(a+b)^3 (a-b)^3 - Simplify the Expression

To simplify the expression, we can use the formula for the cube of a binomial, which states that (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.\Using this formula, we can simplify (a+b)^3 as follows:\(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.\Similarly, we can use the formula for the cube of a binomial, which states that (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.\Using this formula, we can simplify (a-b)^3 as follows:\(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.\Now, to simplify (a+b)^3 (a-b)^3, we can multiply the two expressions together:\(a+b)^3 (a-b)^3 = (a^3 + 3a^2b + 3ab^2 + b^3)(a^3 - 3a^2b + 3ab^2 - b^3).\Expanding this expression, we have:\(a^3 + 3a^2b + 3ab^2 + b^3)(a^3 - 3a^2b + 3ab^2 - b^3) = a^6 - 3a^5b + 3a^4b^2 - a^3b^3 + 3a^5b - 9a^4b^2 + 9a^3b^3 - 3a^2b^4 + 3a^4b^2 - 9a^3b^3 + 9a^2b^4 - 3ab^5 + a^3b^3 - 3a^2b^4 + 3ab^5 - b^6.\Simplifying further, we can combine like terms:\a^6 + 3a^5b - 3a^5b - 9a^4b^2 + 3a^4b^2 + 9a^3b^3 - 9a^3b^3 - 3a^2b^4 + 3a^2b^4 - 3ab^5 + 3ab^5 + a^3b^3 - a^3b^3 - b^6.\This simplifies to:\a^6 - 9a^4b^2 + 9a^3b^3 - 3ab^5 - b^6.\Therefore, (a+b)^3 (a-b)^3 simplifies to a^6 - 9a^4b^2 + 9a^3b^3 - 3ab^5 - b^6.