1. Stating the problem: Simplify the expression $$\frac{(5\sqrt{3}+\sqrt{50})(5-\sqrt{24})}{\sqrt{75}-5\sqrt{2}}$$. 2. Recall the rules: - Simplify radicals where possible. - Use distributive property to expand products. - Rationalize denominators if needed. 3. Simplify radicals: $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$ $$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$$ $$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ 4. Substitute back: $$\frac{(5\sqrt{3} + 5\sqrt{2})(5 - 2\sqrt{6})}{5\sqrt{3} - 5\sqrt{2}}$$ 5. Factor numerator and denominator: Numerator: Expand using distributive property: $$ (5\sqrt{3})(5) + (5\sqrt{3})(-2\sqrt{6}) + (5\sqrt{2})(5) + (5\sqrt{2})(-2\sqrt{6}) $$ Calculate each term: $$ 25\sqrt{3} - 10\sqrt{18} + 25\sqrt{2} - 10\sqrt{12} $$ Simplify radicals inside terms: $$ \sqrt{18} = 3\sqrt{2}, \quad \sqrt{12} = 2\sqrt{3} $$ So: $$ 25\sqrt{3} - 10 \times 3\sqrt{2} + 25\sqrt{2} - 10 \times 2\sqrt{3} = 25\sqrt{3} - 30\sqrt{2} + 25\sqrt{2} - 20\sqrt{3} $$ Combine like terms: $$ (25\sqrt{3} - 20\sqrt{3}) + (25\sqrt{2} - 30\sqrt{2}) = 5\sqrt{3} - 5\sqrt{2} $$ Denominator: $$ 5\sqrt{3} - 5\sqrt{2} = 5(\sqrt{3} - \sqrt{2}) $$ 6. The expression becomes: $$ \frac{5\sqrt{3} - 5\sqrt{2}}{5(\sqrt{3} - \sqrt{2})} $$ 7. Cancel common factor 5: $$ \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = 1 $$ Final answer: $$1$$