Factor using the formula for the sum or difference of two cubes.
x^6 - 27y^3
The first term is x^2, cubed. Call that a
The second term is 3y, cubed. Call that b
Apply the formula that
(a^3 - b^3) = (a-b)(a^2 + ab +b^2)
To factor the expression x^6 - 27y^3 using the formula for the sum or difference of two cubes, we need to recognize that x^6 is a cube of x^2, and 27y^3 is a cube of 3y. The formula for the sum of two cubes is:
a^3 + b^3 = (a + b)(a^2 - ab + b^2),
and the formula for the difference of two cubes is:
a^3 - b^3 = (a - b)(a^2 + ab + b^2).
Using these formulas, we can rewrite the expression x^6 - 27y^3 as:
(x^2)^3 - (3y)^3.
Now, we can apply the formula by identifying a in this case as x^2 and b as 3y. Plugging those values into the formula for the difference of two cubes, we get:
(x^2)^3 - (3y)^3 = (x^2 - 3y)((x^2)^2 + (x^2)(3y) + (3y)^2).
Simplifying this expression further, we have:
(x^2 - 3y)(x^4 + 3x^2y + 9y^2).
So, x^6 - 27y^3 can be factored as (x^2 - 3y)(x^4 + 3x^2y + 9y^2).