Intuitive explanation of $(a^b)^c = a^{bc}$

What is an intuitive explanation for the rule that $(a^b)^c = a^{bc}$. I'm trying to wrap my head around it, but I can't really do it.

When $b$ and $c$ are positive integers, we can make the following argument: $$(a^b)^c=\underbrace{(a^b)\times(a^b)\times\cdots\times(a^b)}_{c\text{ times}}=$$ $$\underbrace{\left(\underbrace{a\times a\times\cdots \times a}_{b\text{ times}}\right)\times\left(\underbrace{a\times a\times\cdots \times a}_{b\text{ times}}\right)\times\cdots\times \left(\underbrace{a\times a\times\cdots \times a}_{b\text{ times}}\right)}_{c\text{ times}}=$$ $$\underbrace{a\times a\times \cdots\times a}_{bc\text{ times}}=a^{bc}$$

I will assume that $b$ and $c$ are positive integers and that $a$ is any "number" (it doesn't really matter much what $a$ is...).

Suppose I have $b \times c$ copies of the number $a$. I can arrange them into a $b \times c$ rectangular array: i.e., with $b$ rows and $c$ columns. When I multiply all $b \times c$ of these $a$'s together, I get $a^{bc}$.

On the other hand, suppose I look at just one column of the array. In this column I have $b$ $a$'s, so the product of all the entries in a column is $a^b$. But now I have $c$ columns altogether, so the product of all the entries is obtained by multiplying the common product of all the entries in a given column by itself $c$ times, or $(a^b)^c$.

Thus $a^{bc} = (a^b)^c$.

If you want to justify this identity when $b$ and $c$ are other things besides positive integers -- e.g. real numbers, or infinite cardinals -- that's another matter: please ask.