Q. 31 - Intermediate Algebra

Question

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

$ⓐ \sqrt[]{x^6} ⓑ \sqrt[]{y^{16}}$

Step-by-Step Solution

Answer

Part(a) $\sqrt[]{x^6}=\left|x\right|$

Part(b)$\sqrt[]{y^{16}}=y^8$

Part(a) Step 1. Given information

$ⓐ \sqrt[]{x^6}$

Simplify using absolute values as necessary. 

Step 2. Explanation

$\begin{array}{l}\text{Since }{\left(a^m\right)}^2=a^{2m}\\ \to {\left(x^3\right)}^2={\text{ x}}^{2*3}\\ \to {\left(x^3\right)}^2={\text{ x}}^6\\ \text{Since the index n is even},\sqrt[n]{a^n}=\left|a\right|.\\ \text{Here n is an even number}\\ \sqrt[]{x^6}=\left|x\right|\end{array}$

Part(b) Given information

$ⓑ \sqrt[]{y^{16}}$

Simplify using absolute values as necessary. 

Step 2. Explanation

$\begin{array}{l}\begin{array}{l}\text{Since }{\left(a^m\right)}^2=a^{2m}\\ \to {\left(y^8\right)}^2={\text{ y}}^{2*3}\\ \to {\left(y^8\right)}^2={\text{ y}}^{16}\\ \text{Since the index n is even},\sqrt[n]{a^n}=\left|a\right|.\\ \text{Here n is an even number}\\ \sqrt[]{y^{16}}=\left|y\right|\\ {\text{In this case the absolute value sign is not needed as y}}^8\text{ is positive.}\\ \sqrt[]{y^{16}}=y^8\\ \end{array}\\ \end{array}$