Chapter 5.1, Problem 19WE

To determine

To find: Simplify the given expression.

Answer to Problem 19WE

  $\frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=\frac{z^2}{x^2}$

Explanation of Solution

Given information:

  $\frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}$

Concept:

  $\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}$

Using law of exponents

  $\frac{a^m}{a^n}=a^{m-n}$

  ${\left(a^n\right)}^m=a^{mn}$

Where $a$ be any value.

  $\frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}$

  $\begin{array}{l}\Rightarrow \frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=\frac{x^2{y}^2{\left(z^2\right)}^2}{{\left(x^2\right)}^2{y}^2{z}^2}\\ \Rightarrow \frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=\frac{x^2\cdot y^2\cdot z^{2\times 2}}{x^{2\times 2}\cdot y^2\cdot z^2}\\ \Rightarrow \frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=\frac{x^2\cdot y^2\cdot z^4}{x^4\cdot y^2\cdot z^2}\\ \Rightarrow \frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=\frac{x^2}{x^4}\cdot \frac{y^2}{y^2}\cdot \frac{z^4}{z^2}\\ \Rightarrow \frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=x^{2-4}\cdot y^{2-2}\cdot z^{4-2}\\ \Rightarrow \frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=x^{-2}\cdot y^0\cdot z^2\\ \Rightarrow \frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=\frac{z^2}{x^2}\end{array}$

Therefore,

  $\frac{{\left(xy{z}^2\right)}^2}{{\left(x^{2}yz\right)}^2}=\frac{z^2}{x^2}$