The problem asks to simplify the expression $3x^6 + 81y^6$.

1. Problem Description

The problem asks to simplify the expression $3x^6 + 81y^6$.

2. Solution Steps

First, factor out the greatest common factor, which is 3:

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

Now, observe that $x^6 = (x^2)^3$ and $27y^6 = (3y^2)^3$. Thus, the expression within the parentheses is a sum of cubes. We can use the sum of cubes formula:

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

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

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

So the complete factored expression is:

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

3. Final Answer

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