Real Numbers to Irrational Powers

In a related question we discussed raising numbers to powers.

I am interested if anybody knows any results for raising numbers to irrational powers.

For instance, we can easily show that there exists an irrational number raised to an irrational power such that the result is a rational number. Observe ${\sqrt 2 ^ {\sqrt 2}}$. Since we do not know if ${\sqrt 2 ^ {\sqrt 2}}$ is rational or not, there are two cases.

1. ${\sqrt 2 ^ {\sqrt 2}}$ is rational, and we are finished.





2. ${\sqrt 2 ^ {\sqrt 2}}$ is irrational, but if we raise it by ${\sqrt 2}$ again, we can see that
   

   $$\left ( \sqrt 2 ^ \sqrt 2 \right ) ^ \sqrt 2 = \sqrt 2 ^ {\sqrt 2 \cdot \sqrt 2} = \sqrt 2 ^ 2 = 2.$$

Either way, we have shown that there exists an irrational number raised to an irrational power such that the result is rational.

Can more be said about raising irrational numbers to irrational powers?

Answer

Some examples:

• Eulers Identity $e^{i \pi} + 1 = 0$




• Gelfond-Schneider Constant: $2^\sqrt{2} = 2.66514414269022518865\cdots$ is trancendental by Gelfond-Schneider.




• $i^i = 0.2078795763507619085469556198\cdots$ is also trancendental




• Ramanujan constant: $e^{\pi \sqrt{163}} = 62537412640768743.999999999999250072597\cdots$.




• $e^\gamma$ where $\gamma$ is the Euler–Mascheroni constant. (okay nobody has proved it's irrational yet, but surely is)