Q. 34 - Intermediate Algebra

Question

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

$ⓐ\sqrt{a^{12}} ⓑ \sqrt{b^{26}}$

Step-by-Step Solution

Answer

Part(a)$\sqrt[]{a^{12}}=a^6$

Part(b)$\sqrt[]{b^{26}}=\left|{\text{b}}^{13}\right|.$

Part(a) Step 1. Given information

$ⓐ\sqrt{a^{12}}$

Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }{\left(a^m\right)}^2=a^{2m}\\ \to {\left(a^6\right)}^2={\text{ a}}^{2*6}\\ \to {\left(a^6\right)}^2={\text{ a}}^{12}\\ \text{Since the index n is even},\sqrt[n]{a^n}=\left|a\right|.\\ \text{Here n is an even number}\\ {\text{In this case the absolute value sign is not needed as a}}^6\text{ is positive.}\\ \sqrt[]{a^{12}}=a^6\end{array}$

Part(b) Step 1. Given information

$ⓑ \sqrt{b^{26}}$

Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }{\left(a^m\right)}^2=a^{2m}\\ \to {\left(b^{13}\right)}^2={\text{ b}}^{2*13}\\ \to {\left(b^{13}\right)}^2={\text{ b}}^{26}\\ \text{Since the index n is even},\sqrt[]{a^n}=\left|a\right|.\\ \sqrt[]{b^{26}}=\left|{\text{b}}^{13}\right|.\end{array}$