1. **Problem:** Simplify the surd expression $\sqrt{50}$. **Solution:** Use the rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$. 2. **Problem:** Rationalize the denominator of $\frac{3}{\sqrt{5}}$. **Solution:** Multiply numerator and denominator by $\sqrt{5}$. $\frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$. 3. **Problem:** Simplify $\sqrt{18} + \sqrt{8}$. **Solution:** Simplify each surd first. $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$, $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$. Add: $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$. 4. **Problem:** Simplify $\frac{\sqrt{45}}{\sqrt{5}}$. **Solution:** Use $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$. $\sqrt{\frac{45}{5}} = \sqrt{9} = 3$. 5. **Problem:** Simplify $\sqrt{32} - \sqrt{18}$. **Solution:** Simplify each surd. $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$, $\sqrt{18} = 3\sqrt{2}$. Subtract: $4\sqrt{2} - 3\sqrt{2} = \sqrt{2}$. 6. **Problem:** Rationalize $\frac{5}{2 + \sqrt{3}}$. **Solution:** Multiply numerator and denominator by the conjugate $2 - \sqrt{3}$. $\frac{5}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{5(2 - \sqrt{3})}{(2)^2 - (\sqrt{3})^2} = \frac{10 - 5\sqrt{3}}{4 - 3} = 10 - 5\sqrt{3}$. 7. **Problem:** Simplify $\sqrt{12} \times \sqrt{3}$. **Solution:** Use $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$. $\sqrt{12} \times \sqrt{3} = \sqrt{36} = 6$. 8. **Problem:** Simplify $\frac{\sqrt{75}}{5}$. **Solution:** Simplify numerator first. $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$. Divide: $\frac{5\sqrt{3}}{5} = \sqrt{3}$. 9. **Problem:** Simplify $\sqrt{20} + 2\sqrt{5}$. **Solution:** Simplify $\sqrt{20} = 2\sqrt{5}$. Add: $2\sqrt{5} + 2\sqrt{5} = 4\sqrt{5}$. 10. **Problem:** Simplify $\sqrt{8} \times \sqrt{2}$. **Solution:** $\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4$. 11. **Problem:** Rationalize $\frac{7}{\sqrt{2} - 1}$. **Solution:** Multiply numerator and denominator by conjugate $\sqrt{2} + 1$. $\frac{7}{\sqrt{2} - 1} \times \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{7(\sqrt{2} + 1)}{(\sqrt{2})^2 - 1^2} = \frac{7(\sqrt{2} + 1)}{2 - 1} = 7\sqrt{2} + 7$. 12. **Problem:** Simplify $\sqrt{50} - \sqrt{18} + \sqrt{8}$. **Solution:** Simplify each surd: $\sqrt{50} = 5\sqrt{2}$, $\sqrt{18} = 3\sqrt{2}$, $\sqrt{8} = 2\sqrt{2}$. Combine: $5\sqrt{2} - 3\sqrt{2} + 2\sqrt{2} = 4\sqrt{2}$. 13. **Problem:** Simplify $\frac{\sqrt{45} + \sqrt{20}}{\sqrt{5}}$. **Solution:** Simplify numerator: $\sqrt{45} = 3\sqrt{5}$, $\sqrt{20} = 2\sqrt{5}$. Sum: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$. Divide by $\sqrt{5}$: $\frac{5\sqrt{5}}{\sqrt{5}} = 5$. 14. **Problem:** Simplify $\sqrt{72} - \sqrt{8}$. **Solution:** Simplify each surd: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$, $\sqrt{8} = 2\sqrt{2}$. Subtract: $6\sqrt{2} - 2\sqrt{2} = 4\sqrt{2}$. 15. **Problem:** Rationalize $\frac{4}{3 - \sqrt{2}}$. **Solution:** Multiply numerator and denominator by conjugate $3 + \sqrt{2}$. $\frac{4}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}} = \frac{4(3 + \sqrt{2})}{9 - 2} = \frac{4(3 + \sqrt{2})}{7} = \frac{12}{7} + \frac{4\sqrt{2}}{7}$.