Question

Simplify each expression:

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

Short 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 oflocalid="1644772739462">(2c−4)3=8c12

Step-by-step solution

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 (−3y)3

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

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

Simplify.

role="math" localid="1644771786081" ${(-3y)}^3={-27y}^3$

Part (b) Step 1. The given expression is (−8m2n3)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 (−4x4)−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,

localid="1644772244344" $$\begin{gathered} {(-4x^4)}^{-2}={(-4)}^{-2}·x^{4·(-2)} \\ {(-4x^4)}^{-2}={(-4)}^{-2}·x^{-8} \end{gathered}$$

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.

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

role="math" localid="1644772776735" ${(-4x^4)}^{-2}=\frac{1}{16x^8}$

Part (d) Step 1. The given expression is (2c−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, $a^{-n}=\frac{1}{a^n}$

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

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