Q. 33 - Intermediate Algebra

Question

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

$ⓐ \sqrt{x^{24}} ⓑ \sqrt{y^{22}}$

Step-by-Step Solution

Answer

Part(a)$\sqrt[]{x^{24}}=x^{12}$

Part(b)$\sqrt[]{y^{22}}=\left|y^{11}\right|$

Part(a) Given information

$\sqrt{x^{24}}$

Simplify using absolute values as necessary. 

Step 2. Explanation

$\begin{array}{l}\text{Since }{\left(a^m\right)}^2=a^{2m}\\ \to {\left(x^{12}\right)}^2={\text{ x}}^{2*12}\\ \to {\left(x^{12}\right)}^2={\text{ x}}^{24}\\ \text{Since the index n is even},\sqrt[]{a^n}=\left|a\right|.\\ \text{Here n is an even number}\\ {\text{In this case the absolute value sign is not needed as x}}^{12}\text{ is positive.}\\ \sqrt[]{x^{24}}=x^{12}\end{array}$

Part(b) Given information

$ⓑ \sqrt{y^{22}}$

Simplify using absolute values as necessary. 

Step 2. Explanation

$\begin{array}{l}\text{Since }{\left(a^m\right)}^2=a^{2m}\\ \to {\left(y^{11}\right)}^2={\text{ y}}^{2*11}\\ \to {\left(y^{11}\right)}^2={\text{ y}}^{22}\\ \text{Since the index n is even},\sqrt[]{a^n}=\left|a\right|.\\ \text{Here n is an even number}\\ \sqrt[]{y^{22}}=\left|y^{11}\right|\end{array}$