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Lets start with x^a
What happens when you take x^a * x^b?
You get x^(a+b), if a and b are positive integers.
Hmm. So, what should x^0 be? Well, that's a nice rule. Can we make it true for x^0 as well as other positive integers?
x^a * x^0 = x^(a+0) for all a and x
x^a * x^0 = x^a for all a and x
x^0 = 1 for all x for which x^a does not equal zero
So, if x is not zero, if you want the x^a*x^b = x^(a+b) rule to work for x^0, x^0 must be zero.
The same lets you work out what happens with negative numbers: you get x^-a = 1/x^a
This is how you get many rules in mathematics: you take the basic definition for positive integers, find a nice property, and extend it to all integers, or even all real numbers, in such a way that the nice property still holds.
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Last edited by JHVH : 10-29-4004 BC at 09:00 PM. Reason: Time for a rest.
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