Chapter 9.4, Problem 54HP

To determine

To compare: $x^{-n}$ and $x^n$ with the help of example.

Answer to Problem 54HP

Both expression mentioned above is not equivalent to each other.

Explanation of Solution

Given information:

The expression mentioned in the problem is $x^{-n}$ and $x^n$ .

Formula used:

For negative powers we used to write $a^{-n}=\frac{1}{a^n}$ .

Calculation:

Consider the statement that $x^{-n}$ and $x^n$ with the help of example.

Recall that for negative exponent we used to write $a^{-n}=\frac{1}{a^n}$ .

Therefore $x^{-n}=\frac{1}{x^n}\Rightarrow \frac{1}{x^n}\ne x^n$ .

Let us take the example:-

Let $x=2$ and $n=3$ .Then $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$ and $2^3=8$ .

Here we observe that $\frac{1}{8}\ne 8$ .Therefore $x^{-n}=\frac{1}{x^n}\Rightarrow \frac{1}{x^n}\ne x^n$ .