Memorize These: 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225 16² = 256, 17² = 289, 18² = 324, 19² = 361, 20² = 400 21² = 441, 22² = 484, 23² = 529, 24² = 576, 25² = 625 26² = 676, 27² = 729, 28² = 784, 29² = 841, 30² = 900
Memorize These: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000 11³ = 1331, 12³ = 1728, 13³ = 2197, 14³ = 2744, 15³ = 3375 16³ = 4096, 17³ = 4913, 18³ = 5832, 19³ = 6859, 20³ = 8000
Q1. Find 23² Solution: 23² = (20+3)² = 20² + 2×20×3 + 3² = 400 + 120 + 9 = 529
Q2. Find 37² Solution: 37² = (40-3)² = 40² – 2×40×3 + 3² = 1600 – 240 + 9 = 1369
Q3. Find 105² Solution: 105² = (100+5)² = 100² + 2×100×5 + 5² = 10000 + 1000 + 25 = 11025
Q4. Find 95² Solution: 95² = (100-5)² = 100² – 2×100×5 + 5² = 10000 – 1000 + 25 = 9025
Q5. Find 72² Solution: 72² = (70+2)² = 70² + 2×70×2 + 2² = 4900 + 280 + 4 = 5184
Q6. Find 88² Solution: 88² = (90-2)² = 90² – 2×90×2 + 2² = 8100 – 360 + 4 = 7744
Q7. Find 64² Solution: 64² = (60+4)² = 60² + 2×60×4 + 4² = 3600 + 480 + 16 = 4096
Q8. Find 116² Solution: 116² = (120-4)² = 120² – 2×120×4 + 4² = 14400 – 960 + 16 = 13456
Q9. Find 85² using (80+5)² Solution: 85² = 80² + 2×80×5 + 5² = 6400 + 800 + 25 = 7225
Q10. Find 76² using (80-4)² Solution: 76² = 80² – 2×80×4 + 4² = 6400 – 640 + 16 = 5776
Q11. Find 25² Solution: For numbers ending in 5: 25² = 2×3×100 + 25 = 625
Q12. Find 35² Solution: 35² = 3×4×100 + 25 = 1225
Q13. Find 45² Solution: 45² = 4×5×100 + 25 = 2025
Q14. Find 65² Solution: 65² = 6×7×100 + 25 = 4225
Q15. Find 75² Solution: 75² = 7×8×100 + 25 = 5625
Q16. Find 85² Solution: 85² = 8×9×100 + 25 = 7225
Q17. Find 125² Solution: 125² = 12×13×100 + 25 = 15625
Q18. Find 11² Solution: Pattern: 11² = 121 (1+1 = 2 in middle)
Q19. Find 111² Solution: 111² = 12321 (1+1+1 = 3, 1+1 = 2 at edges)
Q20. Find 1111² Solution: 1111² = 1234321
Q21. Find 13³ Solution: 13³ = (10+3)³ = 10³ + 3×10²×3 + 3×10×3² + 3³ = 1000 + 900 + 270 + 27 = 2197
Q22. Find 23³ Solution: 23³ = (20+3)³ = 20³ + 3×20²×3 + 3×20×3² + 3³ = 8000 + 3600 + 540 + 27 = 12167
Q23. Find 17³ Solution: 17³ = (20-3)³ = 20³ – 3×20²×3 + 3×20×3² – 3³ = 8000 – 3600 + 540 – 27 = 4913
Q24. Find 27³ Solution: 27³ = (30-3)³ = 30³ – 3×30²×3 + 3×30×3² – 3³ = 27000 – 8100 + 810 – 27 = 19683
Q25. Find 103³ Solution: 103³ = (100+3)³ = 100³ + 3×100²×3 + 3×100×3² + 3³ = 1000000 + 90000 + 2700 + 27 = 1092727
Q26. Find 97³ Solution: 97³ = (100-3)³ = 100³ – 3×100²×3 + 3×100×3² – 3³ = 1000000 – 90000 + 2700 – 27 = 912673
Q27. Find 21³ Solution: 21³ = (20+1)³ = 20³ + 3×20²×1 + 3×20×1² + 1³ = 8000 + 1200 + 60 + 1 = 9261
Q28. Find 19³ Solution: 19³ = (20-1)³ = 20³ – 3×20²×1 + 3×20×1² – 1³ = 8000 – 1200 + 60 – 1 = 6859
Q29. Find 101³ Solution: 101³ = (100+1)³ = 1000000 + 30000 + 300 + 1 = 1030301
Q30. Find 99³ Solution: 99³ = (100-1)³ = 1000000 – 30000 + 300 – 1 = 970299
Q31. Find √144 Solution: 144 = 2⁴ × 3² = (2² × 3)² = 12² √144 = 12
Q32. Find √225 Solution: 225 = 3² × 5² = (3 × 5)² = 15² √225 = 15
Q33. Find √324 Solution: 324 = 2² × 3⁴ = 2² × (3²)² = (2 × 3²)² = 18² √324 = 18
Q34. Find √441 Solution: 441 = 3² × 7² = (3 × 7)² = 21² √441 = 21
Q35. Find √576 Solution: 576 = 2⁶ × 3² = (2³ × 3)² = 24² √576 = 24
Q36. Find √729 Solution: 729 = 3⁶ = (3³)² = 27² √729 = 27
Q37. Find √1024 Solution: 1024 = 2¹⁰ = (2⁵)² = 32² √1024 = 32
Q38. Find √1296 Solution: 1296 = 2⁴ × 3⁴ = (2² × 3²)² = 36² √1296 = 36
Q39. Find √1600 Solution: 1600 = 2⁶ × 5² = (2³ × 5)² = 40² √1600 = 40
Q40. Find √2025 Solution: 2025 = 3⁴ × 5² = (3² × 5)² = 45² √2025 = 45
Q41. Find √484 using division method Solution:
2 2
----
2 | 484
4
----
42| 84
84
----
0
√484 = 22
Q42. Find √676 using division method Solution:
2 6
----
2 | 676
4
----
46| 276
276
----
0
√676 = 26
Q43. Find √1369 using division method Solution:
3 7
----
3 | 1369
9
----
67| 469
469
----
0
√1369 = 37
Q44. Find √1764 using division method Solution: √1764 = 42
Q45. Find √2304 using division method Solution: √2304 = 48
Q46. Find √2916 using division method Solution: √2916 = 54
Q47. Find √3249 using division method Solution: √3249 = 57
Q48. Find √3844 using division method Solution: √3844 = 62
Q49. Find √4624 using division method Solution: √4624 = 68
Q50. Find √5329 using division method Solution: √5329 = 73
Q51. Find ∛216 Solution: 216 = 2³ × 3³ = (2 × 3)³ = 6³ ∛216 = 6
Q52. Find ∛343 Solution: 343 = 7³ ∛343 = 7
Q53. Find ∛512 Solution: 512 = 2⁹ = (2³)³ = 8³ ∛512 = 8
Q54. Find ∛1000 Solution: 1000 = 10³ ∛1000 = 10
Q55. Find ∛1331 Solution: 1331 = 11³ ∛1331 = 11
Q56. Find ∛1728 Solution: 1728 = 2⁶ × 3³ = (2² × 3)³ = 12³ ∛1728 = 12
Q57. Find ∛2197 Solution: 2197 = 13³ ∛2197 = 13
Q58. Find ∛2744 Solution: 2744 = 14³ ∛2744 = 14
Q59. Find ∛3375 Solution: 3375 = 3³ × 5³ = (3 × 5)³ = 15³ ∛3375 = 15
Q60. Find ∛4096 Solution: 4096 = 2¹² = (2⁴)³ = 16³ ∛4096 = 16
Q61. Find ∛4913 Solution:
Q62. Find ∛5832 Solution:
Q63. Find ∛6859 Solution:
Q64. Find ∛8000 Solution:
Q65. Find ∛9261 Solution:
Q66. Find ∛10648 Solution:
Q67. Find ∛12167 Solution:
Q68. Find ∛13824 Solution:
Q69. Find ∛15625 Solution:
Q70. Find ∛17576 Solution:
Q71. Find √0.16 Solution: √0.16 = √(16/100) = √16/√100 = 4/10 = 0.4
Q72. Find √0.25 Solution: √0.25 = √(25/100) = 5/10 = 0.5
Q73. Find √0.36 Solution: √0.36 = √(36/100) = 6/10 = 0.6
Q74. Find √0.64 Solution: √0.64 = √(64/100) = 8/10 = 0.8
Q75. Find √0.81 Solution: √0.81 = √(81/100) = 9/10 = 0.9
Q76. Find √1.44 Solution: √1.44 = √(144/100) = 12/10 = 1.2
Q77. Find √2.25 Solution: √2.25 = √(225/100) = 15/10 = 1.5
Q78. Find √4.84 Solution: √4.84 = √(484/100) = 22/10 = 2.2
Q79. Find √6.25 Solution: √6.25 = √(625/100) = 25/10 = 2.5
Q80. Find √10.24 Solution: √10.24 = √(1024/100) = 32/10 = 3.2
Q81. Find ∛0.008 Solution: ∛0.008 = ∛(8/1000) = ∛8/∛1000 = 2/10 = 0.2
Q82. Find ∛0.027 Solution: ∛0.027 = ∛(27/1000) = 3/10 = 0.3
Q83. Find ∛0.064 Solution: ∛0.064 = ∛(64/1000) = 4/10 = 0.4
Q84. Find ∛0.125 Solution: ∛0.125 = ∛(125/1000) = 5/10 = 0.5
Q85. Find ∛0.216 Solution: ∛0.216 = ∛(216/1000) = 6/10 = 0.6
Q86. Find ∛1.728 Solution: ∛1.728 = ∛(1728/1000) = 12/10 = 1.2
Q87. Find ∛2.197 Solution: ∛2.197 = ∛(2197/1000) = 13/10 = 1.3
Q88. Find ∛3.375 Solution: ∛3.375 = ∛(3375/1000) = 15/10 = 1.5
Q89. Find ∛4.096 Solution: ∛4.096 = ∛(4096/1000) = 16/10 = 1.6
Q90. Find ∛8.000 Solution: ∛8.000 = ∛8 = 2
Q91. Find √50 (approximate) Solution: Since 7² = 49 and 8² = 64, √50 ≈ 7.1
Q92. Find √80 (approximate) Solution: Since 8² = 64 and 9² = 81, √80 ≈ 8.9
Q93. Find √200 (approximate) Solution: Since 14² = 196 and 15² = 225, √200 ≈ 14.1
Q94. Find √300 (approximate) Solution: Since 17² = 289 and 18² = 324, √300 ≈ 17.3
Q95. Find √500 (approximate) Solution: Since 22² = 484 and 23² = 529, √500 ≈ 22.4
Q96. Find √98 (simplify) Solution: √98 = √(49 × 2) = √49 × √2 = 7√2
Q97. Find √128 (simplify) Solution: √128 = √(64 × 2) = √64 × √2 = 8√2
Q98. Find √162 (simplify) Solution: √162 = √(81 × 2) = √81 × √2 = 9√2
Q99. Find √288 (simplify) Solution: √288 = √(144 × 2) = √144 × √2 = 12√2
Q100. Find √450 (simplify) Solution: √450 = √(225 × 2) = √225 × √2 = 15√2
Q101. If √x = 8, find x Solution: x = 8² = 64
Q102. If ∛x = 5, find x Solution: x = 5³ = 125
Q103. Find (√49)² Solution: (√49)² = (7)² = 49
Q104. Find √(15²) Solution: √(15²) = 15
Q105. Find ∛(12³) Solution: ∛(12³) = 12
Q106. Find √25 × √36 Solution: √25 × √36 = 5 × 6 = 30
Q107. Find √(25 × 36) Solution: √(25 × 36) = √900 = 30
Q108. Find √144 ÷ √9 Solution: √144 ÷ √9 = 12 ÷ 3 = 4
Q109. Find √(144 ÷ 9) Solution: √(144 ÷ 9) = √16 = 4
Q110. Find ∛8 × ∛27 Solution: ∛8 × ∛27 = 2 × 3 = 6
Q111. A square field has area 1024 sq.m. Find the side length. Solution: Side = √1024 = 32 m
Q112. Find the number whose square is 2025. Solution: Number = √2025 = 45
Q113. A square has perimeter 48 cm. Find its area. Solution: Side = 48/4 = 12 cm, Area = 12² = 144 sq.cm
Q114. If area of square is 441 sq.cm, find perimeter. Solution: Side = √441 = 21 cm, Perimeter = 4×21 = 84 cm
Q115. 361 students are arranged in square formation. How many rows? Solution: Rows = √361 = 19
Q116. A square plot has area 625 sq.m. Find cost of fencing at ₹25 per meter. Solution: Side = √625 = 25 m, Perimeter = 100 m, Cost = 100×25 = ₹2500
Q117. Find the smallest number to be added to 8200 to make it a perfect square. Solution: √8200 ≈ 90.55, Next perfect square = 91² = 8281 Number to add = 8281 – 8200 = 81
Q118. Find the largest 4-digit perfect square. Solution: √9999 ≈ 99.99, So largest = 99² = 9801
Q119. Find the smallest 3-digit perfect square. Solution: √100 = 10, So smallest = 10² = 100
Q120. A rectangular field 40m × 25m is to be converted to square of same area. Find side. Solution: Area = 40×25 = 1000 sq.m, Side = √1000 ≈ 31.6 m
Q121. A cube has volume 1728 cu.cm. Find the edge length. Solution: Edge = ∛1728 = 12 cm
Q122. Find the number whose cube is 2197. Solution: Number = ∛2197 = 13
Q123. A cube has surface area 1734 sq.cm. Find edge length. Solution: 6a² = 1734, a² = 289, a = √289 = 17 cm
Q124. Volume of cube is 3375 cu.cm. Find total surface area. Solution: Edge = ∛3375 = 15 cm, Surface area = 6×15² = 1350 sq.cm
Q125. 4096 small cubes of edge 1 cm form a large cube. Find edge of large cube. Solution: Edge = ∛4096 = 16 cm
Q126. A cube has edge 20 cm. Find volume and surface area. Solution: Volume = 20³ = 8000 cu.cm, Surface area = 6×20² = 2400 sq.cm
Q127. How many unit cubes fit in a cube of edge 12 cm? Solution: Number = 12³ = 1728
Q128. Find the smallest number to be subtracted from 2700 to make it a perfect cube. Solution: ∛2700 ≈ 13.9, Previous perfect cube = 13³ = 2197 Number to subtract = 2700 – 2197 = 503
Q129. A cubical tank holds 8000 liters. Find the edge length. Solution: Edge = ∛8000 = 20 cm = 0.2 m
Q130. Find the largest 3-digit perfect cube. Solution: ∛999 ≈ 9.99, So largest = 9³ = 729
Q131. Between which two perfect squares does 50 lie? Solution: 7² = 49 < 50 < 64 = 8²
Q132. Between which two perfect squares does 120 lie? Solution: 10² = 100 < 120 < 121 = 11²
Q133. Between which two perfect cubes does 100 lie? Solution: 4³ = 64 < 100 < 125 = 5³
Q134. Between which two perfect cubes does 2000 lie? Solution: 12³ = 1728 < 2000 < 2197 = 13³
Q135. Find √40 to one decimal place. Solution: 6² = 36 < 40 < 49 = 7², so √40 ≈ 6.3
Q136. Find ∛70 to one decimal place. Solution: 4³ = 64 < 70 < 125 = 5³, so ∛70 ≈ 4.1
Q137. Between which two integers does √85 lie? Solution: 9² = 81 < 85 < 100 = 10², so 9 < √85 < 10
Q138. Between which two integers does ∛250 lie? Solution: 6³ = 216 < 250 < 343 = 7³, so 6 < ∛250 < 7
Q139. Estimate √150 to nearest integer. Solution: 12² = 144 < 150 < 169 = 13², closer to 144, so √150 ≈ 12
Q140. Estimate ∛500 to nearest integer. Solution: 7³ = 343 < 500 < 512 = 8³, closer to 512, so ∛500 ≈ 8
Q141. Find 13² – 12² Solution: Using a² – b² = (a+b)(a-b): 13² – 12² = (13+12)(13-12) = 25×1 = 25
Q142. Find 25² – 24² Solution: 25² – 24² = (25+24)(25-24) = 49×1 = 49
Q143. Find 101² – 99² Solution: 101² – 99² = (101+99)(101-99) = 200×2 = 400
Q144. Find 1² + 2² + 3² + … + 10² Solution: Using formula n(n+1)(2n+1)/6: 10×11×21/6 = 385
Q145. Find 1³ + 2³ + 3³ + … + 10³ Solution: Using formula [n(n+1)/2]²: [10×11/2]² = 55² = 3025
Q146. Find 15² + 15² – 2×15² Solution: 15² + 15² – 2×15² = 0
Q147. Find 20² – 19² + 18² – 17² Solution: (20²-19²) + (18²-17²) = (39×1) + (35×1) = 39 + 35 = 74
Q148. Find (√5)² + (√7)² – (√12)² Solution: 5 + 7 – 12 = 0
Q149. Find ∛8 + ∛27 + ∛64 Solution: 2 + 3 + 4 = 9
Q150. Find √16 + √25 – √9 Solution: 4 + 5 – 3 = 6
Q151. Find ⁴√16 Solution: ⁴√16 = ⁴√(2⁴) = 2
Q152. Find ⁴√81 Solution: ⁴√81 = ⁴√(3⁴) = 3
Q153. Find ⁴√625 Solution: ⁴√625 = ⁴√(5⁴) = 5
Q154. Find ⁶√64 Solution: ⁶√64 = ⁶√(2⁶) = 2
Q155. Find ⁵√32 Solution: ⁵√32 = ⁵√(2⁵) = 2
Q156. Find ⁵√243 **Solution