Chapter 7, Problem 25CT

To determine

Simplify the given equation.

Answer to Problem 25CT

  $\boxed{{\left(a^{-3}\right)}^2=\frac{1}{a^6}}$

Explanation of Solution

Formula used:

  $a^{-m}=\frac{1}{a^m}$

  ${\left(a^m\right)}^n=a^{m\times n}$

Given information:

  ${\left(a^{-3}\right)}^2$

As we know that $a^{-m}=\frac{1}{a^m}$ and ${\left(a^m\right)}^n=a^{m\times n}$ .

Now evaluating the above expression ${\left(a^{-3}\right)}^2$ using the identity $a^{-m}=\frac{1}{a^m}$ and ${\left(a^m\right)}^n=a^{m\times n}$:

  $\begin{array}{l}{\left(a^{-3}\right)}^2\\ \\ {\left(a^{-3}\right)}^2=\left(a^{-3\times 2}\right)\\ \\ {\left(a^{-3}\right)}^2=\left(a^{-6}\right)\\ \\ \boxed{{\left(a^{-3}\right)}^2=\frac{1}{a^6}}\end{array}$