Q. 38 - Intermediate Algebra

Question

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

$ⓐ \sqrt[6]{r^{12}} ⓑ \sqrt[3]{s^{30}}$

Step-by-Step Solution

Answer

Part(a)$\sqrt[6]{r^{12}}=r^2$

Part(b)$\sqrt[3]{s^{30}}=s^{10}$

Part(a) Step 1. Given information

$ⓐ \sqrt[6]{r^{12}}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(r^2\right)}^6={\text{ r}}^{12}\\ \Rightarrow \sqrt[6]{r^{12}}\\ \Rightarrow \sqrt[6]{{\left(r^2\right)}^6}\\ \text{Since the index n is even = }\sqrt[n]{a^n}=\left|a\right|.\\ {\text{Since r}}^2{\text{is positive, we do not need an r}}^2\text{absolute value sign. }\\ \Rightarrow \sqrt[6]{r^{12}}=r^2\end{array}$

Part(b) Step 1. Given inforamtion

$ⓑ \sqrt[3]{s^{30}}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(s^{10}\right)}^3={\text{ s}}^{30}\\ \Rightarrow \sqrt[3]{s^{30}}\\ \Rightarrow \sqrt[3]{{\left(s^{10}\right)}^3}\\ \text{Since the index n is odd = }\sqrt[n]{a^n}=a.\\ \Rightarrow \sqrt[3]{s^{30}}=s^{10}\end{array}$