Q. 29 - Intermediate Algebra

Question

In the following exercises, simplify using absolute values as necessary. 
$ⓐ \sqrt[4]{y^4} ⓑ \sqrt[7]{m^7}$

Step-by-Step Solution

Answer

Part(a)$\sqrt[4]{y^4}=\left|y\right|$

Part(b)$\sqrt[7]{m^7}=m$

Part(a) Step 1. Given information

$ⓐ \sqrt[4]{y^4}$

Simplify using absolute values as necessary. 

Step 2. Explanation

$\begin{array}{l}\text{We use the absolute value to be sure to get the positive root.}\\ \text{Since the index n is even},\sqrt[n]{a^n}=\left|a\right|.\\ \text{Here n=4 is an even number}\\ \sqrt[4]{y^4}=\left|y\right|\end{array}$

Part(b) Step 1. Given information

$\left(b\right) \sqrt[7]{m^7}$

Simplify using absolute values as necessary. 

Step 2. Explanation

$\begin{array}{l}\text{This is an odd indexed root so there is no need for an absolute value sign.}\\ \text{Since the index n is odd},\sqrt[n]{a^n}=a.\\ \text{Here n = 7 is an odd number}\\ \sqrt[7]{m^7}=m\\ \end{array}$