The other day I was trying to find a method for cubing numbers similar to one I found for squaring numbers. I found that to find the square of a positive integer n, just sum up the first n odd integers.
$\sum_{t=1}^n 2t-1 = n^2$
Similarly, I found a method for cubing numbers
$\sum_{t=1}^n 3t^2-3t+1 = n^3$
Inside that, I realized I could condense 3t^2 to my sum I found earlier for squaring numbers, and I'd have a nested sigma sum. What I noticed at this point was that all I was doing was writing out in long hand the reduction of multiplication (and exponentiation) to the sum of 1, n times, which makes sense because after all, multiplication is just repeated addition. Also, the number of nested sigma sums was related to the power I was raising the original number to, which is also intuitive because it's just another series of additions.
What I'm curious about is if there is a pattern to this "reduction to summation" that I did. If I wanted to reduce a^b to a summation with terms that are at most of degree (b-1), how is there a repeating pattern that I could follow/extrapolate from the given sums that I have so far?