Let $z, w \in\mathbb{C}$. One definition of the complex exponential is $z^w \equiv \exp\left(w\mathrm{Log}(z)\right) = \exp\left(w\left(\mathrm{ln}(|z|) + i\mathrm{Arg}(z)\right)\right)$, which allows us to better see where the branch cuts of e.g. root functions are.
We know that $z^2$ is an entire function on $\mathbb{C}$, but plugging in $w = 2$ to the prior formula gives $z^2 = \exp\left(2\mathrm{Log}(z)\right) = \exp\left(2\left(\mathrm{ln}(|z|) + i\mathrm{Arg}(z)\right)\right)$, an expression which in is not continuous on the non-positive real numbers due to argument function. Am I missing something here, or do we just have to resort to other kind of interpretation of exponentiating when exponentiating non-negative integers?
You actually get to the same result of analyticity, because you have:
$e^{2 \text{Log}(z)} = e^{2 \ln{|z|} + 2i \arg(z)} = e^{2 \ln{|z|}} \cdot e^{2i \arg(z)} = |z|^2 e^{i 2\theta}$
and see how the factor $e^{i 2\theta}$ is indeed continuous everywhere, you just got to the polar form of a square of a complex number.
The log formula is valid for $z\neq 0$, so you just analyze that point apart, and you end up with an entire function