1. Problem: Simplify $\frac{\sqrt{2}}{\sqrt{3}}$. Work: Multiply numerator and denominator by $\sqrt{3}$ to rationalize the denominator. Work: $\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{2}\sqrt{3}}{\sqrt{3}\sqrt{3}}$. Final answer: $\frac{\sqrt{6}}{3}$. 2. Problem: Simplify $\frac{\sqrt[3]{8}}{\sqrt[3]{6}}$. Work: Combine as one cube root $\sqrt[3]{\frac{8}{6}}$ and simplify the fraction. Work: $\sqrt[3]{\frac{8}{6}}=\sqrt[3]{\frac{4}{3}}$. Final answer: $\sqrt[3]{\frac{4}{3}}$. 3. Problem: Simplify $\frac{\sqrt[4]{36}}{\sqrt[4]{6}}$. Work: Combine as one fourth root $\sqrt[4]{\frac{36}{6}}$ and simplify inside. Work: $\sqrt[4]{\frac{36}{6}}=\sqrt[4]{6}$. Final answer: $\sqrt[4]{6}$. 4. Problem: Simplify $\frac{\sqrt[3]{4}}{\sqrt[3]{6}}$. Work: Combine as $\sqrt[3]{\frac{4}{6}}$ and reduce the fraction. Work: $\sqrt[3]{\frac{4}{6}}=\sqrt[3]{\frac{2}{3}}$. Final answer: $\sqrt[3]{\frac{2}{3}}$. 5. Problem: Simplify $\frac{7}{\sqrt{6}+\sqrt{5}}$. Work: Rationalize by multiplying top and bottom by $\sqrt{6}-\sqrt{5}$. Work: $\frac{7}{\sqrt{6}+\sqrt{5}}=\frac{7(\sqrt{6}-\sqrt{5})}{(\sqrt{6}+\sqrt{5})(\sqrt{6}-\sqrt{5})}$. Work: Denominator is $6-5=1$ so the result is $7(\sqrt{6}-\sqrt{5})$. Final answer: $7(\sqrt{6}-\sqrt{5})$. 6. Problem: Simplify $\frac{4\sqrt{6}-3\sqrt{21}}{\sqrt{3}}$. Work: Divide each term by $\sqrt{3}$: $\frac{4\sqrt{6}}{\sqrt{3}}-\frac{3\sqrt{21}}{\sqrt{3}}$. Work: Simplify radicands: $4\sqrt{\frac{6}{3}}-3\sqrt{\frac{21}{3}}=4\sqrt{2}-3\sqrt{7}$. Final answer: $4\sqrt{2}-3\sqrt{7}$. 7. Problem: Simplify $\frac{\sqrt{2}}{\sqrt[3]{2}}$. Work: Write as powers of 2: $2^{1/2}/2^{1/3}=2^{1/2-1/3}=2^{1/6}$. Final answer: $\sqrt[6]{2}$. 8. Problem: Simplify $\frac{\sqrt{5}}{\sqrt{15}}$. Work: Combine under one root $\sqrt{\frac{5}{15}}=\sqrt{\frac{1}{3}}$. Work: Rationalize: $\sqrt{\frac{1}{3}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$. Final answer: $\frac{\sqrt{3}}{3}$. 9. Problem: Simplify $\frac{\sqrt[3]{3x^{2}b}}{\sqrt[3]{25xy^{2}}}$. Work: Combine as $\sqrt[3]{\frac{3x^{2}b}{25xy^{2}}}$ and cancel one $x$. Work: $\sqrt[3]{\frac{3xb}{25y^{2}}}$ is the simplified cube root. Final answer: $\sqrt[3]{\frac{3xb}{25y^{2}}}$. 10. Problem: Simplify $\frac{\sqrt{2}}{2+\sqrt{3}}$. Work: Rationalize by multiplying top and bottom by $2-\sqrt{3}$. Work: $\frac{\sqrt{2}(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}=\frac{\sqrt{2}(2-\sqrt{3})}{4-3}=\sqrt{2}(2-\sqrt{3})$. Work: Distribute if desired: $2\sqrt{2}-\sqrt{6}$. Final answer: $2\sqrt{2}-\sqrt{6}$. 11. Problem: Simplify $\frac{1}{\sqrt{x}}$. Work: Rationalize by multiplying top and bottom by $\sqrt{x}$ to get $\frac{\sqrt{x}}{x}$. Final answer: $\frac{\sqrt{x}}{x}$. 12. Problem: Simplify $\frac{\sqrt[3]{108}}{\sqrt[3]{2}}$. Work: Combine as $\sqrt[3]{\frac{108}{2}}=\sqrt[3]{54}$. Work: Factor $54=27\cdot 2$ so $\sqrt[3]{54}=\sqrt[3]{27\cdot2}=3\sqrt[3]{2}$. Final answer: $3\sqrt[3]{2}$. 13. Problem: Simplify $\frac{5\sqrt{63}}{6\sqrt{7}}$. Work: Factor inside roots: $\frac{5}{6}\cdot\sqrt{\frac{63}{7}}=\frac{5}{6}\sqrt{9}$. Work: $\frac{5}{6}\cdot 3=\frac{15}{6}=\frac{5}{2}$. Final answer: $\frac{5}{2}$. 14. Problem: Simplify $\sqrt{\frac{400}{20}}$. Work: Simplify inside: $\frac{400}{20}=20$. Work: $\sqrt{20}=\sqrt{4\cdot5}=2\sqrt{5}$. Final answer: $2\sqrt{5}$. 15. Problem: Simplify $\frac{1}{\sqrt{5}}$. Work: Rationalize: multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to get $\frac{\sqrt{5}}{5}$. Final answer: $\frac{\sqrt{5}}{5}$. 16. Problem: Simplify $\frac{6\sqrt{28}}{3\sqrt{4}}$. Work: Reduce numeric coefficient $\frac{6}{3}=2$ and inside roots $\sqrt{\frac{28}{4}}=\sqrt{7}$. Work: $2\sqrt{7}$. Final answer: $2\sqrt{7}$. 17. Problem: Simplify $\frac{\sqrt{80}}{\sqrt{5}}$. Work: Combine as $\sqrt{\frac{80}{5}}=\sqrt{16}=4$. Final answer: $4$. 18. Problem: Simplify $\frac{20\sqrt{46}}{5\sqrt{23}}$. Work: Reduce numeric coefficient $\frac{20}{5}=4$ and combine roots $\sqrt{\frac{46}{23}}=\sqrt{2}$. Work: $4\sqrt{2}$. Final answer: $4\sqrt{2}$. 19. Problem: Simplify $\frac{1}{2+\sqrt{5}}$. Work: Rationalize by multiplying by $2-\sqrt{5}$ to get $\frac{2-\sqrt{5}}{4-5}=\frac{2-\sqrt{5}}{-1}=\sqrt{5}-2$. Final answer: $\sqrt{5}-2$. 20. Problem: Simplify $\frac{10\sqrt{18}}{2\sqrt{9}}$. Work: Reduce numeric coefficient $\frac{10}{2}=5$ and simplify roots $\sqrt{\frac{18}{9}}=\sqrt{2}$. Work: $5\sqrt{2}$. Final answer: $5\sqrt{2}$. 21. Problem: Simplify $\frac{\tfrac{5\sqrt{96}}{6}}{\tfrac{2\sqrt{24}}{5}}$. Work: Division by a fraction multiplies by its reciprocal: $\frac{5\sqrt{96}}{6}\cdot\frac{5}{2\sqrt{24}}=\frac{25}{12}\cdot\sqrt{\frac{96}{24}}$. Work: $\sqrt{\frac{96}{24}}=\sqrt{4}=2$ so the result is $\frac{25}{12}\cdot 2=\frac{25}{6}$. Final answer: $\frac{25}{6}$. 22. Problem: Simplify $\frac{3}{\sqrt{3}-1}$. Work: Rationalize by multiplying by $\sqrt{3}+1$: $\frac{3(\sqrt{3}+1)}{3-1}=\frac{3(\sqrt{3}+1)}{2}$. Final answer: $\frac{3(\sqrt{3}+1)}{2}$. 23. Problem: Simplify $\frac{\sqrt{25}}{\sqrt{625}}$. Work: Combine as $\sqrt{\frac{25}{625}}=\sqrt{\frac{1}{25}}=\frac{1}{5}$. Final answer: $\frac{1}{5}$. 24. Problem: Simplify $\frac{\sqrt[3]{3}}{\sqrt[3]{5}}$. Work: Combine as $\sqrt[3]{\frac{3}{5}}$. Final answer: $\sqrt[3]{\frac{3}{5}}$. 25. Problem: Simplify $\frac{\sqrt{6}}{3}$. Work: This is already simplified; it can be written as $\frac{1}{3}\sqrt{6}$. Final answer: $\frac{\sqrt{6}}{3}$. 26. Problem: Simplify $\frac{\sqrt{x}}{x}$. Work: For $x>0$ this equals $\frac{1}{\sqrt{x}}$ because $\frac{\sqrt{x}}{x}=\frac{\sqrt{x}}{\sqrt{x}\sqrt{x}}=\frac{1}{\sqrt{x}}$. Final answer: $\frac{1}{\sqrt{x}}$. 27. Problem: Simplify $7\sqrt{6}-7\sqrt{5}$. Work: Factor out 7 to get $7(\sqrt{6}-\sqrt{5})$. Final answer: $7(\sqrt{6}-\sqrt{5})$. 28. Problem: Simplify $4$. Work: Already simplified. Final answer: $4$. 29. Problem: Simplify $5\sqrt{2}$. Work: Already simplified. Final answer: $5\sqrt{2}$. 30. Problem: Simplify $\frac{\sqrt[3]{36}}{3}$. Work: This is $\frac{1}{3}\sqrt[3]{36}$ and there is no cube factor to extract from 36. Final answer: $\frac{\sqrt[3]{36}}{3}$. 31. Problem: Simplify $\sqrt{2}$. Work: Already simplified. Final answer: $\sqrt{2}$. 32. Problem: Simplify $\frac{5}{2}$. Work: Already simplified. Final answer: $\frac{5}{2}$. 33. Problem: Simplify $\frac{\sqrt[3]{15bxy}}{5y}$. Work: This is already in simplified form as $\frac{1}{5y}\sqrt[3]{15bxy}$ unless variables allow cancellation. Final answer: $\frac{\sqrt[3]{15bxy}}{5y}$. 34. Problem: Simplify $\frac{1}{5}$. Work: Already simplified. Final answer: $\frac{1}{5}$. 35. Problem: Simplify $\frac{\sqrt{5}}{5}$. Work: Already simplified. Final answer: $\frac{\sqrt{5}}{5}$. 36. Problem: Simplify $\sqrt{6}$. Work: Already simplified. Final answer: $\sqrt{6}$. 37. Problem: Simplify $\frac{3\sqrt{3}+3}{2}$. Work: Factor 3 from numerator to get $\frac{3(\sqrt{3}+1)}{2}$. Final answer: $\frac{3(\sqrt{3}+1)}{2}$. 38. Problem: Simplify $\frac{\sqrt{18}}{3}$. Work: $\sqrt{18}=\sqrt{9\cdot2}=3\sqrt{2}$ so $\frac{3\sqrt{2}}{3}=\sqrt{2}$. Final answer: $\sqrt{2}$. 39. Problem: Simplify $2\sqrt{2}-\sqrt{6}$. Work: Factor $\sqrt{2}$ if desired: $\sqrt{2}(2-\sqrt{3})$. Final answer: $2\sqrt{2}-\sqrt{6}$. 40. Problem: Simplify $2\sqrt{7}$. Work: Already simplified. Final answer: $2\sqrt{7}$. 41. Problem: Simplify $\frac{\sqrt[3]{75}}{5}$. Work: This is $\frac{1}{5}\sqrt[3]{75}$ and 75 has no perfect cube factor. Final answer: $\frac{\sqrt[3]{75}}{5}$. 42. Problem: Simplify $4\sqrt{2}$. Work: Already simplified. Final answer: $4\sqrt{2}$. 43. Problem: Simplify $-2+\sqrt{5}$. Work: This is already simplified and can be written as $\sqrt{5}-2$. Final answer: $\sqrt{5}-2$. 44. Problem: Simplify $2\sqrt{5}$. Work: Already simplified. Final answer: $2\sqrt{5}$. 45. Problem: Simplify $\frac{\sqrt{3}}{3}$. Work: Already simplified; it is the rationalized form of $\frac{1}{\sqrt{3}}$. Final answer: $\frac{\sqrt{3}}{3}$. 46. Problem: Simplify $4\sqrt{2}-3\sqrt{7}$. Work: Already simplified; no like radicals to combine. Final answer: $4\sqrt{2}-3\sqrt{7}$. 47. Problem: Simplify $3\sqrt{2}$. Work: Already simplified. Final answer: $3\sqrt{2}$. 48. Problem: Simplify $\frac{25}{6}$. Work: Already simplified. Final answer: $\frac{25}{6}$.