Exponentiation: notation

I've always wondered why we use the notation,

a^(-1) = 1/a

So after reading Euler, I tried it out, and figured it out.

a^b * a = a^(b+1)
a^b = a^(b+1) / a
a^(-1) = a^(-1+1) / = a^0 / a = 1/a