Q. 41 - Intermediate Algebra

Question

In the following exercises, simplify using absolute values as necessary 

$ⓐ \sqrt{121m^{20}} ⓑ -\sqrt{64a^2}$

Step-by-Step Solution

Answer

Part(a)$\sqrt[]{121m^{20}}=11m^{10}$

Part(b)$-\sqrt[]{64a^2}=-8\left|a\right|$

Part(a) Step 1. Given information

$ⓐ \sqrt{121m^{20}}$

Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(11\left(m^{10}\right)\right)}^2={\text{ 121m}}^{20}\\ \Rightarrow \sqrt[]{121m^{20}}\\ \Rightarrow \sqrt[]{{\left(11\right(m^{10}\left)\right)}^2}\\ \text{Since the index n is even = }\sqrt[n]{a^n}=\left|a\right|.\\ {\text{Since m}}^{10}{\text{is positive, we do not need an m}}^{10}\text{absolute value sign.}\\ \Rightarrow \sqrt[]{121m^{20}}=11m^{10}\end{array}$

Part(b) Step 1. Given information

$ⓑ -\sqrt{64a^2}$

Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(8a\right)}^2={\text{ 64a}}^2\\ \Rightarrow -\sqrt[]{64a^2}\\ \Rightarrow \sqrt[]{{\left(8a\right)}^2}\\ \text{Since the index n is even = }\sqrt[n]{a^n}=\left|a\right|.\\ \Rightarrow -\sqrt[]{64a^2}=-8\left|a\right|\end{array}$