We are given the expression $3x^6 + 81y^6$ and asked to factor it.

1. Problem Description

We are given the expression $3x^6 + 81y^6$ and asked to factor it.

2. Solution Steps

First, we factor out the greatest common factor (GCF), which is $3$:

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

We can rewrite $x^6$ as $(x^2)^3$ and $27y^6$ as $(3y^2)^3$. So we have a sum of cubes.

The sum of cubes factorization formula is:

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

Applying this formula to $x^6 + 27y^6 = (x^2)^3 + (3y^2)^3$, we have $a = x^2$ and $b = 3y^2$.

Then,

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

Therefore, $3x^6 + 81y^6 = 3(x^6 + 27y^6) = 3(x^2 + 3y^2)(x^4 - 3x^2y^2 + 9y^4)$.

3. Final Answer

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