1. **State the problem:** Simplify or factor the expression $x^5 - y^5$. 2. **Formula used:** The difference of fifth powers can be factored using the formula for the difference of powers: $$a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \cdots + ab^{n-2} + b^{n-1})$$ 3. **Apply the formula:** For $n=5$, we have: $$x^5 - y^5 = (x - y)(x^4 + x^3y + x^2y^2 + xy^3 + y^4)$$ 4. **Explanation:** This factorization breaks down the difference of two fifth powers into a product of a linear term $(x - y)$ and a quartic polynomial. This is useful for simplifying expressions or solving equations involving $x^5 - y^5$. 5. **Final answer:** $$x^5 - y^5 = (x - y)(x^4 + x^3y + x^2y^2 + xy^3 + y^4)$$