I wrote my own graphing software, and I thought it had a bug in it, because when I graphed x^x, the curve starts out at (1,1), dips down a little, then rises again to pass through (1,1) as expected, before travelling to the wild blue yonder. It seemed like this function has no range in the negative numbers. But then I was thinking (-1)^(-1) is just 1/-1=-1. Lo and behold, if I stare real close, it did plot a pixel there. And another one at (-2, 0.25). And others that seemed to asymptotically approach 0. I guess a graphing calculator would have been more obvious since the pixels are much bigger. But why no pixels inbetween? Playing with the calculator, I realized that only (-integer)^(-integer) has a valid solution. Is there a plain English explanation as to why non-integers fail? It also got me to thinking.... 0^0. Why is this 1? It seems like it should be indeterminate, like 0/0. Zero divided by anything is 0, and anything divided by 0 is infinity, setting up a paradox for 0/0, which is what I thought defined an indeterminate. 0^anything = 0. anything^0=1, so 0^0 should be another paradox, hence indeterminate. But the calculator sees things differently.