How do you factorise the given equation {x^6} - {y^6}?

To factorize the expression x^6 - y^6, we can use the difference of squares formula.

The difference of squares states that:

a^2 - b^2 = (a - b)(a + b)

Now, x^6 - y^6 can be written as:

(x^3)^2 - (y^3)^2

This is a difference of squares, so we can apply the formula:

(x^3 - y^3)(x^3 + y^3)

Next, we can factor each of these terms further. We recognize that both x^3 - y^3 and x^3 + y^3 are special cases of factorization:

x^3 - y^3 is a difference of cubes, and the formula for factoring the difference of cubes is:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)

So, we can factor x^3 - y^3 as:

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

x^3 + y^3 is a sum of cubes, and the formula for factoring the sum of cubes is:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)

So, we can factor x^3 + y^3 as:

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

Now, combining all the factors, we get:

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

Thus, the factorized form of x^6 - y^6 is:

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