Two Proofs of a⁰ = 1

1. A quotient of two equal exponential variables aⁿ equals one because all factors cancel. A quotient of all variables aⁿ and a^m, where m < n will have all factors cancel leaving the exponent n-m, if aⁿ is the numerator and a^m is the denominator. Therefore if m = n all factors will cancel leaving an exponential with n-n = 0 factors or a⁰. But as previously stated a quotient of two equal exponentials equals 1. Therefore a⁰ = 1. Q. E. D.

1. Exponentials of the same base are unique in that aⁿ has one and only one value for each value of the exponent. Therefore a⁰ has one and only one value. Assume that a⁰ does not equal 1. Then there will be some real number y = a^x that will have two exponential representations in the same base. But this contradicts the uniqueness of exponential numbers in the same base. So a⁰ must be equal to 1.