Prove a^(m+n) = a^m + a^n with mathematical induction

[gluttony]

prove by induction that, 1) a^(m+n) = a^m + a^n

2) (a^m)^n=a^(mn)

2. Homework Equations -- mathematical induction
3. The Attempt at a Solution
the equations hold for a = 1.
let's say the equation hold for a = s; then s^(m+n) = s^m + s^n
now for a = s+1; (s+1)^(m+n) = (binomial expansion of s+1)

how can i prove that (s+1)^(m+n) = (s+1)^m * (s+1)^n

 

[D H]

prove by induction that, 1) a^(m+n) = a^m + a^n

You can't prove that because it's not true. 2²⁺¹=2³=8, 2²+2¹=4+2=6, and 8 is obviously not equal to 6.

What you can prove is that a^{m^n} = a^m*aⁿ.

3. The Attempt at a Solution
the equations hold for a = 1.
let's say the equation hold for a = s; then s^(m+n) = s^m + s^n
now for a = s+1; (s+1)^(m+n) = (binomial expansion of s+1)

how can i prove that (s+1)^(m+n) = (s+1)^m * (s+1)^n

Wrong approach. Try starting with n=0 as your base case rather than a=1. And of course you do want to prove the corrected thesis.