Chapter 4.2, Problem 41WE

To determine

For any positive integers $m,n\text{ and }r$

  ${\left({\left(a^m\right)}^n\right)}^r=a^{mnr}$ .

Explanation of Solution

Given information: For $\left(x\right)$ belongs to real there exists three positive integers $m,n\text{ and }r$ .

Proof: For any positive integers $m,n\text{ and }r$ the result ${\left({\left(a^m\right)}^n\right)}^r=a^{mnr}$ can be proved as shown,

  $\begin{array}{c}{\left({\left(a^m\right)}^n\right)}^r={\left({\left(a\right)}^{mn}\right)}^r\mathrm{......}\left\{{\left(x^m\right)}^n={\left(x\right)}^{mn}\right\}\\ ={\left(a\right)}^{mnr}\mathrm{......}\left\{{\left(x^m\right)}^n={\left(x\right)}^{mn}\right\}\\ =a^{mnr}\end{array}$

Hence prove that ${\left({\left(a^m\right)}^n\right)}^r=a^{mnr}$ if $m,n\text{ and }r$ are positive integers.