Chapter 3, Problem 19REAE

Solution

To write the expression $\sqrt{\frac{18x^5{y}^4}{49x{z}^3}}$ in simplest form.

  $\frac{3x^2{y}^2}{7z}\sqrt{\frac{2}{z}}$

Given:

The expression $\sqrt{\frac{18x^5{y}^4}{49x{z}^3}}$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to simplify the given expression $\sqrt{\frac{18x^5{y}^4}{49x{z}^3}}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}\sqrt{\frac{18x^5{y}^4}{49x{z}^3}}=\frac{\sqrt{18x^5{y}^4}}{\sqrt{49x{z}^3}}\\ =\frac{\sqrt{18}\cdot \sqrt{x^5}\cdot \sqrt{y^4}}{\sqrt{49}\cdot \sqrt{x}\cdot \sqrt{z^3}}\\ =\frac{\sqrt{9\cdot 2}\cdot \sqrt{x^4\cdot x}\cdot \sqrt{{\left(y^2\right)}^2}}{\sqrt{7^2}\cdot \sqrt{x}\cdot \sqrt{z^2\cdot z}}\\ =\frac{\sqrt{9}\cdot \sqrt{2}\cdot \sqrt{x^4}\cdot \sqrt{x}\cdot y^2}{7\cdot \sqrt{x}\cdot \sqrt{z^2}\cdot \sqrt{z}}\\ =\frac{\sqrt{3^2}\cdot \sqrt{2}\cdot \sqrt{{\left(x^2\right)}^2}\cdot \sqrt{x}\cdot y^2}{7\cdot \sqrt{x}\cdot z\cdot \sqrt{z}}\\ =\frac{3\cdot \sqrt{2}\cdot x^2\cdot \sqrt{x}\cdot y^2}{7z\cdot \sqrt{x\cdot z}}\\ =\frac{3x^2{y}^2\cdot \sqrt{2x}}{7z\cdot \sqrt{xz}}\\ =\frac{3x^2{y}^2}{7z}\sqrt{\frac{2x}{xz}}\\ =\frac{3x^2{y}^2}{7z}\sqrt{\frac{2}{z}}\end{array}$

Therefore, the simplified expression is $\frac{3x^2{y}^2}{7z}\sqrt{\frac{2}{z}}$ .