1. What is the simplest value of \(\cfrac{\sqrt{8} + \sqrt{12}}{\sqrt{32} + \sqrt{48}}\)? (a) \(\cfrac{1}{3}\) (b) \(\cfrac{1}{4}\) (c) \(\cfrac{1}{2}\) (d) \(\cfrac{1}{\sqrt2}\)

2. If \(a = \frac{\sqrt{3}}{2}\), then the value of \(\sqrt{1+a} + \sqrt{1-a}\) will be — (a) \(a\) (b) \(3a\) (c) \(2a\) (d) \(4a\)

3. Find the square root of: \(33 - 4\sqrt{35}\) (a) \(\pm(\sqrt{7}-\sqrt{5})\) (b) \(\pm(2\sqrt{7}-\sqrt{5})\) (c) (3\sqrt{7}-\sqrt{5})\) (d) \(\pm(4\sqrt{7}-\sqrt{5})\)

4. What is the value of \(\sqrt{12} + \sqrt{50} + 5\sqrt{3} - \sqrt{147} - \sqrt{32}\)? (a) \(\sqrt3\) (b) \(\sqrt2\) (c) \(2\sqrt2\) (d) \(3\sqrt3\)

5. If \( 2\sqrt{6} \) is a rationalizing factor of \( \sqrt{2x} \), what is the value of \( x \) ? (a) 2 (b) 3 (c) 6 (d) √6

6. What is the value of \(\cfrac{3\sqrt{8} - 2\sqrt{12} + \sqrt{20}}{3\sqrt{18} - 2\sqrt{27} + \sqrt{45}}\)? (a) \(\cfrac{3}{2}\) (b) \(\cfrac{1}{2}\) (c) \(\cfrac{2}{3}\) (d) \(\cfrac{13}{12}\)

7. What is the value of \[ \frac{3\sqrt{8} - 2\sqrt{12} + \sqrt{20}}{3\sqrt{18} - 2\sqrt{27} + \sqrt{45}}? \] (a) \(\cfrac{3}{2}\) (b) \(\cfrac{1}{2}\) (c) \(\cfrac{1}{3}\) (d) \(\cfrac{2}{3}\)

8. Among \( \sqrt[3]{2}, \sqrt[4]{3}, \sqrt[6]{5} \), the smallest number is — (a) \(\sqrt[3]{2}\) (b) \(\sqrt[4]{3}\) (c) \(\sqrt[6]{5}\) (d) three are same

9. What is the value of \[ \sqrt{10 + \sqrt{25 + \sqrt{108 + \sqrt{154 + \sqrt{225}}}}} \] (a) 3 (b) 4 (c) 5 (d) 6

10. If \(x = \sqrt{3} + \sqrt{2}\), then the value of \(x^3 + \frac{1}{x^3}\) is — (a) \(18\sqrt2\) (b) \(18\sqrt3\) (c) \(18\sqrt5\) (d) \(18\sqrt6\)

11. If \(a = \frac{1}{5 + 2\sqrt{6}}\), then what is the value of \(a^2 - \frac{1}{a^2}\)? (a) \(4\sqrt6\) (b) -\(4\sqrt6\) (c) \(40\sqrt6\) (d) -\(40\sqrt6\)

12. The simplest value of \[ \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \ldots + \frac{1}{\sqrt{99} + \sqrt{100}} \] is — (a) 100 (b) 99 (c) 9 (d) 1

13. If \(\sqrt{x} + \sqrt{y} = \sqrt{18 + 6\sqrt{5}}\), what is the value of \(x\)? (a) 8 (b) 15 (c) 6 (d) 12

14. If \(x = 3 + \sqrt{8}\) and \(y = 3 - \sqrt{8}\), then find the value of \(x^{-3} + y^{-3}\). (a) 199 (b) 195 (c) 198 (d) 201

15. What is the value of \(\sqrt{12+\sqrt{12+\sqrt{12+\cdots}}}\) to infinity? (a) 3 (b) 4 (c) 12 (d) None of the above

16. What is the simplest value of \((1 + \sqrt{2} + \sqrt{3})(1 - \sqrt{2} + \sqrt{3})\)? (a) \(2(\sqrt3-2)\) (b) \(2(\sqrt3-1)\) (c) \(2(1+\sqrt3)\) (d) \(1+\sqrt3\)

17. If \(x = \frac{\sqrt{3}}{2}\), then what is the value of \(\frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}\)? (a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

18. If \(x=\cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(xy=1\), then the value of \(\cfrac{x^2+xy+y^2}{x^2-xy+y^2}\) is — (a) \(\cfrac{11}{12}\) (b) \(\cfrac{12}{11}\) (c) \(\cfrac{13}{12}\) (d) \(\cfrac{14}{13}\)

19. If \(x = \cfrac{2\sqrt{15}}{\sqrt{5} + \sqrt{3}}\), then what is the value of \(\cfrac{x + \sqrt{5}}{x - \sqrt{5}} + \cfrac{x + \sqrt{3}}{x - \sqrt{3}}\)? (a) -1 (b) 0 (c) 1 (d) None of the above

20. If \(a = 3 + 2\sqrt{2}\), then what is the value of \[ \frac{a^6 + a^4 + a^2 + 1}{a^3}? \] (a) 100 (b) 200 (c) 204 (d) 250

21. The value of (√125 – √5) is (a) √120 (b) √80 (c) √100 (d) 5√5

22. Sure! Here's the English translation of your math question: **If** \(x = \frac{\sqrt{3}}{2}\), then what is the value of \[ \frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}? \] (a) \(2\sqrt3\) (b) \(\cfrac{1}{\sqrt3}\) (c) \(\sqrt3\) (d) \(\sqrt5\)

23. If \(\sqrt{5} - \sqrt{3} = a\), then what is the value of \(\sqrt{5} + \sqrt{3}\)? (a) \(\cfrac{2}{a}\) (b) \(\cfrac{a}{2}\) (c) 0 (d) None of the above

24. If \( x = 2 + \sqrt{3} \), then what is the value of \( x + \frac{1}{x} \)? (a) -4 (b) 2 (c) 4 (d) \(2+\sqrt3\)

25. If \((5 + √3)(5 - √3) = 25 - x^2\), find the value of \(x\). (a) 3 (b) √3 (c) -√3 (d) \(\pm\)√3

26. If \(x = \sqrt{7 + 4√3}\), find the value of \(x - \cfrac{1}{x}\). (a) 2 (b) 2√3 (c) 4 (d) 2-√3

27. If \(\sqrt{10} - 3 = k\), then what is the value of \(\sqrt{10} + 3\)? (a) \(2k\) (b) \(\cfrac{1}{k}\) (c) \(\cfrac{1}{2k}\) (d) \(-\cfrac{1}{k}\)

28. If \(2 + b\sqrt{3} = \cfrac{1}{2+\sqrt{3}}\), then \(b\) = ? (a) 1 (b) -1 (c) 0 (d) 2

29. If \(a+\cfrac{1}{a}=\sqrt{3}\), then the value of \(a^3+\cfrac{1}{a^3}\) will be — (a) 1 (b) 0 (c) -1 (d) 3

30. If \(x - \cfrac{1}{x} = \sqrt{5}\), then the value of \(x^4 + \cfrac{1}{x^4}\) is — (a) 45 (b) 46 (c) 47 (d) 48

31. (a) 24 (b) 80 (c) 16 (d) 8

32. If \(p+q=\sqrt{13}\) and \(p−q=\sqrt{5}\), then the value of \(pq\) is— (a) 2 (b) 18 (c) 9 (d) 8

33. If \(x = 9 + 4\sqrt{5}\), then the value of \(\sqrt{x} - \frac{1}{\sqrt{x}}\) will be — (a) 4 (b) 3 (c) 2 (d) 1

34. What is the square root of \(\sqrt{75} - \sqrt{72}\)? (a) \(\pm\sqrt[4]{3}(\sqrt3-\sqrt2)\) (b) \(\pm\sqrt3(\sqrt3-\sqrt2)\) (c) \(\pm(\sqrt3-\sqrt2)\) (d) \(\pm(\sqrt1-\sqrt2)\)

35. If \(x = \frac{\sqrt{a^2 + b^2} + \sqrt{a^2 - b^2}}{\sqrt{a^2 + b^2} - \sqrt{a^2 - b^2}}\), then — (a) \(b^2x^2-2a^2x-b^2=0\) (b) \(b^2x^2-2a^2x+b^2=0\) (c) \(b^2x^2+2a^2x+b^2=0\) (d) None of the above

36. If \(x = 7 + 4\sqrt{3}\), then find the value of \(\cfrac{x^3}{x^6 + 7x^3 + 1}\). (a) \(\cfrac{1}{2737}\) (b) \(\cfrac{1}{2730}\) (c) \(\cfrac{1}{2710}\) (d) \(\cfrac{1}{2709}\)

37. If \(x = 2 + \sqrt{3}\), then the value of \(x + \frac{1}{x}\) will be \(2\sqrt{3}\). True / False

38. \(2 - \sqrt{3}\) and \(\cfrac{1}{2 + \sqrt{3}}\) are equivalent expressions. True / False

39. If \(a + b : \sqrt{ab} = 4 : 1\), then what is the ratio of \(a : b\)?

40. If \(a + b : \sqrt{ab} = 2 : 1\), then find the ratio \(a : b\).

41. If \(2x = \sqrt{5} + 1\), then show that \(x^2 - x - 1 = 0\).

42. If \(x = \cfrac{1}{2 + \sqrt{3}}\) and \(y = \cfrac{1}{2 - \sqrt{3}}\), then what is the value of \(\cfrac{1}{1 + x} + \cfrac{1}{1 + y}\)?

43. Find the value of \(\sqrt{8} \times 3 \times \sqrt{2}\).

44. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \frac{1}{x}\), then find the value of: \[ (x + \frac{1}{x})^2 + \left( \frac{1}{y} - y \right)^2 \]

45. Given \(x = 3 + \sqrt{3}\) and \(y = 6\), find the value of \((x + y)^2\).

46. Simplify: \[ \frac{1}{\sqrt{2} + \sqrt{3}} - \frac{\sqrt{3} + 1}{2 + \sqrt{3}} + \frac{\sqrt{2} + 1}{3 + 2\sqrt{2}} \]

47. If \(x (4 - \sqrt{3}) = y (4 + \sqrt{3}) = 1\), then the value of \(x^2 + y^2\) will be _____

48. If \(x = \cfrac{4\sqrt{15}}{\sqrt{5} + \sqrt{3}}\), then find the value of \[ \cfrac{x + \sqrt{20}}{x - \sqrt{20}} + \cfrac{x + \sqrt{12}}{x - \sqrt{12}}. \]

49. Given \(a = \cfrac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \cfrac{\sqrt{5} - 1}{\sqrt{5} + 1}\), find the value of \(\cfrac{a^2 + ab + b^2}{a^2 - ab + b^2}\).

50. Simplify: \(\cfrac{4\sqrt{3}}{2 - \sqrt{2}} - \cfrac{30}{4\sqrt{3} - \sqrt{18}} - \cfrac{\sqrt{18}}{3 - \sqrt{12}}\)

51. Find the simplest value of: \(\sqrt7(\sqrt5 − \sqrt2) − \sqrt5(\sqrt7 − \sqrt2) + \cfrac{2\sqrt2}{\sqrt5 + \sqrt7}\)

52. If \(a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}\) and \(b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}\), then what is the value of \(\frac{a^2 + ab + b^2}{a^2 - ab + b^2}\)?

53. If the sum and product of two quadratic surds are rational numbers, then the surds are-------

54. Simplify: \[ \frac{\sqrt{5}}{\sqrt{3} + \sqrt{2}} - \frac{3\sqrt{3}}{\sqrt{2} + \sqrt{5}} + \frac{2\sqrt{2}}{\sqrt{3} + \sqrt{5}} \]

55. \(7\sqrt{11}\) is a _____ number.

56. If \(a =\cfrac{\sqrt{5}+1}{\sqrt{5}-1}\) and \(ab = 1\), then find the value of \(\left(\cfrac{a}{b}+\cfrac{b}{a}\right)\).

57. Find the simplest value of the following expression: \[ \cfrac{\sqrt{5}}{\sqrt{3}+\sqrt{2}} - \cfrac{3\sqrt{3}}{\sqrt{2}+\sqrt{5}} + \cfrac{2\sqrt{2}}{\sqrt{3}+\sqrt{5}} \]

58. If \(m + \cfrac{1}{m} = \sqrt{3}\), then find the simplest value of: (a) \(m^2 + \cfrac{1}{m^2}\), and (b) \(m^3 + \cfrac{1}{m^3}\).

59. \((5 + \sqrt{3})(5 - \sqrt{3}) = 25 - x^2\); find the value of \(x\).

60. Rationalize the denominator of the expression: \[ \cfrac{12}{\sqrt{15} - 3} \]

61. What is the value of \( \sqrt{125} - \sqrt{5} \)?

62. If \(x = \sqrt{3} + \sqrt{2}\), find the value of \(x^3 + \cfrac{1}{x^3}\).

63. If \(m = \sqrt{\cfrac{n}{n + \cfrac{1}{2}}}\) and \(m = \cfrac{1}{2}\), then what is the value of \(n\)?

64. If \(x = \sqrt{3} + \sqrt{2}\) and \(y = \cfrac{1}{\sqrt{3} + \sqrt{2}}\), then find the value of \((x + y)^2 + (x - y)^2\).

65. If \(x = 3 + 2\sqrt{2}\), then find the value of \(\left(\sqrt{x} + \cfrac{1}{\sqrt{x}}\right)\).

66. If \(x = \sqrt{3} + \frac{1}{\sqrt{3}}\) and \(y = \sqrt{3} - \frac{1}{\sqrt{3}}\), find the value of \(\frac{x^2}{y} + \frac{y^2}{x}\).

67. If \(x = 2 + \sqrt{3}\) and \(x + y = 4\), then find the simplest value of \(xy + \frac{1}{xy}\).

68. How much should be subtracted from √72 to get √32?

69. If \(a + b = 3\) and \(a - b = \sqrt{5}\), then find the value of \(ab\).

70. If \( x = \cfrac{\sqrt{3} + 1}{\sqrt{3} - 1} \) and \( xy = 1 \), determine the value of \( \cfrac{x^3 - y^3}{x^3 + y^3} \).

71. Simplify: \[ \frac{3\sqrt{7}}{\sqrt{2} + \sqrt{5}} - \frac{5\sqrt{5}}{\sqrt{2} + \sqrt{7}} + \frac{2\sqrt{2}}{\sqrt{5} + \sqrt{7}} \]

72. Which is greater between \((\sqrt{15} + \sqrt{3})\) and \((\sqrt{10} + \sqrt{8})\)?

73. If \[ x = \frac{2\sqrt{15}}{\sqrt{5} + \sqrt{3}} \] then find the value of \[ \frac{x + \sqrt{3}}{x - \sqrt{3}} + \frac{x + \sqrt{5}}{x - \sqrt{5}} \]

74. If \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] then show that \[ \frac{x^2 + xy + y^2}{x^2 - xy + y^2} = \frac{12}{11} \]

75. Which is greater between \(2 + \sqrt{7}\) and \(\sqrt{6} + \sqrt{5}\)?

76. Given: \[ x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} \quad \text{and} \quad xy = 1 \] Find the value of: \[ \frac{x^2 + 3xy + y^2}{x^2 - 3xy + y^2} \]