Question 103883
Let's work it out
{{{(a+b)^3=(a+b)*(a+b)^2}}}
{{{(a+b)^3=(a+b)*(a^2+2ab+b^2)}}}
{{{(a+b)^3=a*(a^2+2ab+b^2)+b*(a^2+2ab+b^2)}}}
{{{(a+b)^3=(a^3+2a^2b+ab^2)+(ba^2+2ab^2+b^3)}}}
{{{(a+b)^3=a^3+3a^2b+3ab^2+b^3}}}
Now let's substitute. 
{{{(a+b)^3=a^3+b^3}}} Your original equation.
{{{cross(a^3)+3a^2b+3ab^2+cross(b^3)=cross(a^3)+cross(b^3)}}} Substitute for the expanded cubic amd remove like terms from both sides.

{{{3a^2b+3ab^2=0}}} Simplify. 
{{{(3a^2b+3ab^2)/(3ab)=0}}} Divide both sides by 3ab
{{{(3a^2b)/3ab+(3ab^2)/3ab=0}}} Distribute. 
{{{a+b=0}}} Simplify
So whenever a+b=0, then you know that 
{{{(a+b)^3=a^3+b^3}}}
For example,
{{{a=3}}} and {{{b=-3}}}
{{{a+b=0}}} and {{{(a+b)^3=0}}}
{{{a^3+b^3=(27)+(-27)=0}}}