Why do this problem?
It gives practice in working with inequalities.
As we know $n$ is a positive integer learners can investigate
$n^{{1\over n}}$ for for different values of $n$ and make
conjectures about where the maximum value occurs.
Possible approach
You need to find a local maximum for a small value of $n$ and then
prove that this is the only maximum value. Clearly it is impossible
to check all values of $n$. One method of proving the result uses
the Binomial theorem.