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

Question

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?