Cubes and Cube Roots - Worksheets

CBSE Worksheet-1
Class 08 - Mathematics (Cube and cube roots)

---

General Instructions: All questions are compulsory. Q.1 to Q.2 carries one mark each. Q.3 to Q.7 carries two marks each. Q.8 and Q.9 carries three marks each. Q.10 to Q.12 carries four marks each.

---

1. What is the value of 103 - 93?
2. What is cube of 0.3?
3. State whether the following statements are True or False:
   1. Each prime factor appears 3 times in its cube.
   2. The cube of a negative number is always positive.
   3. The cube of 0.4 is 0.064.
   4. The cube root of 8000 is 200.
4. Fill in the blanks.
   1. The next two numbers in the pattern: 1, 8, 27, 64, 125, ___, ____.
   2. The cube of an odd number is always ______.
   3. The cube of a negative integer is _______.
   4. The last digit of a number which has 3 as last digit in its cube is ______.
5. Match the following:

   Column A	Column B
   (a) (-m)3	(p) -m
   (b) −m3−−−−√3$\sqrt[3]{-m^3}$	(q) -m3
   (c) (m)3	(r) m
   (d) m3−−−√3$\sqrt[3]{m^3}$	(s) m3

6. Find the value of the following: 73 – 63
7. Find if 15625 is a perfect cube ?
8. Find the smallest number by which 675 must be multiplied to obtain a perfect cube.
9. Find the cube root 4913 without factorisation.
10. The difference of two perfect cubes is 189. If the cube root of the smaller of the two numbers is 3, find the cube root of the larger number.
11. Find the cube root of 166375 by estimation method with steps.
12. Shyamala grew 400 plants in her garden. How many more plants should she grow, so that the total number is a perfect cube?

CBSE Worksheet-1
Class 08 - Mathematics (Cube and cube roots)
Solution

---

1. 103 - 93 can be written as 1+3×10×9=1+270=271$1+3\times 10\times 9=1+270=271$
   (or) 1000 -729 = 271.
2. (0.3)3 = 0.3 × 0.3 × 0.3 = 0.027.
3. 1. True
   2. False, negative.
   3. True
   4. False, 20.
4. 1. 216 and 343;
   2. odd
   3. negative
   4. 7
5. 1. -m3;
   2. -m;
   3. m3;
   4. m
6. 73 – 63
   73 – 63 = (7-6)(72+7*6+62) ( a​​​​​​3 - b​​​​​​3 )= ( a- b ) (a​​​​​​2 +ab + b​​​​​​2)
   =(1)(49+42+36)
   = 127
7. 
   
   By prime factorisation,
   15625 = 5 × 5 × 5 × 5 × 5 × 5 [grouping the factors in triplets]
   = 53 × 53 [by laws of exponents]
   = (5 × 5)3
   = 253 which is a perfect cube.
   All the terms form triplets
   Therefore, 15625 is a perfect cube.
8. 
   By prime factorisation,
   675 = 3 × 3 × 3 × 5 × 5 [grouping the factors in triplets]
   The prime factor 5 does not appear in a group of three.
   Therefore, 675 is not a perfect cube. To make it a cube, we need one more 5. In that case,
   675 × 5 = 3 × 3 × 3 × 5 × 5 × 5
   3375= 33 × 53
   = (3 × 5)3
   = 153 which is a perfect cube.
   Hence, the smallest number by which 675 must be multiplied to obtain a perfect cube is 5.
   The resulting perfect cube is 3375 (= 153).
9. Cube root of 4913 = 17
   Cube root of 4913
   The given number is 4913.
   Step 1: From groups of three starting from the rightmost digit of 4913.
   In this case, one group i.e., 913 has three digits whereas 4 has only one digit.
   Step 2: Take 913
   The digit 3 is at its axis place we take the axis place of the required cube root as 7.
   Step 3: Take the other group, i.e., 4 cube of 1 is 1 and cube of 2 is 8. 4 has between 1 and 8.
   The smaller number among 1 and 2 is 1.
   The one's place of 1 is 1 itself.
   Take 1 as ten's place of the cube root of 4913.
   Thus, 4913−−−−√3=17$\sqrt[3]{4913}=17$.
10. Given the difference of two perfect cubes =189.
    Let the smaller cube number be x.
    Then the other cube (larger cube) = 189 + x
    Given cube root of the smaller cube = 3
    i.e., x−−√3=3$\sqrt[3]{x}=3$
    Therefore, x = 33 = 27.
    Larger cube = 189 + 27 = 216
    Cube root of the larger cube number = 216−−−√3$\sqrt[3]{216}$ = 6.
11. Step-1: Group the given number of three digits starting from the right most digit of the number. 166 375
    Step-2: The first group 375 will give the ones digit of the required cube root. The number 375 ends with 5. We know that 5 comes at the units place of a number only when its cube root ends in 5.
    So, we get 5 at the units place of cube root.
    Step-3: In the second group 166,
    166 lies between two successive cubes, 53 and 63.
    53 = 125 and 63 = 216
    i.e., 125 < 166 < 216. So between the cube numbers 125 and 216, the smaller one is 125, whose cube root is 5.
    So, 5 is the tens digit of the required cube root.
    Therefore, 166375−−−−−−√3=55$\sqrt[3]{166375}=55$.
12. No.of plants in the garden = 400
    400 is not a perfect cube. So, some number should be added to 400 to make it a perfect cube.
    400 lies between 73 = 343 and 83 = 512.
    Therefore, to make 400 to 512, the number that should be added to 400 is
    512 - 400 = 112.
    So, 112 more plants should be grown, to make total number 512, which is a perfect cube.