Raising a Power of Another Power.

The title means that if you raise a number to a certain power, and then use raise the resulting number to another power, what do you get?

In symbols, we get $$\left(a^m\right)^n = a^{mn}\,.$$

Explanation

The $a^m$ expression inside means this: $$\underbrace{a \times a \times \dotsb \times a}_{\text{\(a\) appearing \(m\) times}}\,.$$

The $({\phantom{a}})^n$ part means to repeat “$a^m$” $n$ times: $$\text{repeat \(n\) times} \left\{ \vphantom{ \begin{align} &a\\ &a\\ &\vdots\\ &a \end{align} } \right. \begin{align} &\overbrace{a \times a \times \dotsb \times a}^{\text{\(m\) times}}\\ \times &a \times a \times \dotsb \times a\\ &\phantom{a \times a}\vdots\\ \times &a \times a \times \dotsb \times a\\ \vphantom{a} \end{align} %\right.$$

So $\left(a^m\right)^n$ is the same as multiplying $n$ rows of $m$ lots of $a$, with the total of $m \times n$ lots of $a$ being multiplied together.

This is the same as raising $a$ to the power of $m+n$.

Example

Take $13^4$ as an example. This is the same as $$13 \times 13 \times 13 \times 13.$$

Now, raise that whole thing to the power of $2$ and it’ll become $$\begin{align} &(13 \times 13 \times 13 \times 13)\\ \times &(13 \times 13 \times 13 \times 13) = 13^{4 \times 2} \end{align}$$

If instead of squaring, you cubed it: $$\begin{align} &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13 = 13^{4 \times 3} = 13^{12} \end{align}$$

And, to the power of five: $$\begin{align} &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13\\ \times &13 \times 13 \times 13 \times 13 = 13^{4 \times 5} = 13^{20} \end{align}$$