# Polynomials in irrational powers and their roots

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?

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