Simplification of $n^{1/\sqrt{\log n}}$

Question

I would like to simplify this function, how can we do it ?

Answer 1

You may write, for $n >1$,

$$ n^{1/\sqrt{\log n}}=e^{\log n/\sqrt{\log n}}=e^{\sqrt{\log n}} $$

using $$ a^b=e^{b \log a}, \qquad a>0, \,b>0. $$

Answer 2

$$n=e^{\log n}$$

$$n^{1/(\log n)^{1/2}} =e^{(\log n)^{1/2}}$$