Q. 50 - Intermediate Algebra

Question

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

$ⓐ \sqrt{225x^2{y}^2{z}^2} ⓑ\sqrt{36r^2{s}^{20}} ⓒ \sqrt[3]{125y^{18}{z}^{27}}$

Step-by-Step Solution

Answer

Part(a)$\sqrt{225x^2{y}^2{z}^2}=15\left|xyz\right|$

Part(b)$\sqrt{36r^2{s}^{20}}=6\left|r\right|s^{10}$

Part(c)$\sqrt[3]{125y^{18}{z}^{27}}=5y^6{z}^9$

Part(a) Step 1. Given information

$ⓐ \sqrt{225x^2{y}^2{z}^2}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since}\\ \Rightarrow {\left(15xyz\right)}^2=225x^2{y}^2{z}^2\\ \Rightarrow \sqrt[]{225x^2{y}^2{z}^2}\\ \Rightarrow \sqrt{{\left(15xyz\right)}^2}\\ \Rightarrow 15\left|xyz\right|\end{array}$

Part(b) Step 1. Given information

$ⓑ\sqrt{36r^2{s}^{20}}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since,}\\ \Rightarrow {\left(6r{s}^{10}\right)}^2=36r^2{s}^{20}\\ \Rightarrow \sqrt[]{36r^2{s}^{20}}\\ \Rightarrow \sqrt[]{{\left(6r{s}^{10}\right)}^2}\\ \Rightarrow 6\left|r\right|s^{10}\end{array}$

Part(c) Step 1. Given information

$ⓒ \sqrt[3]{125y^{18}{z}^{27}}$

 Simplify using absolute values as necessary.

Step 2. Explanation

$\begin{array}{l}\text{Since,}\\ \Rightarrow {\left(5y^6{z}^9\right)}^3=125y^{18}{z}^{27}\\ \Rightarrow \sqrt[3]{125y^{18}{z}^{27}}\\ \Rightarrow \sqrt[3]{{\left(5y^6{z}^9\right)}^3}\\ \Rightarrow 5y^6{z}^9\end{array}$