x^6-y^6
fist look at as a difference of squares:
(x^3)2 - (y^3)2
=(x^3 + y^3)(x^3 - y^3)
Now you have the sum of cubes and the difference of cubes, I assume you know the formulas for those. So you can keep going
(x+y)(x^2-xy+y^2)(x-y)(x^2+xy+y^2) would be your final answer.
To simplify the expression x^6 - y^6, you can factor it using the difference of squares formula. The difference of squares formula states that a^2 - b^2 can be factored as (a + b)(a - b).
In this case, x^6 - y^6 can be rewritten as (x^3)^2 - (y^3)^2. Now we can see that it matches the pattern of the difference of squares formula.
So, we can factor x^6 - y^6 as (x^3 + y^3)(x^3 - y^3).
Now, we have a sum of cubes in the first term (x^3 + y^3), and we have a difference of cubes in the second term (x^3 - y^3).
The formulas for sum of cubes and difference of cubes are:
Sum of Cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Difference of Cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Using these formulas, we can simplify further:
(x^3 + y^3)(x^3 - y^3) = (x + y)(x^2 - xy + y^2)(x - y)(x^2 + xy + y^2)
This is the final answer for x^6 - y^6.