Let $z,w$ complex numbers. Than one get: $$\begin{align*} (\mathrm{e}^z)^w&= (\exp(z\cdot \log(e))^w\\ &= \exp(w\cdot\log(\exp(z\cdot\log(e)))\\ &= \exp(w\cdot\log(\exp(z)))\\ &= \exp(w\cdot (z+2\pi \mathrm{i}k)) \qquad \forall k\in \mathbb{N}\\ &= \mathrm{e}^{w\cdot z+2\pi \mathrm{i}k\cdot w} \end{align*}$$
I'm not sure if this is correct. Any hints?
On
For complex $a, b$, we define $a^b$ as the multivalued function $\exp(b \log(a))$, where $\log(a)$ is any of the branches of the logarithm of $a$.
However, $e^b$ is conventionally taken to be $\exp(b)$, which is not multivalued.
Under this interpretation,
$$(e^z)^w = \exp(w \log(e^z)) = \exp(w (z+2 \pi i k)) = \exp(w z + 2 \pi i k w)$$
where $k$ is any integer (that's $\mathbb Z$, not $\mathbb N$).
As most of the comments indicate this is more about fiddling with the definitions and trying not to get it wrong, since a complex number to a complex power is not very well defined. I'd suggest \begin{align} (e^z)^w & = e^{w\log {e^z}} && \text{Using the definition } s^t= e^{t\log s}\text{ for }s, t \in \Bbb{C} \\ & =e^{w(z+2n\pi i)} && \log {e^z} = z+2n\pi i \\ & = e^{wz+2n\pi i w} \end{align}