Problems and Theorems in Analysis · 2

If the Euler (Mascheroni) constant $C=-\Gamma '(1)$ were a rational number, then $\Gamma '(n+1)$ would have to be an integer for all integers $n$ from some point onward.

The number

is irrational.

The numbers $a$ and $b$ are positive; $a$ may be rational or irrational and the same holds for $b$, and so there are four possible cases. Show by means of examples that $a^b$ can be rational or irrational in each of these four cases (eight examples are required).