Question 937500
Ex.1:  5^3 / 5^2 = 125 / 25 = 5
       5^3 / 5^2 = 5 ^(3-2) = 5^1 = 5

Ex.2:  5^3 / 5^3 = 125 / 125 = 1
       5^3 / 5^3 = 5 ^(3-3) = 5^0 = 
       ... and the result must be 1. So 5^0 =1.

The rule is: x^b / x^c = x^(b-c).

In order to generalize this rule for the case b=c, it must be defined that x^0 = 1 (x is any number different from 0). In mathematics, usefulness and consistency are very important. This convention allows us to extend definitions of power that would otherwise require treating 0 as a special case.

This method can also explain the definition: x^(-b) = 1 / x^b.

Ex.  5^2 / 5^4 = 25 / 625 = 1 / 5^2
     5^2 / 5^4 = 5 ^(2-4) = 5^(-2)