What kind of number is sqrt(2)^sqrt(2)? I have noted sqrt(2)^sqrt(2) = 2^(sqrt(2)/2) = 2^(1/sqrt(2)), i.e. a rational number to an irrational power. Now, 1/sqrt(2) is less than 1, but greater than zero. So, given that 2^x is an increasing function, 2^(1/sqrt(2)) is less than 2^1, but greater than 2^0. Also, I tried the following. Let a,b be positive integers such that 2^(1/sqrt(2)) = a/b. What I want to do is reach a contradicition for otherwise I would assume the number in question is rational.