Question 864944
First, lengthen the exponent:
{{{(a+b)(a+b)(a+b)-(a-b)(a-b)(a-b)}}}
To find (a+b)(a+b)(a+b), find all the multiplications and add them:
a*a*a
a*a*b
a*b*a
a*b*b
b*a*a
b*a*b
b*b*a
b*b*b
for a total of {{{a^3+a^2b+a^2b+ab^2+a^2b+ab^2+ab^2+b^3}}}, or {{{a^3+3a^2b+3ab^2+b^3}}}

For (a-b)(a-b)(a-b), it's the same, except the terms with exactly 1 or 3 b's will be negative.
The total of (a-b)(a-b)(a-b) is {{{a^3-3a^2b+3ab^2-b^3}}}
 
Because you're subtracting the second, your number is
{{{(a^3+3a^2b+3ab^2+b^3)-(a^3-3a^2b+3ab^2-b^3)}}}
Distribute the -1
{{{a^3+3a^2b+3ab^2+b^3-a^3+3a^2b-3ab^2+b^3}}}
Combine like terms
{{{6a^2b+2b^3}}}