Factorise {a^6} - {b^6}

To factorize the expression a^6 - b^6, we can follow these steps:

Recognize the difference of squares: a^6 - b^6 is a difference of squares because it can be written as (a^3)^2 - (b^3)^2.

Apply the difference of squares formula: The difference of squares formula is: x^2 - y^2 = (x - y)(x + y). Here, we have: (a^3)^2 - (b^3)^2 = (a^3 - b^3)(a^3 + b^3).

Factorize further: Now, each of the factors (a^3 - b^3) and (a^3 + b^3) can be factorized as the sum and difference of cubes.

a^3 - b^3 can be factorized using the difference of cubes formula: x^3 - y^3 = (x - y)(x^2 + xy + y^2). So, a^3 - b^3 = (a - b)(a^2 + ab + b^2).

a^3 + b^3 can be factorized using the sum of cubes formula: x^3 + y^3 = (x + y)(x^2 - xy + y^2). So, a^3 + b^3 = (a + b)(a^2 - ab + b^2).

Final factorized form: Now, combining all the factors together, we get the complete factorization of a^6 - b^6:

a^6 - b^6 = (a - b)(a + b)(a^2 + ab + b^2)(a^2 - ab + b^2).

This is the fully factorized form of the expression a^6 - b^6.