The process of explanation by example (though I agree with others that it is really intrinsic in the meaning of a power - id like to hear impendia's explanation):

a^(n+1)=a^n * a

a^(n+2)=a^n * a^2

a^(n+3)=a^n * a^3

a^(n+0)=a^n * a^0

therefore

a^(n+m)=a^n * a^m