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.

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.
