For powers:
a^(m+n) = a^m * a^n
Suppose you're trying to compute a^20
a^20 = a^10* a^10
a^10 = a^5 * a^5
a^5 = a^2 * a^2 * a
a^2 = a * a
If you could make your program do that, you could compute a^2, store it in a variable, and work your way up from there, storing each needed power in a variable. It might not seem more efficient, but in the example above, it changes 19 multiplications into a mere 5 (assuming I counted correctly) Cuts down on overhead incurred from using recursion, too. Just an idea, but it seems like it would help a lot.
edit:
This becomes much more useful as the exponents get bigger. It doesn't give any benefit for exponents of 2 and 3. The number of multiplications needed after this method / the number needed to "brute force" it seems to be approximated fairly well by
Y=2X^(-2/3) for even powers (x represents the power, y represents the return of my method)
For odd powers it doesn't perform quite as well (because of the extra *a) and it's better represented by
Y=1.5X^(-.4)
I feel like a geek now.
