Answer: a^3 + 3a^2b + 3ab^2 + b^3
========================================================
Explanation:
Cubing something means we're multiplying it by itself three times
Something like m^3 means m*m*m
So (a+b)^3 means (a+b)*(a+b)*(a+b)
Let's focus on (a+b)*(a+b) for now. We would use the FOIL rule to find that
(a+b)*(a+b) = a^2 + 2ab + b^2
So we can say
(a+b)*(a+b)*(a+b) = (a+b)(a^2 + 2ab + b^2)
-------------------------
Now let
c = a+b
that way we then have,
(a+b)(a^2 + 2ab + b^2) = c(a^2 + 2ab + b^2)
From here, distribute that 'c' through to each term
c(a^2 + 2ab + b^2)
a^2*c + 2abc + b^2*c
a^2( c ) + 2ab( c ) + b^2( c )
Next, replace c with a+b and apply distribution three more times
a^2( c ) + 2ab( c ) + b^2( c )
a^2( a+b ) + 2ab( a+b ) + b^2( a+b )
a^3 + a^2b + 2a^2b + 2ab^2 + ab^2 + b^3
a^3 + 3a^2b + 3ab^2 + b^3
You could also apply the Binomial Theorem to get the same result. The binomial coefficients can be computed through the combination formula, or through using Pascal's Triangle.
