1. Let's start with the first problem: Simplify $$\frac{(5\sqrt{3}+\sqrt{50})(5-\sqrt{24})}{\sqrt{75}-5\sqrt{2}}$$. 2. First, simplify the radicals inside the expression: - $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$ - $$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$$ - $$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ 3. Substitute these back: $$\frac{(5\sqrt{3} + 5\sqrt{2})(5 - 2\sqrt{6})}{5\sqrt{3} - 5\sqrt{2}}$$ 4. Factor 5 from numerator's first term and denominator: Numerator: $$5(\sqrt{3} + \sqrt{2})(5 - 2\sqrt{6})$$ Denominator: $$5(\sqrt{3} - \sqrt{2})$$ 5. Cancel 5 from numerator and denominator: $$\frac{(\sqrt{3} + \sqrt{2})(5 - 2\sqrt{6})}{\sqrt{3} - \sqrt{2}}$$ 6. Expand numerator using distributive property: $$ (\sqrt{3} + \sqrt{2})(5 - 2\sqrt{6}) = 5\sqrt{3} + 5\sqrt{2} - 2\sqrt{3} \times \sqrt{6} - 2\sqrt{2} \times \sqrt{6} $$ 7. Simplify the products inside: - $$\sqrt{3} \times \sqrt{6} = \sqrt{18} = 3\sqrt{2}$$ - $$\sqrt{2} \times \sqrt{6} = \sqrt{12} = 2\sqrt{3}$$ 8. Substitute back: $$5\sqrt{3} + 5\sqrt{2} - 2 \times 3\sqrt{2} - 2 \times 2\sqrt{3} = 5\sqrt{3} + 5\sqrt{2} - 6\sqrt{2} - 4\sqrt{3}$$ 9. Combine like terms: - $$5\sqrt{3} - 4\sqrt{3} = \sqrt{3}$$ - $$5\sqrt{2} - 6\sqrt{2} = -\sqrt{2}$$ So numerator is $$\sqrt{3} - \sqrt{2}$$ 10. Now the expression is: $$\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}}$$ 11. Since numerator and denominator are the same, the value is: $$1$$ Final answer: $$1$$