1. **Stating the problem:** Simplify the expression $$80 \sqrt{10} + \sqrt{40}$$. 2. **Formula and rules:** Recall that $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ and that we can simplify square roots by factoring out perfect squares. 3. **Simplify each term:** - $$\sqrt{40} = \sqrt{4 \cdot 10} = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10}$$ 4. **Rewrite the expression:** $$80 \sqrt{10} + 2 \sqrt{10}$$ 5. **Combine like terms:** Since both terms have $$\sqrt{10}$$, factor it out: $$\sqrt{10} (80 + 2) = 82 \sqrt{10}$$ 6. **Final answer:** $$\boxed{82 \sqrt{10}}$$