Chapter 5.2, Problem 44WE

To determine

To find: By counterexample that the given expressions are not equivalent.

Explanation of Solution

Given information:

  ${\left(xy\right)}^{-1}$ and $\frac{x}{y}$

  ${\left(xy\right)}^{-1}$ and $\frac{x}{y}$

Taking $x=2$ and $y=2$

We get

  $\begin{array}{l}{\left(xy\right)}^{-1}={\left(2\cdot 2\right)}^{-1}\\ \Rightarrow {\left(xy\right)}^{-1}={\left(4\right)}^{-1}\\ \Rightarrow {\left(xy\right)}^{-1}=\frac{1}{4}\\ \Rightarrow {\left(xy\right)}^{-1}=0.25\end{array}$

  $\begin{array}{l}\frac{x}{y}=\frac{2}{2}\\ \Rightarrow \frac{x}{y}=1\end{array}$

Therefore,

  ${\left(xy\right)}^{-1}=0.25$ and $\frac{x}{y}=1$

Which are not equivalent.