=

  (a*a*a*a*a) * (a*a*a*a*a*a*a)

      ^              ^

   n times        m times

    X^N * X^M
    = N copies of X, multiplied by M copies of X
    = N+M copies of X multiplied together
    = X^(N+M)

x^n = exp((log x)n). By definition, exp(n+m)=exp(n)exp(m). By definition, a(b+c) = ab + ac (for the log x thing). QED.

By the way, (the infamous calculus book) Baby Rudin has the poor reader show this property holds in exponentiation for reals, starting with integers and via rationals, as an exercise on its first chapter. Insane difficulty for me, even though the author practically holds your hand along the way! Cool read, though.

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