Q. 5.32 - Intermediate Algebra

Question

Simplify each expression:

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

Step-by-Step Solution

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 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 3 5 − 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 3 5 − 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 - a b − 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 - a b − 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

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