real analysis - limit of a polynomial over a $log$ exponential

I want to know the limit of

$$\frac{n^{\frac{1}{k}}}{(8c)^{\frac{\log n}{\log \log n}}}$$

as $n \to \infty$ for some constants $k > 1$, $c > 1$.

I conjecture (based on numerical calculations) that this limit should be infinity, but I am not sure how to prove it. Can someone help me?

Answer

Hint : write everything in an exponential as

$$u_n=\exp\left[\ln n\left(\frac1k-\frac{\ln(8c)}{\ln\ln n}\right)\right].$$
The answer should be straightforward from this point.