>>14315194> I thought a number to a power means a number multiplied by itself, the power number of times.Not quite, it means taking a number, and multiplying it by itself the power minus one number of times.
To calculate 5^3, you start with 5, and then you multiply it by 5 two times and end up with 125.
To calculate 0^3, you start with 0, and then you multiply it by 0 two times and end up with 0.
To calculate 0^0, you start with 0, and then you multiply it by 0 minus one times...
What does that even mean? How can I do something minus one times?
You could stop there and say that this definition only works for positive powers and leave 0^0 undefined.
Or, you could argue that doing something minus one times means doing the inverse one time (that's how we get negative exponents).
But multiplying by 0 doesn't have an inverse (dividing by 0 is not a thing in standard arithmetic).
So you can't multiply by 0 minus one times.
So 0^0, using this definition, is still undefined.
You could define exponentiation a bit differently to avoid this edge case.
Since multiplying anything by 1 doesn't change anything, you could instead define a^b as starting with 1 and multiplying by a, b times.
With this definition, to calculate 5^3, you start with 1, and then you multiply it by 5 three times and end up with 125.
To calculate 0^0, you start with 1, and then you multiply it by 0 zero times and end up with 1.
So 0^0, using this definition, is 1.
If feel this definition is more intuitive.
Why should 4^0 (multiplying no copies of 4 together) be any different than 0^0 (multiplying no copies of 0 together)?
In both cases, you are multiplying no numbers together.
But at the end of the day, which definition is best depends on what you want to do with it.
The only situation I know of where leaving 0^0 undefined/undetermined is convenient is when doing limits since x^y is not continuous at (0, 0).
