\frac{2}{3}\right)}^{-4}ⓑ{\left(-\frac{m}{n}\right)}^{-2}$

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

Quotient to a Negative Power Property

If $a$and $b$are real numbers, $a\ne 0,b\ne 0$and n is an integer, then${\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^n$

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.

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

Simplifying

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

${\left(\frac{2}{3}\right)}^{-4}=\frac{81}{16}$

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

Quotient to a Negative Power Property

If $a$and $b$are real numbers, $a\ne 0,b\ne 0$and n is an integer, then${\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^n$

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.

${\left(-\frac{m}{n}\right)}^{-2}={\left(-\frac{n}{m}\right)}^2$

Simplifying

${\left(-\frac{m}{n}\right)}^{-2}=\frac{n^2}{m^2}$