Chapter 7.9, Problem 55WE

To determine

To calculate: The simplified form of the expression ${\left(\frac{a^{-6}}{a^{-2}}\right)}^{-1}$ .

Answer to Problem 55WE

The simplified form of the expression ${\left(\frac{a^{-6}}{a^{-2}}\right)}^{-1}$ is $a^4$ .

Explanation of Solution

Given information:

The expression ${\left(\frac{a^{-6}}{a^{-2}}\right)}^{-1}$ .

Formula used:

Law of exponent for quotient of powers is $a^m÷a^n=a^{m-n}$ where a is a nonzero integers and n and m are positive integers.

Law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Calculation:

Consider the expression ${\left(\frac{a^{-6}}{a^{-2}}\right)}^{-1}$ .

Recall that law of exponent for quotient of powers is $a^m÷a^n=a^{m-n}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}{\left(\frac{a^{-6}}{a^{-2}}\right)}^{-1}={\left(a^{-6-\left(-2\right)}\right)}^{-1}\\ ={\left(a^{-4}\right)}^{-1}\end{array}$

Recall that law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}{\left(\frac{a^{-6}}{a^{-2}}\right)}^{-1}={\left(a^{-6-\left(-2\right)}\right)}^{-1}\\ ={\left(a^{-4}\right)}^{-1}\\ =a^{-4\left(-1\right)}\\ =a^4\end{array}$

Thus, the simplified form of the expression ${\left(\frac{a^{-6}}{a^{-2}}\right)}^{-1}$ is $a^4$ .