Factor the expression $3x^6 + 81y^6$.

1. Problem Description

Factor the expression $3x^6 + 81y^6$.

2. Solution Steps

First, factor out the greatest common factor (GCF) which is

3. $3x^6 + 81y^6 = 3(x^6 + 27y^6)$.

Notice that we have the sum of cubes. We can rewrite the expression inside the parentheses as

$x^6 + 27y^6 = (x^2)^3 + (3y^2)^3$.

Recall the sum of cubes formula:

$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$.

In this case, we have $a=x^2$ and $b=3y^2$.

Applying the sum of cubes formula, we get:

$(x^2)^3 + (3y^2)^3 = (x^2 + 3y^2)((x^2)^2 - (x^2)(3y^2) + (3y^2)^2) = (x^2 + 3y^2)(x^4 - 3x^2y^2 + 9y^4)$.

Substituting this back into our original expression, we have:

$3(x^6 + 27y^6) = 3(x^2 + 3y^2)(x^4 - 3x^2y^2 + 9y^4)$.

3. Final Answer

$3(x^2+3y^2)(x^4-3x^2y^2+9y^4)$