Solve $y = x^N + z^N$ for N? Is there a way to solve equations of the form 
$y = x^N+z^N$ for N?
I've been looking and I don't think there is a way, but I'm not sure if there's some obscure algebra rule that will help me here.

I think you can get an approximate solution by applying a multidimensional Newton-Raphson. You start by an initial guess, approximate the function by its tangent plane, get a better solution and iterate until reaching a solution within a reasonable interval of error. 
http://en.wikipedia.org/wiki/Newton%27s_method
