Q. 5.40 - Intermediate Algebra

Question

Simplify each expression:

$ⓐ {\left(\frac{-2}{q}\right)}^3 ⓑ {\left(\frac{w}{x}\right)}^{-4} ⓒ {\left(\frac{x{y}^3}{3z^2}\right)}^2 ⓓ {\left(\frac{2m^{-2}}{n^{-2}}\right)}^3$

Step-by-Step Solution

Answer

The simplified expression of ${\left(\frac{-2}{q}\right)}^3=\frac{-8}{q^3}$

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

The simplified expression of ${\left(\frac{x{y}^3}{3z^2}\right)}^2=\frac{x^2{y}^6}{9z^4}$

The simplified expression of ${\left(\frac{2m^{-2}}{n^{-2}}\right)}^3=\frac{8n^6}{m^6}$

Step 1. Given Information

In the given question we have to simplify each expression:

$ⓐ {\left(\frac{-2}{q}\right)}^3 ⓑ {\left(\frac{w}{x}\right)}^{-4} ⓒ {\left(\frac{x{y}^3}{3z^2}\right)}^2 ⓓ {\left(\frac{2m^{-2}}{n^{-2}}\right)}^3$

Part (a) Step 1. The given expression is − 2 q 3

Use Quotient to a Power Property, ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

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

Simplify.

${\left(\frac{-2}{q}\right)}^3=\frac{-8}{q^3}$

Part (b) Step 1. The given expression is w x − 4

Raise the numerator and denominator to the power.

${\left(\frac{w}{x}\right)}^{-4}=\frac{w^{-4}}{x^{-4}}$

Use the definition of negative exponent.

${\left(\frac{w}{x}\right)}^{-4}=\frac{x^4}{1}·\frac{1}{w^4}$

Multiply.

${\left(\frac{w}{x}\right)}^{-4}=\frac{x^4}{w^4}$

Part (c) Step 1. The given expression is x y 3 3 z 2 2

Use Quotient to a Power Property, ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

${\left(\frac{x{y}^3}{3z^2}\right)}^2=\frac{x^2{\left(y^3\right)}^2}{{\left(3\right)}^2{\left(z^2\right)}^2}$

Use the Product to a Power Property, ${\left(ab\right)}^m=a^m{b}^m$

${\left(\frac{x{y}^3}{3z^2}\right)}^2=\frac{x^2\left(y^{3·2}\right)}{9(z^{2·2})}$

${\left(\frac{x{y}^3}{3z^2}\right)}^2=\frac{x^2{y}^6}{9z^4}$

Part (d) Step 1. The given expression is 2 m − 2 n − 2 3

Use Quotient to a Power Property, ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

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

Use the Product to a Power Property, ${\left(ab\right)}^m=a^m{b}^m$

${\left(\frac{2m^{-2}}{n^{-2}}\right)}^3=\frac{{\left(2\right)}^3\left(m^{-2·3}\right)}{\left(n^{-2·3}\right)}$

${\left(\frac{2m^{-2}}{n^{-2}}\right)}^3=\frac{8m^{-6}}{n^{-6}}$

Part (d) Step 2. Simplify using the Power Property, ( a m ) n = a m · n

Use the definition of negative exponent.

${\left(\frac{2m^{-2}}{n^{-2}}\right)}^3=\frac{8n^6}{1}·\frac{1}{m^6}$

Simplify.

${\left(\frac{2m^{-2}}{n^{-2}}\right)}^3=\frac{8n^6}{m^6}$