Rational numbers from irrational numbers Can any real function of irrational numbers  be rational ? If possible, please give examples. Special case  $ e^ {\pi i } = -1 $ seems to restrict solutions entirely to complex numbers. The question is right now vague as I do not know how to  exclude "obviously  derived " examples.

A well-known example.  
Is $m:=\sqrt{2}^{\sqrt{2}}$ rational?  If yes, that is your answer.  
If no, then $m^\sqrt{2}$ is your answer: $m$ and $\sqrt{2}$ are irrational, but
$$
m^\sqrt{2} = \left(\sqrt{2}^\sqrt{2}\right)^\sqrt{2}
=\sqrt{2}^{(\sqrt{2}\cdot\sqrt{2})} = \sqrt{2}^2 = 2
$$
is rational.
