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Number system 1surds & indices

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Number system 1surds & indices

  1. 1. SURDS & INDICES
  2. 2. If 32x = 81, what is the value of x? 1 2 3 4 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 2
  3. 3. 34 = 81. 2x = 4 X = 2. 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 3 If 32x = 81, what is the value of x?
  4. 4. 5 + 2 – 5 + 2 Either 1 or 2 None of these. 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 4 What is the conjugate of 5–2?
  5. 5. 1. 5 + 2 2. – 5 + 2 3. Either 1 or 2 4. None of these. 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 5 Conjugate is – 5 – 2
  6. 6. 2 – 2 2 – 3 3 – 2 2 – 2 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 6
  7. 7. 1 3+ 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 7 = 1 3+ 2 x 3− 2 3− 2
  8. 8. 1 3+ 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 8 = 1 3+ 2 x 3− 2 3− 2 = 3− 2 3−2
  9. 9. 1 3+ 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 9 = 1 3+ 2 x 3− 2 3− 2 = 3− 2 3−2 = 3 − 2
  10. 10. 1 3+ 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 10 = 1 3+ 2 x 3− 2 3− 2 = 3− 2 3−2 = 3 − 2 1 2 + 3 𝑋 2 − 3 2 − 3
  11. 11. 1 3+ 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 11 = 1 3+ 2 x 3− 2 3− 2 = 3− 2 3−2 = 3 − 2 1 2 + 3 𝑋 2 − 3 2 − 3 = 2 − 3 4 − 3
  12. 12. 1 3+ 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 12 = 1 3+ 2 x 3− 2 3− 2 = 3− 2 3−2 = 3 − 2 1 2 + 3 𝑋 2 − 3 2 − 3 = 2 − 3 4 − 3 =2− 3
  13. 13. 1 3+ 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 13 = 1 3+ 2 x 3− 2 3− 2 = 3− 2 3−2 = 3 − 2 1 2 + 3 𝑋 2 − 3 2 − 3 = 2 − 3 4 − 3 =2− 3 Adding we get 2 – 2
  14. 14. What is the positive square root of 48 − 27? 3 1 2 3 1 8 3 1 4 None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 14
  15. 15. 48 − 27 = 16(3) − 9(3) 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 15
  16. 16. 48 − 27 = 16(3) − 9(3) 4 3 − 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 16
  17. 17. 48 − 27 = 16(3) − 9(3) 4 3 − 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 17 3
  18. 18. What is the positive square root of 7 + 4 3 3+1 2 + 2 3 2 + 3 2 + 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 18
  19. 19. What is the positive square root of 7 + 4 3 3+1 2 + 2 3 2 + 3 2 + 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 19 7 + 4 3 = 7 + 2 3(4)
  20. 20. What is the positive square root of 7 + 4 3 3+1 2 + 2 3 2 + 3 2 + 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 20 7 + 4 3 = 7 + 2 3(4) 3 + 4 + 2 3(4) = 3 + 4 2
  21. 21. What is the positive square root of 72 − 40 2 5−1 4 2 5−1 4 2 3 − 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 21 4 2 5 − 3
  22. 22. What is the positive square root of 72 − 40 2 5−1 4 2 5−1 4 2 3 − 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 22 4 2 5 − 3 72 − 40
  23. 23. What is the positive square root of 72 − 40 2 5−1 4 2 5−1 4 2 3 − 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 23 4 2 5 − 3 72 − 40 36(2) − 20(2)
  24. 24. What is the positive square root of 72 − 40 2 5−1 4 2 5−1 4 2 3 − 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 24 4 2 5 − 3 72 − 40 36(2) − 20(2) 2 36 − 20
  25. 25. What is the positive square root of 72 − 40 2 5−1 4 2 5−1 4 2 3 − 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 25 4 2 5 − 3 72 − 40 36(2) − 20(2) 2 36 − 20 2 6 − 2 5
  26. 26. What is the positive square root of 72 − 40 2 5−1 4 2 5−1 4 2 3 − 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 26 4 2 5 − 3 72 − 40 36(2) − 20(2) 2 36 − 20 2 6 − 2 5 2 5 + 1 − 2 5
  27. 27. 5 3 2 3−6 1 3+2 2 3 = 31 3 + 21 3 31 3 − 21 3 21 3 − 31 3 32 3 + 22 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 27
  28. 28. 5 3 2 3−6 1 3+2 2 3 = 31 3 + 21 3 31 3 − 21 3 21 3 − 31 3 32 3 + 22 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 28 5 3 2 3−6 1 3+2 2 3 5 3 1 3 2 −(2) 1 3(3) 1 3+ 2 1 3 2
  29. 29. 5 3 2 3−6 1 3+2 2 3 = 31 3 + 21 3 31 3 − 21 3 21 3 − 31 3 32 3 + 22 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 29 5 3 2 3−6 1 3+2 2 3 5 3 1 3 2 −(2) 1 3(3) 1 3+ 2 1 3 2 5 3 1 3 2 −(2) 1 3(3) 1 3+ 2 1 3 2 𝑥 3 1 3+2 1 3 3 1 3+2 1 3
  30. 30. 5 3 2 3−6 1 3+2 2 3 = 32 3 + 21 3 31 3 − 21 3 21 3 − 31 3 32 3 + 22 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 30 5 3 2 3−6 1 3+2 2 3 5 3 1 3 2 −(2) 1 3(3) 1 3+ 2 1 3 2 5 3 1 3 2 −(2) 1 3(3) 1 3+ 2 1 3 2 𝑥 3 1 3+2 1 3 3 1 3+2 1 3 5 3 1 3+2 1 3 3+2
  31. 31. What is the rationalizing factor of 41 3 − 31 3 4 2 3 + 12 + 3 2 3 4 2 3 − 12 + 3 2 3 4 2 3 + 12 1 3 + 3 2 3 4 4 3 + 12 2 3 + 3 4 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 31
  32. 32. What is the rationalizing factor of 41 3 − 31 3 4 2 3 + 12 + 3 2 3 4 2 3 − 12 + 3 2 3 4 2 3 + 12 1 3 + 3 2 3 4 4 3 + 12 2 3 + 3 4 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 32 41 3 − 31 3 is of the form a – b
  33. 33. What is the rationalizing factor of 41 3 − 31 3 4 2 3 + 12 + 3 2 3 4 2 3 − 12 + 3 2 3 4 2 3 + 12 1 3 + 3 2 3 4 4 3 + 12 2 3 + 3 4 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 33 41 3 − 31 3 is of the form a – b a2+ab+ b2 should be multiplied to get a3−b3 4 2 3 + 4 1 33 1 3 + 3 2 3
  34. 34. Simplify: x a a+b+c.x b a+b+c. x c a+b+c 1 x 0 xabc 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 34
  35. 35. Which is greater? 1 8+ 5 , 1 2+ 5 1 8 + 5 1 2 + 5 Cannot be determined None of these. 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 35
  36. 36. Which is greater? 1 8+ 5 , 1 5+ 2 1 8+ 5 1 5+ 2 Cannot be determined None of these. 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 36 8 > 5
  37. 37. Which is greater? 1 8+ 5 , 1 5+ 2 1 8+ 5 1 5+ 2 Cannot be determined None of these. 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 37 8 > 5 5 > 2
  38. 38. Which is greater? 1 8+ 5 , 1 5+ 2 1 8+ 5 1 5+ 2 Cannot be determined None of these. 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 38 8 > 5 5 > 2 8 + 5 > 5 + 2
  39. 39. Which is greater? 222 , 322 222 322 Cannot be determined Both are equal 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 45
  40. 40. –6 – 3 5 5 − 2 – 5 − 2 None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 46 What is the rationalizing factor of 6 + 3 5
  41. 41. –6 – 3 5 5 − 2 – 5 − 2 None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 47 What is the rationalizing factor of 6 + 3 5 6 + 3 5
  42. 42. –6 – 3 5 5 − 2 – 5 − 2 None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 48 What is the rationalizing factor of 6 + 3 5 6 + 3 5 3(2) + 3 5
  43. 43. –6 – 3 5 5 − 2 – 5 − 2 None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 49 What is the rationalizing factor of 6 + 3 5 6 + 3 5 3(2) + 3 5 3(2 + 5)
  44. 44. –6 – 3 5 5 − 2 – 5 − 2 None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 50 What is the rationalizing factor of 6 + 3 5 6 + 3 5 3(2) + 3 5 3(2 + 5) 5 − 2 is the factor req.
  45. 45. –6 – 3 5 5 − 2 – 5 − 2 None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 51 What is the rationalizing factor of 6 + 3 5 6 + 3 5 3(2) + 3 5 3(2 + 5) 5 − 2 is the factor req.
  46. 46. –6 – 3 5 5 − 2 – 5 − 2 None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 52 What is the rationalizing factor of 6 + 3 5 6 + 3 5 3(2) + 3 5 3(2 + 5) 5 − 2 is the factor req.
  47. 47. What is the value of x if 2 𝑥 𝑥 = 16 0 1 3 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 53
  48. 48. If 2 𝑥 3 𝑦 = 216 and x and y are integers then x,y? 3,2 2,2 3,4 3,3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 54
  49. 49. If 3 1 23 3 23 5 23 7 2 = 9 𝑥 , what is the value of x? 4 3 2 1 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 55
  50. 50. (17)3.5 x (17)? = 178 2.29 2.75 4.25 4.5 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 56
  51. 51. If 𝑎 𝑏 𝑥−1 = 𝑏 𝑎 𝑥−3 ,then the value of x is 2 1 1 2 7 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 57
  52. 52. If 𝑎 𝑏 𝑥−1 = 𝑏 𝑎 𝑥−3 ,then the value of x is 2 1 1 2 7 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 58 𝑎 𝑏 𝑥−1 = 𝑏 𝑎 𝑥−3 𝑎 𝑏 𝑥−1 = 𝑎 𝑏 −(𝑥−3)
  53. 53. If 𝑎 𝑏 𝑥−1 = 𝑏 𝑎 𝑥−3 ,then the value of x is 2 1 1 2 7 2 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 59 𝑎 𝑏 𝑥−1 = 𝑏 𝑎 𝑥−3 𝑎 𝑏 𝑥−1 = 𝑎 𝑏 −(𝑥−3) x – 1 = 3 – x Solving x = 2
  54. 54. Given 100.48 = x, 100.7 = y and xz = y2, value of z = ? 1.45 1.88 2.9 3.7 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 60
  55. 55. Given 100.48 = x, 100.7 = y and xz = y2, value of z = ? 1.45 1.88 2.9 3.7 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 61 xz = y2 100.48z = 102*0.7
  56. 56. Given 100.48 = x, 100.7 = y and xz = y2, value of z = ? 1.45 1.88 2.9 3.7 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 62 xz = y2 100.48z = 102*0.7 0.48z = 1.4
  57. 57. Given 100.48 = x, 100.7 = y and xz = y2, value of z = ? 1.45 1.88 2.9 3.7 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 63 xz = y2 100.48z = 102*0.7 0.48z = 1.4 Z = 140 48 = 35 12 ≈ 2.9
  58. 58. If 5a = 3125, then the value of 5a – 3 is? 25 125 625 1625 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 64
  59. 59. If 3x – y = 27 and 3x + y = 243, then x =? 0 2 4 6 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 65
  60. 60. If 3x – y = 27 and 3x + y = 243, then x = ? 0 2 4 6 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 66 3x – y = 27 = 33 3x + y = 243 = 35
  61. 61. If 3x – y = 27 and 3x + y = 243, then x = ? 0 2 4 6 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 67 3x – y = 27 = 33 3x + y = 243 = 35 x – y = 3
  62. 62. If 3x – y = 27 and 3x + y = 243, then x = ? 0 2 4 6 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 68 3x – y = 27 = 33 3x + y = 243 = 35 x – y = 3 x + y = 5
  63. 63. If 3x – y = 27 and 3x + y = 243, then x = ? 0 2 4 6 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 69 3x – y = 27 = 33 3x + y = 243 = 35 x – y = 3 x + y = 5 Solving x = 4
  64. 64. 2560.16 x 2560.09 = ? 4 16 64 256.25 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 70
  65. 65. 2560.16 x 2560.09 = ? 4 16 64 256.25 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 71 2560.16 x 2560.09 = 2560.25
  66. 66. 2560.16 x 2560.09 = ? 4 16 64 256.25 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 72 2560.16 x 2560.09 = 2560.25 256 = 44
  67. 67. 1 1+x b−a +x c−a + 1 1+x a−b +x c−b + 1 1+x b−c +x a−c = ? 0 1 Xa – b – c None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 73
  68. 68. 1 1+x b−a +x c−a + 1 1+x a−b +x c−b + 1 1+x b−c +x a−c = ? 0 1 Xa – b – c None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 74 1 1+ xb xa + xc xa + 1 1+ xa xb + xc xb + 1 1+ xb xc + xa xc
  69. 69. 1 1+x b−a +x c−a + 1 1+x a−b +x c−b + 1 1+x b−c +x a−c = ? 0 1 Xa – b – c None of these 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 75 1 1+ xb xa + xc xa + 1 1+ xa xb + xc xb + 1 1+ xb xc + xa xc xa xa+xb+xc + xb xa+xb+xc + xc xa+xb+xc
  70. 70. 257.5 x 52.5 ÷1251.5 = 5? 8.5 13 16 17.5 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 76
  71. 71. 243 n 5 X 32n+1 9nX3n−1 1 2 9 3n 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 77
  72. 72. 243 n 5 X 32n+1 9nX3n−1 1 2 9 3n 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 78 35 n 5 X32n+1 32 n X 3n−1
  73. 73. 243 n 5 X 32n+1 9nX3n−1 1 2 9 3n 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 79 35 n 5 X32n+1 32 n X 3n−1 = 3 𝑛+2𝑛+1 32𝑛+𝑛−1
  74. 74. 243 n 5 X 32n+1 9nX3n−1 1 2 9 3n 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 80 35 n 5 X32n+1 32 n X 3n−1 = 3n+2n+1 32n+n−1 = 33n + 1 – 3n + 1
  75. 75. 1 1+a n−m + 1 1+a m−n = ? 0 1 2 1 am+n 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 81
  76. 76. 1 1+a n−m + 1 1+a m−n = ? 0 1 2 1 am+n 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 82 1 1+an am + 1 1+am an =
  77. 77. 1 1+a n−m + 1 1+a m−n = ? 0 1 2 1 am+n 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 83 1 1+an am + 1 1+am an = 𝑎 𝑚 𝑎 𝑚+𝑎 𝑛 + 𝑎 𝑛 𝑎 𝑚+𝑎 𝑛
  78. 78. If mn = 121, the value of (m – 1)n+1 is 1 10 121 1000 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 84
  79. 79. xb xc b + c − a xc xa c +a −b xa xb a +b −c xabc 1 xab+bc+ca xa+b+c 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 85
  80. 80. xb xc b + c − a xc xa c +a −b xa xb a +b −c xabc 1 xab+bc+ca xa+b+c 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 86 xb xc b + c − a xc xa c +a −b xa xb a +b −c
  81. 81. xb xc b + c − a xc xa c +a −b xa xb a +b −c xabc 1 xab+bc+ca xa+b+c 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 87 xb xc b + c − a xc xa c +a −b xa xb a +b −c = x(b - c)(b + c - a) . x(c - a)(c + a - b) . x(a - b)(a + b - c)
  82. 82. xb xc b + c − a xc xa c +a −b xa xb a +b −c xabc 1 xab+bc+ca xa+b+c 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 88 xb xc b + c − a xc xa c +a −b xa xb a +b −c = x(b - c)(b + c - a) . x(c - a)(c + a - b) . x(a - b)(a + b - c) = x(b - c)(b + c) – (b-c)(a) . x(c - a)(c + a) – (c-a)(b) . x(a - b)(a + b – (a-b)(c)
  83. 83. xb xc b + c − a xc xa c +a −b xa xb a +b −c xabc 1 xab+bc+ca xa+b+c 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 89 xb xc b + c − a xc xa c +a −b xa xb a +b −c = x(b - c)(b + c - a) . x(c - a)(c + a - b) . x(a - b)(a + b - c) = x(b - c)(b + c) – (b-c)(a) . x(c - a)(c + a) – (c-a)(b) . x(a - b)(a + b – (a-b)(c) =𝑥 𝑏2−𝑐2+𝑐2−𝑎2+𝑎2−𝑏2 ∗ 𝑥 𝑏𝑎−𝑐𝑎+𝑏𝑐−𝑎𝑏+𝑎𝑐−𝑏𝑐
  84. 84. If x = 3 + 2 2, then 𝑥 − 1 𝑥 is 1 2 2 2 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 90
  85. 85. If x = 3 + 2 2, then 𝑥 − 1 𝑥 is 1 2 2 2 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 91 X = 3 + 2 2
  86. 86. If x = 3 + 2 2, then 𝑥 − 1 𝑥 is 1 2 2 2 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 92 X = 3 + 2 2 𝑥 − 1 𝑥 2 = 𝑥 + 1 𝑥 − 2
  87. 87. If x = 3 + 2 2, then 𝑥 − 1 𝑥 is 1 2 2 2 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 93 X = 3 + 2 2 𝑥 − 1 𝑥 2 = 𝑥 + 1 𝑥 − 2 = 3 + 2 2 + 1 3 + 2 2 − 2
  88. 88. If x = 3 + 2 2, then 𝑥 − 1 𝑥 is 1 2 2 2 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 94 X = 3 + 2 2 𝑥 − 1 𝑥 2 = 𝑥 + 1 𝑥 − 2 = 3 + 2 2 + 1 3 + 2 2 − 2 = 3 + 2 2 + 1 3 + 2 2 𝑥 3 − 2 2 3 − 2 2
  89. 89. If x = 3 + 2 2, then 𝑥 − 1 𝑥 is 1 2 2 2 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 95 X = 3 + 2 2 𝑥 − 1 𝑥 2 = 𝑥 + 1 𝑥 − 2 = 3 + 2 2 + 1 3 + 2 2 − 2 = 3 + 2 2 + 1 3 + 2 2 𝑥 3 − 2 2 3 − 2 2 = 3 + 2 2 + (3 - 2 2) − 2
  90. 90. If x = 3 + 2 2, then 𝑥 − 1 𝑥 is 1 2 2 2 3 3 12-Jun-15 C.S.VEERARAGAVAN 98948 34264 veeraa1729@gmail.com 96 X = 3 + 2 2 𝑥 − 1 𝑥 2 = 𝑥 + 1 𝑥 − 2 = 3 + 2 2 + 1 3 + 2 2 − 2 = 3 + 2 2 + 1 3 + 2 2 𝑥 3 − 2 2 3 − 2 2 = 3 + 2 2 + (3 - 2 2) − 2 = 4

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  • aanandkumarsahu

    Jun. 12, 2015
  • SumitKaushik31

    Apr. 12, 2020

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