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### Factor x^4 - 81y^4

This expression can be factored as a difference of two squares because 81 is a square number. Rewrite 81 as x^4 as (x^2)^2, 9^2 = (3^2)^2 and y^4 as (y^2)^2,

(x^2)^2 - (3^2)^2 * (y^2)^2

Combine (3^2)^2 and (y^2)^2,

(x^2)^2 - ((3y)^2)^2

Applying the difference of squares formula,

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

Applying the difference of squares formula on the second parenthesis,

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

Simplify by rewriting (3y)^2 as 9y^2,

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

Thus the expression x^4 - 81y^4 is completely factored to (x^2 + 9y^2)(x + 3y)(x - 3y)