Q. 32 - Intermediate Algebra

Question

In the following exercises, simplify using absolute values as necessary .

$ⓐ \sqrt{a^{14}} ⓑ \sqrt{w^{24}}$

Step-by-Step Solution

Answer

Part(a)$\sqrt[]{{\text{a}}^{14}}=\left|{\text{a}}^7\right|$

Part(b)$\sqrt[]{w^{24}}=w^{12}$

Part(a) Step 1. Given information

$ⓐ \sqrt{a^{14}}$

Simplify using absolute values as necessary. 

Step 2. Explanation

$\begin{array}{l}\text{Since }{\left(a^m\right)}^2=a^{2m}\\ \to {\left(a^7\right)}^2={\text{ a}}^{2*7}\\ \to {\left(a^7\right)}^2={\text{ a}}^{14}\\ \text{Since the index n is even},\sqrt[n]{a^n}=\left|a\right|.\\ \text{Here n is an even number}\\ \sqrt[]{{\text{a}}^{14}}=\left|{\text{a}}^7\right|\end{array}$

Part(b) Step 1. Given information

$ⓑ \sqrt{w^{24}}$

Simplify using absolute values as necessary. 

Step 2. Explanation

$\begin{array}{l}\text{Since }{\left(a^m\right)}^2=a^{2m}\\ \to {\left(w^{12}\right)}^2={\text{ a}}^{2*12}\\ \to {\left(w^{12}\right)}^2={\text{ w}}^{24}\\ \text{Since the index n is even},\sqrt[n]{a^n}=\left|a\right|.\\ \text{Here n is an even number}\\ \text{In this case the absolute value sign is not needed as }{w}^{12}\text{ is positive.}\\ \sqrt[]{w^{24}}=w^{12}\end{array}$