Question

Simplify each expression:

$ⓐ{\left(\frac{3}{5}\right)}^{-3}ⓑ{\left(-\frac{a}{b}\right)}^{-4}$

Short answer

The simplified expression of${\left(\frac{3}{5}\right)}^{-3}=\frac{125}{27}$

The simplified expression of${\left(-\frac{a}{b}\right)}^{-4}=\frac{b^4}{a^4}$

Step-by-step solution

Step 1. Given Information 

In the given question we have to simplify each expression:

$ⓐ{\left(\frac{3}{5}\right)}^{-3}ⓑ{\left(-\frac{a}{b}\right)}^{-4}$

Part (a) Step 1. The given expression is 35−3

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 35−3

Using the Quotient to a Negative Exponent Property,

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

${\left(\frac{3}{5}\right)}^{-3}={\left(\frac{5}{3}\right)}^3$

Simplifying

${\left(\frac{3}{5}\right)}^{-3}=\frac{5^3}{3^3}$

${\left(\frac{3}{5}\right)}^{-3}=\frac{125}{27}$

Part (b) Step 1. The given expression is -ab−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 (b) Step 2. The expression is -ab−4

Using the Quotient to a Negative Exponent Property,

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

${\left(-\frac{a}{b}\right)}^{-4}={\left(-\frac{b}{a}\right)}^4$

Simplifying

${\left(-\frac{a}{b}\right)}^{-4}=\frac{b^4}{a^4}$