Question

Simplify each expression:

$ⓐ{\left(2wx\right)}^5ⓑ{(-11p{q}^3)}^0ⓒ{\left(2b^3\right)}^{-4}ⓓ{\left(8a^{-4}\right)}^2$

Short answer

The simplified expression of${\left(2wx\right)}^5={32w^{5}x}^5$

The simplified expression of${(-11p{q}^3)}^0=1$

The simplified expression of${\left(2b^3\right)}^{-4}=\frac{1}{16b^{12}}$

The simplified expression oflocalid="1661181494626" ${\left(8a^{-4}\right)}^2=\frac{64}{a^8}$

Step-by-step solution

Step 1. Given Information

In the given question we have to simplify each expression:

$ⓐ{\left(2wx\right)}^5ⓑ{(-11p{q}^3)}^0ⓒ{\left(2b^3\right)}^{-4}ⓓ{\left(8a^{-4}\right)}^2$

Part (a) Step 1. The given expression is (2wx)5

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

${\left(2wx\right)}^5={2^5{w}^{5}x}^5$

Simplify.

${\left(2wx\right)}^5={32w^{5}x}^5$

Part (b) Step 1. The given expression is (−11pq3)0

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

${(-11p{q}^3)}^0={{(-11)}^0{\left(p\right)}^0\left(q^3\right)}^0$

Simplify.

localid="1644769778317" ${(-11p{q}^3)}^0=1·1·1$

Multiply.

${(-11p{q}^3)}^0=1$

Part (c) Step 1. The given expression is (2b3)−4

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

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

Use the Power Property, ${\left(a^m\right)}^n=a^{m·n}$

$$\begin{gathered} {\left(2b^3\right)}^{-4}={\left(2\right)}^{-4}·b^{3·(-4)} \\ {\left(2b^3\right)}^{-4}={\left(2\right)}^{-4}·b^{-12} \end{gathered}$$

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

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

Simplify.

$$\begin{gathered} {\left(2b^3\right)}^{-4}=\frac{1}{16}·\frac{1}{b^{12}} \\ {\left(2b^3\right)}^{-4}=\frac{1}{16b^{12}} \end{gathered}$$

Part (d) Step 1. The given expression is (8a−4)2

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

localid="1644769549931" ${\left(8a^{-4}\right)}^2={{\left(8\right)}^2\left(a^{-4}\right)}^2$

${\left(8a^{-4}\right)}^2={\left(8\right)}^2·a^{(-4)·2}$

Simplify.

localid="1661181641718" ${\left(8a^{-4}\right)}^2=64·a^{-8}$

Rewrite $a^{-8}$using, $a^{-n}=\frac{1}{a^n}$

localid="1661181650572" ${\left(8a^{-4}\right)}^2=64·\frac{1}{a^8}$

localid="1661181657892" ${\left(8a^{-4}\right)}^2=\frac{64}{a^8}$