system
Given the following post, respond as if you're commenting on Reddit:
human
Why does X^0=1?
gpt
Exponentiation is repeated multiplication. This only makes sense when the exponent is a positive integer. You can't multiply something by itself -3/4 times, that doesn't make sense. However, after defining exponentiation to be repeated multiplication, we find that that doesn't really matter. The only important thing about exponentiation is the additive rule. That is, if N and M are positive integers, then 

* x^(N+M)=x^(N)\*x^(M)

This is literally the only reason why we care about exponentiation: it satisfies this property. 

As it stands, we only know how to do things like x^(1), x^(2), x^(3) and so on. How can we make sense of x^(0)? Right now it means nothing, we could just say that x^(0)=-32 and be done with it. But if we did that then the additive rule, the only thing that makes exponentiation meaningful, will be lose.

So we want to make x^(0) be "Whatever number it takes to preserve the additive rule". That's what it *actually* is. But we can be more specific. For instance, we have x^(1+0) = x^(1)=x. But on the other hand, we want the additive rule to still be true even if we include zero and what we get is x^(1+0)=x^(1)\*x^(0)=x\*x^(0). What this means is that we have both

* x^(1+0)=x

* x^(1+0)=x\*x^(0)

So *if* x^(0) is whatever value it needs to be to preserve the additive rule, *then* it must satisfy

* x\*x^(0)=x

So, *if x is not zero*, we can divide both side by x to get 

* x^(0)=1

This means that if we say that x^(0) is whatever value it takes to preserve the additive rule, then that value must be 1 whenever x is not zero.

If x is zero, then the equation x=x\*x^(0) is satisfied for *any* value for x^(0). In real life, there are different situations where 0^(0) seems to take on different values, in fact it can take on *any* value. There may even be situations where 0^(0) seems to equal -32. But this is okay, because the additive rule is still satisfied for x=0, so 0^(0) does not have to be any specific value. For this reason, unless context forces you to choose a value, we say that 0^(0) is an undefined quantity. It has no value. Not zero, not 1, not -32. It has the same value as (Balloons)^(Iridium), which is nothing because it is nonsense.