Q. 37 - Intermediate Algebra

Question

In the following exercises, simplify using absolute values as necessary 

$ⓐ \sqrt[4]{m^8} ⓑ \sqrt[5]{n^{20}}$

Step-by-Step Solution

Answer

Part(a)$\sqrt[4]{m^8}=m^2$

Part(b)$\sqrt[5]{n^{20}}=n^4$

Part(a) Step 1. Given information

$ⓐ \sqrt[4]{m^8}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(m^2\right)}^4={\text{ m}}^8\\ \Rightarrow \sqrt[4]{m^8}\\ \Rightarrow \sqrt[4]{{\left(m^2\right)}^4}\\ \text{Since the index n is even = }\sqrt[n]{a^n}=\left|a\right|.\\ {\text{Since m}}^2{\text{ is positive, we do not need an m}}^2\text{absolute value sign. }\\ \Rightarrow \sqrt[4]{m^8}=m^2\end{array}$

Part(b) Step 1. Given information

$ⓑ \sqrt[5]{n^{20}}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(n^4\right)}^5={\text{ n}}^{20}\\ \Rightarrow \sqrt[5]{n^{20}}\\ \Rightarrow \sqrt[5]{{\left(n^4\right)}^5}\\ \text{Since the index n is odd = }\sqrt[n]{a^n}=\text{ a}.\\ \Rightarrow \sqrt[5]{n^{20}}=n^4\end{array}$