1. **State the problem:** Expand and simplify the expression $$\sqrt{8} (\sqrt{50} - 10\sqrt{10})$$. 2. **Recall the properties of square roots:** - $$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$$ - Simplify square roots by factoring out perfect squares. 3. **Simplify each square root:** - $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$ - $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$ 4. **Rewrite the expression:** $$2\sqrt{2} (5\sqrt{2} - 10\sqrt{10})$$ 5. **Distribute $$2\sqrt{2}$$:** $$2\sqrt{2} \times 5\sqrt{2} - 2\sqrt{2} \times 10\sqrt{10}$$ 6. **Multiply inside each term:** - $$2 \times 5 = 10$$ - $$\sqrt{2} \times \sqrt{2} = \sqrt{4} = 2$$ - So, first term: $$10 \times 2 = 20$$ - For second term: - $$2 \times 10 = 20$$ - $$\sqrt{2} \times \sqrt{10} = \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ - So, second term: $$20 \times 2\sqrt{5} = 40\sqrt{5}$$ 7. **Combine the terms:** $$20 - 40\sqrt{5}$$ **Final answer:** $$20 - 40\sqrt{5}$$