1. **State the problem:** Simplify the expression $$\sqrt{24} + \sqrt{150} + \sqrt{54}$$ without using a calculator. 2. **Recall the rule:** The square root of a product can be written as the product of square roots: $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. We use this to simplify each term by factoring out perfect squares. 3. **Simplify each term:** - $$\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$$ - $$\sqrt{150} = \sqrt{25 \times 6} = \sqrt{25} \times \sqrt{6} = 5\sqrt{6}$$ - $$\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}$$ 4. **Combine like terms:** Since all terms have $$\sqrt{6}$$, add their coefficients: $$2\sqrt{6} + 5\sqrt{6} + 3\sqrt{6} = (2 + 5 + 3)\sqrt{6} = 10\sqrt{6}$$ 5. **Final answer:** $$\boxed{10\sqrt{6}}$$