TITLE: Is this expression always irrational?
QUESTION [5 upvotes]: Is it right that
$$\sqrt[a]{2^{2^n}+1}$$
for every $$a>1,n \in \mathbb N $$ 
is always irrational?

REPLY [5 votes]: As lhf notes, we just have to see that $2^{2^n}+1$ can never be a perfect power. However, since $2^{2^n}$ is a perfect power, using Mihailescu's theorem the only pair of perfect powers differing by $1$ is $8$ and $9$, but $2^{2^n}$ can't be equal to $8$ for $n$ natural. So your expression is, indeed, always irrational.
(I guess I should note that using full Mihailescu's theorem here might be an overkill - 2 being a prime should let us give a relatively simple argument... but hey, whatever works :) )