When solving an equation with irrational (or algebraic in the case) powers, are the roots likely to be transcendental or algebraic, or does it vary?
As an example, I was trying to figure out if $(x + 1)^{\sqrt{2}}=x^2 - 2 x + 2$ had an exact solution. I tried it myself and then let Wolfram Alpha work on it, only getting approximate solutions (0.31375... and 3. something)
I was curious if these solutions are algebraic, and thus the solution of some polynomial, or transcendental.
Also, are there any known techniques for solving this kind of thing?