1. **State the problem:** Simplify the expression $$\sqrt{500} + \sqrt{20} + 11\sqrt{5}$$. 2. **Recall the rule:** The square root of a product can be written as the product of square roots: $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. 3. **Simplify each square root term:** - $$\sqrt{500} = \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10\sqrt{5}$$ - $$\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5}$$ 4. **Rewrite the expression with simplified terms:** $$10\sqrt{5} + 2\sqrt{5} + 11\sqrt{5}$$ 5. **Combine like terms:** Since all terms have $$\sqrt{5}$$, add the coefficients: $$10 + 2 + 11 = 23$$ 6. **Final simplified expression:** $$23\sqrt{5}$$