1. **State the problem:** Simplify the expression $a^3 - b^3$. 2. **Recall the formula:** The difference of cubes can be factored using the identity: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ 3. **Apply the formula:** Substitute the given expression into the formula: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ 4. **Explanation:** This factorization works because when you expand $(a - b)(a^2 + ab + b^2)$, you get: $$a \cdot a^2 + a \cdot ab + a \cdot b^2 - b \cdot a^2 - b \cdot ab - b \cdot b^2 = a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3 = a^3 - b^3$$ 5. **Final answer:** $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$