Q. 48 - Intermediate Algebra

Question

In the following exercises, simplify using absolute values as necessary 

$ⓐ \sqrt{196a^2{b}^2} ⓑ \sqrt{81p^{24}{q}^6} ⓒ\sqrt[3]{27p^{45}{q}^9}$

Step-by-Step Solution

Answer

Part(a)$\sqrt{196a^2{b}^2}=16\left|ab\right|$

Part(b)$\sqrt{81p^{24}{q}^6}=9p^{12}\left|q^3\right|$

Part(c)$\sqrt[3]{27p^{45}{q}^9}=3p^{15}{q}^3$

Part(a) Step 1. Given information

$ⓐ \sqrt{196a^2{b}^2}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since}\\ \Rightarrow {\left(16ab\right)}^2=196a^2{b}^2\\ \Rightarrow \sqrt[]{196a^2{b}^2}\\ \Rightarrow \sqrt{{\left(16ab\right)}^2}\\ \Rightarrow 16\left|ab\right|\end{array}$

Part(b) Step 1. Given information

$ⓑ \sqrt{81p^{24}{q}^6}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since,}\\ \Rightarrow {\left(9p^{12}{q}^3\right)}^2=81p^{24}{q}^6\\ \Rightarrow \sqrt[]{81p^{24}{q}^6}\\ \Rightarrow \sqrt[]{{\left(9p^{12}{q}^3\right)}^2}\\ \Rightarrow 9p^{12}\left|q^3\right|\end{array}$

Part(c) Step 1. Given information

$\sqrt[3]{27p^{45}{q}^9}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since,}\\ \Rightarrow {\left(3p^{15}{q}^3\right)}^3=27p^{45}{q}^9\\ \Rightarrow \sqrt[3]{27p^{45}{q}^9}\\ \Rightarrow \sqrt[3]{{\left(3p^{15}{q}^3\right)}^3}\\ \Rightarrow 3p^{15}{q}^3\end{array}$