1. **State the problem:** Simplify the expression $$\frac{(5\sqrt{3}+\sqrt{50})(5-\sqrt{24})}{\sqrt{75}-5\sqrt{2}}$$. 2. **Rewrite radicals in simplest form:** - \(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\) - \(\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}\) - \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\) So the expression becomes: $$\frac{(5\sqrt{3} + 5\sqrt{2})(5 - 2\sqrt{6})}{5\sqrt{3} - 5\sqrt{2}}$$ 3. **Factor out common terms in denominator:** $$5(\sqrt{3} - \sqrt{2})$$ 4. **Expand numerator:** $$(5\sqrt{3})(5) = 25\sqrt{3}$$ $$(5\sqrt{3})(-2\sqrt{6}) = -10\sqrt{18} = -10 \times 3\sqrt{2} = -30\sqrt{2}$$ $$(5\sqrt{2})(5) = 25\sqrt{2}$$ $$(5\sqrt{2})(-2\sqrt{6}) = -10\sqrt{12} = -10 \times 2\sqrt{3} = -20\sqrt{3}$$ Sum these: $$25\sqrt{3} - 30\sqrt{2} + 25\sqrt{2} - 20\sqrt{3} = (25\sqrt{3} - 20\sqrt{3}) + (-30\sqrt{2} + 25\sqrt{2}) = 5\sqrt{3} - 5\sqrt{2}$$ 5. **Rewrite numerator and denominator:** Numerator: $$5\sqrt{3} - 5\sqrt{2} = 5(\sqrt{3} - \sqrt{2})$$ Denominator: $$5(\sqrt{3} - \sqrt{2})$$ 6. **Simplify fraction:** $$\frac{5(\sqrt{3} - \sqrt{2})}{5(\sqrt{3} - \sqrt{2})} = 1$$ **Final answer:** $$1$$