Cubes and Cube Roots Worksheet

Multiple Choice Questions

1. Find the perfect cube number of the smallest odd number
   (a)1
   (b)8
   (c)27
   (d)54|
   
   
2. Which of the following can’t be the cube of a number given by 2n, where ‘n’ is a natural number
   (a)1728
   (b)2744
   (c)4096
   (d)4913
   
   
3. 17576 is the cube of
   (a)24
   (b)25
   (c)26
   (d)27
   
   
4. 729 can be represented as the sum of  _______ consecutive odd numbers
   (a)6
   (b)7
   (c)8
   (d)9
   
   
5. Which of the following options is not a perfect cube
   (a)4×6×9
   (b)25×16×20
   (c)27×4×8
   (d)7×14×28
   
   
6. Which of the following options is the cube root of number 195112
   (a)52
   (b)54
   (c)56
   (d)58
   
   
7. The smallest number to be added or subtracted to convert 279 to perfect cube is
   (a)61
   (b)62
   (c)63
   (d)64
   
   
8. Which of the following statement is false
   (a)Cube root of two digit number is a one digit number
   (b)Cube root of three digit number is a two digit number
   (c)Cube of a one digit number can be a one digit number
   (d)Cube of a two digit number can be a six digit number
   
   
9. Which of the following option can be the cube root of (2n+1)3, where ‘n’ is a natural number
   (a)12
   (b)13
   (c)14
   (d)16
   
   
10. The smallest number to be multiplied or divided to convert 6750 to perfect cube is
    (a)5
    (b)4
    (c)2
    (d)3
    
    
    Subjective Questions
    
    
11. Cube of a number ends with the same digit as the number itself. So the number will have which digit in its unit’s place 
    
    
12. Ram wanted to buy a new smartphone which costs 193 rupees. If he had 203 rupees with him, then find the amount left with him after the purchase of the smartphone.
    
    
13. Ravi and Rani are solving homework given by their teacher, wherein they were asked to find out three, four-digit cube numbers. Now Ravi writes down the following numbers, 4096, 5834 and 9261. Now Rani realizes that Ravi’s one number is wrong. So identify the number and suggest how you can convert it into a perfect cube.
    
    
14. Arjun is fascinated by the way a cube number can be represented using the sum of odd consecutive numbers as shown
    13 = 1
    23 = 3 + 5
    33 = 7 + 9 + 11
    43 = 13+15+17+19
    53 = 21 + 23 + 25 + 27 + 29
    Now he decides to represent 343 as the sum of odd consecutive number. Help him to find the solution
    
    
15. Ganesh and Niraj were arguing on whether 1512 is a perfect cube number or not. Ganesh says that 1512 is a perfect cube and Niraj says that it is not. Decide who is correct in their observation.
    
    
16. Now Raj and Raju were finding the cube root of 9261. Raj came up with the answer 21, now Raju is not convinced that the answer is correct. Help out Raj in convincing convince Raju.
    
    
17. “If two perfect cubes are multiplied together, then the resulting product is also an perfect cube”. Explain this statement with an example
    
    
18. Without using prime factorization, prove that 4913 is not the cube of 18. 
    
    
19. Prove that 270000 is not a perfect cube without using prime factorization.
    
    
20. Niraj likes to play video games and give below are the points earned or lost by him on a given day
    
    Find the difference between points earned by him on consecutive days the letters to form meaningful words based on the clues provided.

Multiple Choice Questions

1. Find the perfect cube number of the smallest odd number
   (a)1
   (b)8
   (c)27
   (d)54
   Correct option: (a) 1
   Solution:
   Smallest odd number is 1 and 1³= 1
   
   
2. Which of the following can’t be the cube of a number given by 2n, where ‘n’ is a natural number
   (a)1728
   (b)2744
   (c)4096
   (d)4913
   Correct option: (d) 4913
   Solution:
   2n is an even number, its cube can’t be odd.
   
   
3. 17576 is the cube of
   (a)24
   (b)25
   (c)26
   (d)27
   Correct option: (c) 26
   Solution:
   Only numbers ending with 7, will have their cubes ending with 3.
   26³ = 17576
   
   
4. 729 can be represented as the sum of  _______ consecutive odd numbers
   (a)6
   (b)7
   (c)8
   (d)9
   Correct option: (d) 9
   Solution:
   729 = 9³
   Now,
   1 = 1³ (one odd number)
   3 + 5 = 2³ (sum of two consecutive odd number)
   7 + 9 + 11 = 3³ (sum of 3 consecutive odd number)
   Hence, n³ can be represented as the sum of n consecutive odd numbers.
   So, 9 odd number needed
   
   
5. Which of the following options is not a perfect cube
   (a)4×6×9
   (b)25×16×20
   (c)27×4×8
   (d)7×14×28
   Correct option: (c) 27×4×8
   Solution:
   27×4×8= 3×3×3×2×2×2×2×2
   Hence, not a perfect cube.
   
   
6. Which of the following options is the cube root of number 195112
   (a)52
   (b)54
   (c)56
   (d)58
   Correct option: (d) 58
   Solution:
   As 195112 ends with 2, so its cube root should end with 8.
   
   
7. The smallest number to be added or subtracted to convert 279 to perfect cube is
   (a)61
   (b)62
   (c)63
   (d)64
   Correct option: (c) 63
   Solution:
   Number 279 lies between cubes 216 and 343
   279 + 64 = 343
   279 – 63 = 216
   So, the smallest number is 63 out of the two.
   
   
8. Which of the following statement is false
   (a)Cube root of two digit number is a one digit number
   (b)Cube root of three digit number is a two digit number
   (c)Cube of a one digit number can be a one digit number
   (d)Cube of a two digit number can be a six digit number
   Correct option: (c) Cube of a one digit number can be a one digit number
   Solution:
   Cube root of 3-digit number is one digit number not two-digit number. Cubes of two-digit number will have digits more than 4-digits.
   
   
9. Which of the following option can be the cube root of (2n+1)3, where ‘n’ is a natural number
   (a)12
   (b)13
   (c)14
   (d)16
   Correct option: (b) 13
   Solution:
   Cube root of an odd number will be an odd number.
   Here ‘2n + 1’ is an odd number given that n is a natural number.
   
   
10. The smallest number to be multiplied or divided to convert 6750 to perfect cube is
    (a)5
    (b)4
    (c)2
    (d)3
    Correct option: (c) 2
    Solution:
    6750 = 5×5×5×3×3×3×2
     If we divide 6750 by 2 we get 3375 which is 15³.
    
    Subjective Questions
    
    
11. Cube of a number ends with the same digit as the number itself. So the number will have which digit in its unit’s place
    Solution:
    Now cubes of 1, 4, 5, 6, 9 and 0 ends with 1, 4, 5, 6, 9 and 0 respectively.
    Hence a number which will have any digit from 1, 4, 5, 6, 9 and 0 in its units place, will have the same digit in the unit’s place of its cube.
    
    
12. Ram wanted to buy a new smartphone which costs 193 rupees. If he had 203 rupees with him, then find the amount left with him after the purchase of the smartphone.
    Solution:
    Amount with Ram = 20³
    Price of smartphone = 19³
    Amount left after the purchase = 20³ – 19³ = 1 + 20 × 19 × 3 = Rs. 1141
    
    
13. Ravi and Rani are solving homework given by their teacher, wherein they were asked to find out three, four-digit cube numbers. Now Ravi writes down the following numbers, 4096, 5834 and 9261. Now Rani realizes that Ravi’s one number is wrong. So identify the number and suggest how you can convert it into a perfect cube.
    Solution:
     
    18³=5832
    So, we can convert 5834 to 5832 by subtracting 2 from it.
    
    
14. Arjun is fascinated by the way a cube number can be represented using the sum of odd consecutive numbers as shown
    13 = 1
    23 = 3 + 5
    33 = 7 + 9 + 11
    43 = 13+15+17+19
    53 = 21 + 23 + 25 + 27 + 29
    Now he decides to represent 343 as the sum of odd consecutive number. Help him to find the solution
    Solution:
     
    Hence,from the pattern it is clear that we need sum of 7
    consecutive odd numbers.
    So,following the pattern
    We get,
    63 = 31 + 33 + 35 + 37 + 39 + 41
    73 = 43 + 45 + 47 + 49 + 51 + 53 + 55
    
    
15. Ganesh and Niraj were arguing on whether 1512 is a perfect cube number or not. Ganesh says that 1512 is a perfect cube and Niraj says that it is not. Decide who is correct in their observation.
    Solution:
    Using prime factorization
     
    1512=2×2×2×3×3×3×7
    Clearly we can see that 1512 is not a perfect cube
    
    
16. Now Raj and Raju were finding the cube root of 9261. Raj came up with the answer 21, now Raju is not convinced that the answer is correct. Help out Raj in convincing convince Raju.
    Solution:
    Using prime factorization
    
    9261=3×3×3×7×7×7
    Clearly we can see that 1512 is not a perfect cube
    
    
17. “If two perfect cubes are multiplied together, then the resulting product is also an perfect cube”. Explain this statement with an example
    Solution:
    Consider two cube numbers 8 and 27
    We can represent them as follows
     8=2³
    27=3³
    Product = 8×27
    =2³ × 3³
    =(2× 3)³
    =(6)³
    Hence, we can say that the product is also a perfect cube number.
    
    
18. Without using prime factorization, prove that 4913 is not the cube of 18. 
    Solution:
    18 has 8 in its unit’s place, we know that cube of 8 ends with digit 2.
    Now, 4913 has 3 in the unit’s place, hence 4913 won’t be the cube of 18.
    
    
19. Prove that 270000 is not a perfect cube without using prime factorization.
    Solution:
    If we cube a number ending with ‘0’, we get ‘0’ in the unit place of its cube.
    Now, a cube number can end with three zeroes and the number of zeroes will always be the multiple of 3. So it can’t never end with four zeroes 
    
    
20. Niraj likes to play video games and give below are the points earned or lost by him on a given day
    
    Find the difference between points earned by him on consecutive days the letters to form meaningful words based on the clues provided.
    Solution:
    Points earned on Monday = 21³
    Points earned on Tuesday = 22³
    Difference = 22³ – 21³ = 1 + 22 × 21 × 3 = 1387
    Points earned on Monday = 22³
    Points earned on Tuesday = 23
    Difference = 23³ – 22³ = 1 + 23 × 22 × 3 = 1518