Tiêu đề 7. CUBE AND CUBE ROOTS (Ex).pdf ✅

7. CUBE AND CUBE ROOTS EXPLANATION 1. 2 8 × 8 × 8 = 512. 2. 1 1 × 1 × 1 = 1. 3. Both A and R are true and R is the correct explanation of A. 4. 343 5. 1729 20683 = 10 3 + 27 3 = 19 3 +24 3 13832 = 20 3 +18 3 = 24 3 + 2 3 4104 = 2 3 + 16 3 = 9 3 + 15 3 1729 = 10 3 + 9 3 12 3 + 1 3 6. A is false but R is true. 7. 100 100 = 2 × 2 × 5 × 5 = 2 2× 5 2 8. Odd. 3 × 3 × 3 = 27 (odd). 9. 12167 Cube of 23 = 23 × 23 × 23 = 12167 10. 5 5 × 5 × 5 = 125. 11. 2 250 = 5 × 5 × 5 × 2 = 5 3 × 2. 12. 729cm 3 The volume of the cubical box is given by: (side) 3 The volume of the cubical box with side 9cm. = (9cm) 3=9cm × 9cm × 9cm = 729cm 3 13. even We know, the multiplication of 3 even numbers, i.e. the cube of an even natural number, will always be even Example, consider the even natural numbers 2 and 4. Then, their cube is 2 3 = 8 and 4 3 = 64, whose units place is even. That is, the cubes are also even. Hence, we can say, cube of even natural number is even. Therefore, option A is correct. 14. 216 121 = 11 × 11 169 = 13 × 13 196 = 7 × 7 × 2 × 2 216 = 2 × 2 × 2 × 3 × 3 × 3 = (2) 3 × (3) 3 = (6) 3 216 = (6) 3 Hence, 216 is a perfect cube. 15. 3 10 3 = 1000 20 3 = 8000 30 3 = 27000 16. 5 Square of 5 = 5 × 5 = 25 Cube of 5 = 5 × 5 × 5 = 125 125 - 25 = 100 17. 8 The unit digit of 242 is 2 Cube of 2 = 2 × 2 × 2 = 8 18. 125 125 = 5 × 5 × 5 = 53. 19. 8 √512 3 2 152 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2 2 1 = √2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 3 = √2 3 × 2 3 × 2 3 3 = 2 × 2 × 2 = 8 20. 1 Multiple Choice Question Easy 21. A is false but R is true. 22. 6 If the digit in one's place of a number is a, then the last digit of its cube will be the last digit of its cube. Thus, if the digit in one's place of a number is 6, then the last digit of its cube will be unit digit of. We know, the cube of 6, i.e. 6 3 = 216, Since the last digit of cube of 6 is 6. 23. 0 0 × 0 × 0 = 0. 24. Odd 25. 4 7
√ 64 343 3 = √ 4 × 4 × 4 7 × 7 × 7 3 = √ 4 3 7 3 3 = 4 7 26. 11 88 = 2 × 2 × 2 × 11 = 2 3 × 11. 27. 7 The prime factorization of 6125 is: 5 × 5 × 5 × 7 × 7 Here the prime factor 7 does not appear in a group of three. To make it a perfect number, we need one more 7 In that case 6125 × 7 = 5 × 5 × 5 × 7 × 7 = 42875 which is a perfect cube. 28. (10y+x) 3 29. Cube of negative numbers is negative. As we know that negative number times a negative number gives a positive number and positive number times a negative number gives a negative number. Example: (-5) 3 = -5 × -5 × -5 = -125 30. 343 343 is a Perfect Cube Number, It is cube of 7 7 3 = 7 × 7 × 7 = 343 31. 10000 Soultion: By prime factorisation method. 10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5 = 2 4 × 5 4 = 2 3 × 2 × 5 3 × 5 32. 5 200 = 2 × 2 × 2 × 5 × 5 = 2 3× 5 × 5. 33. 8 If the digit in one's place of a number is a, then the last digit of its cube will be the last digit of its cube. Thus, if the digit in one's place of a number is 2, then the last digit of its cube will be unit digit of. We know, the cube of 2, i.e. 2 3 = 8, Since the last digit of cube of 2 is 8. 34. A is false but R is true. 35. 4 Prime factorising, we get, 64 = 2 × 2 × 2 × 2 × 2 × 2 = 4 × 4 × 4. Here, the factor 4 occur as triplet. Hence, it is a perfect cube. Therefore, cube root of 64, i.e. √64 3 = 4 36. Both A and R are true and R is the correct explanation of A. 37. There is no perfect cube which ends with 8. 1728 = 12 3 . 38. Both A and R are true and R is the correct explanation of A. 39. 60 The given number is 216000. It can be expressed as: 216000 = √216 × 103 3 = 6 × 10 = 60 40. 3 Let us consider few examples. Cube of 7 = 7 3 = 7 × 7 × 7 = 343 Cube of 17 17 3 = 17 × 17 × 17 = 4913 Cube of 27 27 3 = 27 × 27 × 27 = 19683 From the above examples, we can see that cube of the numbers with 7 at the unit's place end, with 3 at their unit's place. 41. 125 42. Both A and R are true and R is the correct explanation of A. 43. Both A and R are true and R is the correct explanation of A. 44. 7 53 3 = 53 x 53 x 53 3 3 = 3x3x3 = 27 Hence, at the unit place, we will get 7. Recheck: 53 3 = 53 x 53 x 53 = 148877 45. A is false but R is true. 46. 2 108 = 2 × 2 × 3 × 3 × 3 = 2 × 2 × 3 3 . 47. 2.197 Cube of the number 1.3: (1.3) 3=1.3 × 1.3 × 1.3 = 2.197. Hence, option A is correct. 48. Both A and R are true and R is the correct explanation of A. 49. 2 197 1000 (1 3 10 ) 3

factorization 3x3 remains after groping 5' s and 2' s in triplets Hence, it is not a perfect. square 67. 10 10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 ×5 = 2 3 × 2 × 5 3 × 5. 68. A is false but R is true. 69. 7 53 3 = 53 x 53 x 53 3 3 = 3 x 3 x 3 = 27 Hence, at the unit place, we will get 7 Recheck: 53 3 = 53 x 53 x 53 = 148877 70. 6859 71. 2 ∵ Number of zeroes at the end of the cube = 6 ∴ Number of zeroes at the end of the cube root = 6 3 = 2. 72. 13.824 cu m 73. 24 √216 × 64 3 = √216 3 × √64 3 = √2 × 2 × 2 × 3 × 3 × 3 3 × √2 × 2 × 2 × 2 × 2 × 2 3 = √(2) 3 × (3) 3 3 × √(2) 3 × (2) 3 3 = √(6) 3 3 × √(4) 3 3 = 6 × 4 √216 × 64 3 = 24 ∴ √216 × 64 3 = 24 74. − 5 8 The given fraction is − 125 512 It can be expressed as: − 125 512 = − 5 8 × 5 8 × 5 8 75. A is false but R is true. 76. 7 If the digit in one's place of a number is a, then the last digit of its cube will be the last digit of its cube. Thus, if the digit in one's place of a number is 3, then the last digit of its cube will be unit digit of. We know, the cube of 3, i.e. 3 3 = 27, Since the last digit of cube of 3 is 7 77. 6 36 = 2 × 2 × 3 × 3. 78. Both A and R are true but R is not the correct explanation of A. 79. 3 The prime factorisation of 81 will be: 81 = 3 × 3 × 3 × 3 81 = 3 3 × 3 Hence, we need to divide 81 by 3 to get: 81 3 = 27 = 3 3 80. 5 625 = 5 × 5 × 5 × 5 = 5 3 × 5. 81. 3 3 × 3 × 3 = 27. 82. 25 Volume = 5 × 4 × 2 = 5 × 2 × 2 × 2 = 5 × 2 3 . 83. 24 84. 243 85. 0.12 86. A is false but R is true. 87. 0.512 The cube of 0.8 is given as: (0.8) 3 = ( 8 10 ) 3 = 512 1000 = 0.512 88. 0.512 (0.8) 3 = 0.8 × 0.8 × 0.8 = 0.512 89. Both A and R are true but R is not the correct explanation of A. 90. A is false but R is true. 91. -28 92. 22 The prime factorization of 7546 is: 2 × 7 × 7 × 7 × 11 Here, the primes 2 and 11 do not appear in the group of three. So, we need to divide 7546 by 2 × 11 = 22 to make it a perfect cube. ⇒ 7546 22 = 343 = (7) 3 93. Even. 6 × 6 × 6 = 216 (even). 94. Both A and R are true and R is the correct explanation of A. 95. Both A and R are true and R is the correct explanation of A. 96. Cube of that number If a number is raised to the power 3, then it is called the cube of that number. Hence, option A is correct.