i think it's ridiculous to use induction because

a^(m+n) = aaaaaaaaa...aaaaa m+n times

and a^(m)a^(n) = aaaa...aaa (m times) x aaaa....aaaa (n times)

but if you used induction,

a^(m+1) = aaaaa...aaaaa (m times) x a (1 time)

and then if a^(m + n) = a^(m)a^(n), then

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

sorry if it's confusing