Q. 5.38 - Intermediate Algebra

Question

Simplify each expression:

$ⓐ {(-3y)}^3 ⓑ {(-8m^2{n}^3)}^0 ⓒ {(-4x^4)}^{-2} ⓓ {\left(2c^{-4}\right)}^3$

Step-by-Step Solution

Answer

The simplified expression of ${(-3y)}^3={-27y}^3$

The simplified expression of ${(-8m^2{n}^3)}^0=1$

The simplified expression of ${(-4x^4)}^{-2}=\frac{1}{16x^8}$

The simplified expression of style="max-width: none; vertical-align: -18px;" ${\left(2c^{-4}\right)}^3=\frac{8}{c^{12}}$

Step 1. Given Information

In the given question we have to simplify each expression:

$ⓐ {(-3y)}^3 ⓑ {(-8m^2{n}^3)}^0 ⓒ {(-4x^4)}^{-2} ⓓ {\left(2c^{-4}\right)}^3$

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

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

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

Simplify. 

${(-3y)}^3={-27y}^3$

Part (b) Step 1. The given expression is ( − 8 m 2 n 3 ) 0

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

${(-8m^2{n}^3)}^0={{(-8)}^0{\left(m^2\right)}^0\left(n^3\right)}^0$

Simplify 

${(-8m^2{n}^3)}^0=1·1·1$

Multiply. 

${(-8m^2{n}^3)}^0=1$

Part (c) Step 1. The given expression is ( − 4 x 4 ) − 2

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

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

Use the Power Property,

$$\begin{gathered} {(-4x^4)}^{-2}={(-4)}^{-2}·x^{4·(-2)} \\ {(-4x^4)}^{-2}={(-4)}^{-2}·x^{-8} \end{gathered}$$

Use the Definition of a negative exponent, $a^{-n}=\frac{1}{a^n}$

${(-4x^4)}^{-2}=\frac{1}{{(-4)}^2}·\frac{1}{x^8}$

Simplify. 

${(-4x^4)}^{-2}=\frac{1}{16}·\frac{1}{x^8}$

${(-4x^4)}^{-2}=\frac{1}{16x^8}$

Part (d) Step 1. The given expression is ( 2 c − 4 ) 3

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

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

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

Simplify.

${\left(2c^{-4}\right)}^3=8c^{-12}$

using, $a^{-n}=\frac{1}{a^n}$

${\left(2c^{-4}\right)}^3=8\frac{1}{c^{12}}$

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