Q. 39 - Intermediate Algebra

Question

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

$ⓐ \sqrt{49x^2} ⓑ -\sqrt{81x^{18}}$

Step-by-Step Solution

Answer

Part(a)$\sqrt[]{49x^2}=7\left|x\right|$

Part(b)$-\sqrt[]{81x^{18}}=-9\left|x^9\right|$

Part(a) Step 1. Given information

$ⓐ \sqrt{49x^2}$

Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(7x\right)}^2={\text{ 49x}}^2\\ \Rightarrow \sqrt[]{49x^2}\\ \Rightarrow \sqrt[]{{\left(7x\right)}^2}\\ \text{Since the index n is even = }\sqrt[n]{a^n}=\left|a\right|.\\ \Rightarrow \sqrt[]{49x^2}=7\left|x\right|\end{array}$

Part(b) Step 1 Given information

$ⓑ -\sqrt{81x^{18}}$

Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since }\\ \Rightarrow {\left(9\left(x^9\right)\right)}^2={\text{ 81x}}^{18}\\ \Rightarrow -\sqrt[]{81x^{18}}\\ \Rightarrow -\sqrt[]{{\left(9\right(x^9\left)\right)}^2}\\ \text{Since the index n is even = }\sqrt[n]{a^n}=\left|a\right|.\\ \Rightarrow -\sqrt[]{81x^{18}}=-9\left|x^9\right|\\ \end{array}$