1. **State the problem:** Simplify the expression $$\sqrt{40} + 3\sqrt{32} + 7\sqrt{2} - 6\sqrt{10}$$. 2. **Recall the rule:** To simplify square roots, factor the number inside the root into its prime factors and extract perfect squares. 3. **Simplify each term:** - $$\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}$$ - $$3\sqrt{32} = 3 \times \sqrt{16 \times 2} = 3 \times \sqrt{16} \times \sqrt{2} = 3 \times 4 \sqrt{2} = 12\sqrt{2}$$ - $$7\sqrt{2}$$ stays as is. - $$-6\sqrt{10}$$ stays as is. 4. **Rewrite the expression with simplified terms:** $$2\sqrt{10} + 12\sqrt{2} + 7\sqrt{2} - 6\sqrt{10}$$ 5. **Group like terms:** - Terms with $$\sqrt{10}$$: $$2\sqrt{10} - 6\sqrt{10}$$ - Terms with $$\sqrt{2}$$: $$12\sqrt{2} + 7\sqrt{2}$$ 6. **Combine like terms:** - $$2\sqrt{10} - 6\sqrt{10} = (2 - 6)\sqrt{10} = -4\sqrt{10}$$ - $$12\sqrt{2} + 7\sqrt{2} = (12 + 7)\sqrt{2} = 19\sqrt{2}$$ 7. **Final simplified expression:** $$19\sqrt{2} - 4\sqrt{10}$$