Chapter 4.2, Problem 50HP

To determine

To compare : the relation between $x^{-n}$ and $x^n$ .

Answer to Problem 50HP

The relation between $x^{-n}$ and $x^n$ is reciprocal to each other

Explanation of Solution

Given information :

  $x^{-n}\text{ and }{x}^n,\text{ where }x\ne 0$ .

Calculation :

The of expression $x^{-n}$ can also be written as ${\left(\frac{1}{x}\right)}^n$

Comparison between $x^n$ and $x^{-n}$ is that both have power $n$ but one has negative power and other have positive and the contrast or difference between the expression $x^{-n}$ and $x^n$ is that their base isreciprocal to each other.

For example:

let $x=2$ and $n=2$ .

Then the expression $x^{-n}$ is

  $\begin{array}{l}{2}^{-2}\\ \frac{1}{2^2}\\ \frac{1}{4}\end{array}$

The expression $x^n$ is

  $\begin{array}{l}{2}^2\\ \left(2\cdot 2\right)=4\end{array}$

Hence, the relation between $x^{-n}$ and $x^n$ is reciprocal to each other.