Smallest value of $k$ for which a function approaches $0$ as $x$ goes to $\infty$

I was playing around with factorials on desmos and trying to find some inequality between $x!$ and $\sqrt[x]{x}!$. After a bit I formulated the following question:

What is the smallest value of $k$ such that $ \lim_{x\to\infty} \frac{x!}{\left(\sqrt[x]{x}!\right)^{(x^k)}}=0 $?

I know that $2 < k \leq e$, but I don't know what the smallest is. Any help would be appreciated!