1. **Simplify each radical expression:** 2. **Problem 1:** Simplify $\sqrt{32} - 2\sqrt{72} + \sqrt{200}$. 3. First, express each radical in simplest form: $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$$ $$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$$ $$\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}$$ 4. Substitute back: $$4\sqrt{2} - 2(6\sqrt{2}) + 10\sqrt{2} = 4\sqrt{2} - 12\sqrt{2} + 10\sqrt{2}$$ 5. Combine like terms: $$ (4 - 12 + 10)\sqrt{2} = 2\sqrt{2}$$ 6. **Final answer for problem 1:** $2\sqrt{2}$. --- 7. **Problem 2:** Simplify $(4\sqrt{2})^2 - (3\sqrt{3})^2$. 8. Use the rule $(a\sqrt{b})^2 = a^2 \times b$: $$(4\sqrt{2})^2 = 4^2 \times 2 = 16 \times 2 = 32$$ $$(3\sqrt{3})^2 = 3^2 \times 3 = 9 \times 3 = 27$$ 9. Subtract: $$32 - 27 = 5$$ 10. **Final answer for problem 2:** $5$. --- 11. **Problem 3:** Simplify $(\sqrt{3} - \sqrt{4})(\sqrt{3} - \sqrt{4})$. 12. Recognize this as $(a - b)^2 = a^2 - 2ab + b^2$: $$a = \sqrt{3}, b = \sqrt{4} = 2$$ 13. Calculate each term: $$a^2 = (\sqrt{3})^2 = 3$$ $$2ab = 2 \times \sqrt{3} \times 2 = 4\sqrt{3}$$ $$b^2 = 2^2 = 4$$ 14. Substitute: $$3 - 4\sqrt{3} + 4 = 7 - 4\sqrt{3}$$ 15. **Final answer for problem 3:** $7 - 4\sqrt{3}$. --- 16. **Problem 4:** Simplify $(2\sqrt{5} - 1)(4 + 3\sqrt{2})$. 17. Use distributive property: $$2\sqrt{5} \times 4 = 8\sqrt{5}$$ $$2\sqrt{5} \times 3\sqrt{2} = 6\sqrt{10}$$ $$-1 \times 4 = -4$$ $$-1 \times 3\sqrt{2} = -3\sqrt{2}$$ 18. Combine all terms: $$8\sqrt{5} + 6\sqrt{10} - 4 - 3\sqrt{2}$$ 19. **Final answer for problem 4:** $8\sqrt{5} + 6\sqrt{10} - 4 - 3\sqrt{2}$. --- 20. **Problem 5:** Simplify $(3\sqrt{5} - 1)(3\sqrt{5} + 1)$. 21. Recognize as difference of squares: $(a - b)(a + b) = a^2 - b^2$. $$a = 3\sqrt{5}, b = 1$$ 22. Calculate: $$a^2 = (3\sqrt{5})^2 = 9 \times 5 = 45$$ $$b^2 = 1^2 = 1$$ 23. Subtract: $$45 - 1 = 44$$ 24. **Final answer for problem 5:** $44$. --- 25. **Problem 6:** Simplify $\sqrt{24} \times 2\sqrt{2} \times \sqrt{6}$. 26. Simplify radicals: $$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$$ 27. Substitute: $$2\sqrt{6} \times 2\sqrt{2} \times \sqrt{6}$$ 28. Multiply coefficients: $$2 \times 2 = 4$$ 29. Multiply radicals: $$\sqrt{6} \times \sqrt{2} \times \sqrt{6} = \sqrt{6 \times 2 \times 6} = \sqrt{72}$$ 30. Simplify $\sqrt{72}$: $$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$$ 31. Multiply coefficients and radicals: $$4 \times 6\sqrt{2} = 24\sqrt{2}$$ 32. **Final answer for problem 6:** $24\sqrt{2}$.