for the clever mathematician

@Pfuzylogic even zero ?

@espoir Any number

@Pfuzylogic
I will confess to googling this. 😇
[quote]n short, the multiplicative identity is the number 1, because for any other number x, 1*x = x. So, the reason that any number to the zero power is one ibecause any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1.[/quote]

@Pfuzylogic at least you were right

@espoir I am pretty strong on Algebra and the rules. I didn’t know the proof but i did know the rule!

@Pfuzylogic Well done

@Pfuzylogic That quote illustrates some pretty poor reasoning.

How about this:

Given:

x^p, for p > 0

we know that

x^{-p} = 1/(x^p)

so

(x^p) * (x^{-p}) = (x^p) * [1/(x^p)] = 1

but

(x^p) * (x^{-p}) = x^{p + (-p)} = x^0

So

x^0 = 1, for any positive x

Thus

lim_{x ‐> 0+} x^0 = 1

(since δ^0 = 1 for arbitrarily small δ).

@SomeMichGuy
i think your proof would require a post secondary math background. Just keeping it very simple!

@Pfuzylogic The last part--the limit stuff--is from calculus, but the first part is just high school.

Adding exponents (when the base is the same) comes from there, as well as being shown that a number raised to a neg. exponent is just

1/(that number raised to the pos. exponent)

Those two things lead immediately to the general result.

The "limit" thing is also conceptually simple. Since x^0 = 1 is true for even a billionth or a trillionth, etc., it's true for ANY positive number arbitrarily close to 0. So, in the limit as x goes to 0 through positive values, it makes sense that
0^0 = 1.