The number \sqrt{2}^\sqrt{2}, is it rational, an algebraic irrational or is it transcendental?

Recall that a rational number can be expressed as $p/q$ for integers $p, q$ with $q \neq 0$. An algebraic irrational number $\alpha$ is a solution to the equation $P(x) = 0$, where $P(x)$ is a polynomial with rational coefficients. A transcendental number is a real number that is irrational but not an algebraic irrational.

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