Q. 5.39 - Intermediate Algebra

Question

Simplify each expression:

$ⓐ {\left(\frac{p}{10}\right)}^4 ⓑ {\left(\frac{m}{n}\right)}^{-7} ⓒ {\left(\frac{3a{b}^3}{c^2}\right)}^4 ⓓ {\left(\frac{3x}{-2y^3}\right)}^3$

Step-by-Step Solution

Answer

The simplified expression of ${\left(\frac{p}{10}\right)}^4=\frac{p^4}{10000}$

The simplified expression of ${\left(\frac{m}{n}\right)}^{-7}=\frac{n^7}{m^7}$

The simplified expression of ${\left(\frac{3a{b}^3}{c^2}\right)}^4=\frac{81a^4{b}^{12}}{c^8}$

The simplified expression of ${\left(\frac{3x}{-2y^3}\right)}^3=\frac{27x^3}{-8y^9}$

Step 1. Given Information

In the given question we have to simplify each expression:

$ⓐ {\left(\frac{p}{10}\right)}^4 ⓑ {\left(\frac{m}{n}\right)}^{-7} ⓒ {\left(\frac{3a{b}^3}{c^2}\right)}^4 ⓓ {\left(\frac{3x}{-2y^3}\right)}^3$

Part (a) Step 1. The given expression is p 10 4

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

${\left(\frac{p}{10}\right)}^4=\frac{p^4}{{10}^4}$

Simplify.

${\left(\frac{p}{10}\right)}^4=\frac{p^4}{10000}$

Part (b) Step 1. The given expression is m n − 7

Raise the numerator and denominator to the power.

${\left(\frac{m}{n}\right)}^{-7}=\frac{m^{-7}}{n^{-7}}$

Use the definition of negative exponent.

${\left(\frac{m}{n}\right)}^{-7}=\frac{1}{m^7}·\frac{n^7}{1}$

Multiply.

${\left(\frac{m}{n}\right)}^{-7}=\frac{n^7}{m^7}$

Part (c) Step 1. The given expression is 3 a b 3 c 2 4

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

${\left(\frac{3a{b}^3}{c^2}\right)}^4=\frac{{\left(3\right)}^4{\left(a\right)}^4{\left(b^3\right)}^4}{{\left(c^2\right)}^4}$

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

${\left(\frac{3a{b}^3}{c^2}\right)}^4=\frac{81a^4\left(b^{3·4}\right)}{\left(c^{2·4}\right)}$

${\left(\frac{3a{b}^3}{c^2}\right)}^4=\frac{81a^4{b}^{12}}{c^8}$

Part (d) Step 1. The given expression is 3 x − 2 y 3 3

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

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

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

${\left(\frac{3x}{-2y^3}\right)}^3=\frac{27x^3}{(-8)\left(y^{3·3}\right)}$

${\left(\frac{3x}{-2y^3}\right)}^3=\frac{27x^3}{(-8)\left(y^9\right)}$

${\left(\frac{3x}{-2y^3}\right)}^3=\frac{27x^3}{-8y^9}$