Last night, I came up with the following (warning, math ahead):
xn - xn-1 = xn-1(x-1)
Or in practice, for those of you who don't read math notation:
103 - 102 = 102(10-1) or:
1000 - 100 = 100 * 9
I haven't tested for some of the more esoteric cases, but thus far, it works with both negative x and negative n, so I'm fairly confident it will hold for all integer values of x and n. Whether it holds for fractional exponents is yet to be determined, as with non-integer values of x. (It appears it doesn't work with fractional exponents, so it's something to do with the properties of integers, I think.) [Note: I goofed the math, it worked with n=3/2 and x=2, so...maybe it does work with fractional exponents.]
But now, the next question is, why does it work? I haven't managed to figure out that one yet, but I'm sure one of the math geeks on my friends list can manage to give me an explanation.
I'll be out at court all afternoon (whee blah kill me now), so have fun with this.
(And yes, I'm a geek. And maybe even a nerd.)