I tried to google a proof, and i found that it comes from the formula: x^1*(1/x^1) where the two x^1s cancel eachother out. I wasn’t satisfied with the answer, so i am still somewhat confused.
Usually when there’s some math i don’t get, i try to imagine a scenario irl, where a formula would apply. This always helps, but i can’t find one here.
In: 5
We don’t really need to prove things about operations. We made them up. We define them. The reason x to the power of 0 is 1 is because we say so. So there’s nothing really to prove necessarily, you just need to understand why that definition is useful.
When we define exponentiation we start by thinking of it as repeated multiplication. However, this definition really only works when the exponent is a positive integer. What does it mean to multiply something by itself 3.2 times or -2 times or even something like pi times? That’s much harder to define.
So we do a bunch of stuff to fill in what exponentiation means when you have different types of numbers in the exponent. For a negative power we think of that as a reciprocal being raised to a positive power. For a rational power we think of that as a number being raised to the power of the numerator and a root of the denominator. Irrational numbers have a definition to, although getting into that is a lot harder.
When we come up with all these things we get a curve. It just so happens that that curve always goes straight through 1 as the exponent goes to 0 no matter what the base of our exponent is (The notable exception being 0^x). So it is very useful to define x to the power of 0 to be 1.
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