1. The problem is to expand and simplify the expression $(x - y)^2$ using the formula $$(a - b)^2 = a^2 - 2ab + b^2$$ 2. Applying the formula, let $a = x$ and $b = y$: $$ (x - y)^2 = x^2 - 2xy + y^2 $$ 3. This is the expanded and simplified form of $(x - y)^2$. 4. Note that the original expression given as $x^8 - 2xy + y^8$ is incorrect because the powers should be 2, not 8. Final answer: $$(x - y)^2 = x^2 - 2xy + y^2$$