Question

factor completely. \\( x^4 - 64y^6 \\)

Explanation:

Step1: Recognize difference of squares

$x^4 - 64y^6 = (x^2)^2 - (8y^3)^2$

Step2: Apply difference of squares formula

$(x^2 - 8y^3)(x^2 + 8y^3)$

Step3: Factor sum/difference of cubes

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

Step4: Combine all factors

Multiply the factored terms together.

Answer:

$(x - 2y)(x + 2y)(x^2 + 2xy + 4y^2)(x^2 - 2xy + 4y^2)$