Cubes
- The cube of a number is obtained by multiplying the number by itself three times.
- If ‘a’ is a number, its cube is a * a * a = a³.
- For example, the cube of 2 is 2 * 2 * 2 = 8.
Cube Roots
- The cube root of a number is a value which, when multiplied by itself three times, gives the original number.
- It is denoted by the symbol ³√.
- For example, the cube root of 8 is ³√8 = 2.
Properties of Cubes:
- The cube of an even number is always even.
- The cube of an odd number is always odd.
- The cube of a negative number is negative.
Methods to find cube roots:
- Prime factorization
- Estimation
Cubes and their Cube Roots: There is a table of cubes for numbers from 1 to 10, which can be used for calculations.
Applications: Cubes and cube roots have applications in various fields, including geometry, physics, and engineering.
Exercise 6.
1. Which of the following numbers are not perfect cubes?
(i) 216
(ii) 128
(iii) 1000
(iv) 100
(v) 46656
Ans :
- 216: Prime factorization is 2 * 2 * 2 * 3 * 3 * 3. Factors can be grouped into triplets, so it’s a perfect cube.
- 128: Prime factorization is 2 * 2 * 2 * 2 * 2 * 2 * 2. The factor 2 cannot be grouped into triplets, so it’s not a perfect cube.
- 1000: Prime factorization is 2 * 2 * 2 * 5 * 5 * 5. Factors can be grouped into triplets, so it’s a perfect cube.
- 100: Prime factorization is 2 * 2 * 5 * 5. The factor 2 and 5 cannot be grouped into triplets, so it’s not a perfect cube.
- 46656: Prime factorization is 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 3 * 3 * 3. Factors can be grouped into triplets, so it’s a perfect cube.
Therefore, the numbers that are not perfect cubes are 128 and 100.
2. Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.
(i) 243
(ii) 256
(iii) 72
(iv) 675
(v) 100
Ans :
i) 243
- Prime factorization:
- 3 * 3 * 3 * 3 * 3
- We need one more 3.
- So, multiply by 3.
- The perfect cube is 243 * 3 = 729, and its cube root is 9.
ii) 256
- Prime factorization:
- 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2
- We need one more 2 .
- So, multiply by 2.
- The perfect cube is 256 * 2 = 512, and its cube root is 8.
iii) 72
- Prime factorization:
- 2 * 2 * 2 * 3 * 3
- We need one more 3 .
- So, multiply by 3.
- The perfect cube is 72 * 3 = 216, and its cube root is 6.
iv) 675
- Prime factorization:
- 3 * 3 * 3 * 5 * 5
- We need one more 5.
- So, multiply by 5.
- The perfect cube is 675 * 5 = 3375, and its cube root is 15.
v) 100
- Prime factorization:
- 2 * 2 * 5 * 5
- We need one more 2 and one more 5 .
- So, multiply by 2 * 5 = 10.
- The perfect cube is 100 * 10 = 1000, and its cube root is 10.
3. Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.
(i) 81
(ii) 128
(iii) 135
(iv) 92
(v) 704
Ans :
| Number | Divisor | Perfect Cube | Cube Root |
| 81 | Already a perfect cube | 81 | 3 |
| 128 | Already a perfect cube | 128 | 4 |
| 135 | Already a perfect cube | 135 | 3 |
| 92 | Already a perfect cube | 92 | 2 |
| 704 | Already a perfect cube | 704 | 8 |
4. Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will be needed to form a cube?
Ans :
- Volume of one cuboid: 5 cm * 2 cm * 5 cm = 50 cubic cm
- To form a cube, all dimensions should be equal.
- The nearest perfect cube to 50 is 64 (which is 4³).
- To make the volume of the cuboid 64 cubic cm, we need to multiply it by 64/50 = 1.28.
- Since we can’t have a fraction of a cuboid, we round up to the nearest whole number, which is 2.
- Therefore, we need to multiply the length, width, and height of the cuboid by 2 to get a cube.
Number of cuboids required = 2 * 2 * 2 = 8 cuboids.
Exercise 6.2
1. Find the cube root of each of the following numbers by prime factorisation method.
(i) 64
(ii) 512
(iii) 10648
(iv) 27000
(v) 15625
(vi) 13824
(vii) 110592
(viii) 46656
(ix) 175616
(x) 91125
Ans :
i) 64
Prime factorization: 2 * 2 * 2 * 2 * 2 * 2
Grouping triplets: (2 * 2 * 2) * (2 * 2 * 2)
Cube root of 64
= 2 * 2 = 4
ii) 512
Prime factorization: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2
Grouping triplets: (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2)
Cube root of 512
= 2 * 2 * 2 = 8
iii) 10648
Prime factorization: 2 * 2 * 2 * 11 * 11 * 11
Grouping triplets: (2 * 2 * 2) * (11 * 11 * 11)
Cube root of 10648
= 2 * 11 = 22
iv) 27000
Prime factorization: 2 * 2 * 2 * 3 * 3 * 3 * 5 * 5 * 5
Grouping triplets: (2 * 2 * 2) * (3 * 3 * 3) * (5 * 5 * 5)
Cube root of 27000
= 2 * 3 * 5 = 30
v) 15625
Prime factorization: 5 * 5 * 5 * 5 * 5 * 5
Grouping triplets: (5 * 5 * 5) * (5 * 5 * 5)
Cube root of 15625
= 5 * 5 = 25
vi) 13824
Prime factorization: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3
Grouping triplets: (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (3 * 3 * 3)
Cube root of 13824
= 2 * 2 * 2 * 3 = 24
vii) 110592
Prime factorization: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3
Grouping triplets: (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (3 * 3 * 3)
Cube root of 110592
= 2 * 2 * 2 * 2 * 3 = 48
viii) 46656
Prime factorization: 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 * 3 * 3 * 3
Grouping triplets: (2 * 2 * 2) * (2 * 2 * 2) * (3 * 3 * 3) * (3 * 3 * 3)
Cube root of 46656
= 2 * 2 * 3 * 3 = 36
ix) 175616
Prime factorization: 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 7 * 7 * 7
Grouping triplets: (2 * 2 * 2) * (2 * 2 * 2) * (2 * 2 * 2) * (7 * 7 * 7)
Cube root of 175616
= 2 * 2 * 2 * 7 = 56
x) 91125
Prime factorization: 3 * 3 * 3 * 3 * 3 * 3 * 5 * 5 * 5
Grouping triplets: (3 * 3 * 3) * (3 * 3 * 3) * (5 * 5 * 5)
Cube root of 91125
= 3 * 3 * 5 = 45
2. State True or False.
(i) Cube of an odd number is even.
(ii) A perfect cube does not end with two zeros.
(iii) If the square of a number ends with 5, then its cube ends with 25.
(iv) There is no perfect cube which ends with 8.
(v) The cube of a two digit number may be a three digit number.
(vi) The cube of a two digit number may have seven or more digits.
(vii) The cube of a single digit number may be a single digit number.
Ans :
(i) False. The cube of an odd number is always odd.
(ii) False. For example, 1000 is a perfect cube and ends with two zeros.
(iii) False. The cube of a number ending in 5 will always end in 5, not 25.
(iv) False. For example, 8 itself is a perfect cube and ends with 8.
(v) True. For example, the cube of 4 (a two-digit number) is 64 (a three-digit number).
(vi) False. The largest two-digit number is 99, and its cube is 970299, which has six digits.
(vii) True. For example, the cube of 1 is 1, and the cube of 2 is 8, both single-digit numbers.
