Chapter 7.1, Problem 62PPE

a.

To determine

To Simplify: The expression $a^n\cdot a^{-n}$ .

Answer to Problem 62PPE

The required answer is 1.

Explanation of Solution

Given information:

The expression $a^n\cdot a^{-n}$ .

Calculation:

  $\begin{array}{l}{a}^n\cdot a^{-n}\\ =a^n\cdot \frac{1}{a^n}\end{array}$

(Using the definition of negative exponent)

  $\begin{array}{l}=\frac{a^n}{a^n}\\ =1\end{array}$

(Simplify)

b.

To determine

To Explain: The mathematical relationship between. $a^n$ and $a^{-n}$ .

Answer to Problem 62PPE

The required answer is that $a^n$ and $a^{-n}$ are reciprocals of each other..

Explanation of Solution

Given information:

  $a^n$ and $a^{-n}$ .

Calculation:

  $a^{-n}$ can be written as $\frac{1}{a^n}$

Which means it is the reciprocal of $a^n$ .

Therefore, $a^n$ and $a^{-n}$ are reciprocals of each other.