$$$0^0$$$ is often defined to equal $$$1$$$, see https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero

What is the difference between Undefined and Unspecified ?

I think it's one of those.

there are good reasons for wanting to define 0^0=1 , but does it break anything the way defining 0/0 would? In fact it doesn’t. All the usual laws of exponentiation remain valid with this definition.

That’s what we mean when we say that the choice to define 0^0=1 is the right one.

Source

since the blog is more focused on knowing individual opinions, i would first like to give my two cents on it and subsequently elaborate on it.

for me, undefined is the most apt answer, essentially because it pretty much explains the situation very accurately. if we go into the world of limits, it would not be difficult to show that $$${0^0}$$$ approaches several values depending on how you evaluate the limit and the function you start with.

however, if we look at it from a different standpoint, one that favors usability and discourages unnecessary complexity, then 1 is a good enough answer. It's supported by some well-known branches of mathematics, such as set theory and binomial theory.

there's this cool video that explores this question in much more depth.

It depends on what exactly is needed. We can assume $$$0^0 = 1$$$ and write $$$\exp(x) = \displaystyle \sum\limits_{i=0}^\infty \frac{x^i}{i!}$$$ for all $$$x\in\mathbb{R}$$$. On other hand, we can write $$$\mathbf{0}^0 = \mathbf{0}^{-1} \mathbf{0}^1 = \mathbf{0}^{-1}$$$, where $$$\mathbf{0}$$$ is the zero vector of some linear space. Now, considering $$$\mathbf{0}^{-1}$$$ as the inverse in the Drazin sense, we get $$$\mathbf{0}^{-1} = \mathbf{0}$$$ and $$$\mathbf{0}^0 = \mathbf{0}$$$.