calculus - How to prove that $e = lim_{n to infty} (sqrt[n]{n})^{pi(n)} = lim_{n to infty} sqrt[n]{n#}$?

While reading this post, I stumbled across these definitions (Wiki_german)

$$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$

and

$$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$

The last one seems interesting, since $\lim_{n \to \infty} (\sqrt[n]{n})=1$, proven

here.

How to prove these?