### Instruction: Completely factor the expression: $$x^8-256$$.### Response: Let's use difference of squares.
Right, so we can write $x^8-256$ as $(x^4)^2-16^2$.
Now we can factor as $(x^4-16)(x^4+16)$.
And since $x^4-16$ is also a difference of squares, we can factor again as $(x^2-4)(x^2+4)(x^4+16)$.
And since $x^2-4$ is also a difference of squares, we can factor again as $(x-2)(x+2)(x^2+4)(x^4+16)$.