prove that (x+y)^3-(x-y)^3-6y(x^2-y^2)=8y^3

To prove the equation (x+y)^3-(x-y)^3-6y(x^2-y^2)=8y^3, we need to simplify both sides of the equation and show that they are equal.

First, let's simplify the left side of the equation.
Using the formula for the difference of cubes, we can rewrite (x+y)^3 and (x-y)^3 as follows:

(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3

Now, substitute these expressions back into the original equation:

(x+y)^3-(x-y)^3-6y(x^2-y^2) = (x^3 + 3x^2y + 3xy^2 + y^3) - (x^3 - 3x^2y + 3xy^2 - y^3) - 6y(x^2-y^2)

Next, simplify each term separately:

x^3 + 3x^2y + 3xy^2 + y^3 - (x^3 - 3x^2y + 3xy^2 - y^3) - 6y(x^2-y^2)
= x^3 +3x^2y + 3xy^2 + y^3 - x^3 + 3x^2y - 3xy^2 + y^3 - 6y(x^2-y^2)

Combine like terms:

= 6x^2y + 2y^3 - 6xy(x-y)

Finally, simplify further:

= 6x^2y + 2y^3 - 6xy(x-y)
= 6x^2y + 2y^3 - 6xy^2 + 6y^2
= 6x^2y - 6xy^2 + 8y^2
= 2y(3x^2 - 3xy + 4y)

Therefore, we have simplified the left side of the equation to be 2y(3x^2 - 3xy + 4y).

Now, let's simplify the right side of the equation:

8y^3

Since the right side does not contain any variables other than y, we do not need to simplify it further.

Now that both sides of the equation have been simplified, we can see that:

Left side: 2y(3x^2 - 3xy + 4y)
Right side: 8y^3

These two sides are not equal. Therefore, the equation (x+y)^3-(x-y)^3-6y(x^2-y^2)=8y^3 is not true.