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- Hardy - Ramanujan Number
- Cube Numbers
- Prime Factorization of Cubes
- Smallest multiple that is a perfect cube
- Cube Root
- How to Find Cube root

expressed as sum of two cubes in two different ways.

1729 = 1728 + 1 = 12

1729 = 1000 + 729 = 10

1729 is the smallest Hardy– Ramanujan Number. There are an infinitely many such numbers. Few are 4104 (2, 16; 9, 15), 13832 (18, 20; 2, 24), Check it with the numbers given in the brackets

Example

1=1^{3}

8=2^{3}

27=3^{3}

Or if a natural number 1=1

8=2

27=3

The numbers 1, 8, 27, 125 ... are cube numbers. These numbers are also called

8= 2×2×2 (Triplet of prime factor 2)

216 = (2 × 2 × 2) × (3 × 3 × 3) ( Triplet of 2 and 3)

Each prime factor of a number appears three times in the prime factorization of its cube.

Find the smallest multiple that will make 392 perfect cube

392 = 2 × 2 × 2 × 7 × 7

The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect ube. To make its a cube, we need one more 7. In that case

392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744 which is a perfect cube

So we know that

27=3

Cube root of 27

$\sqrt[3]{27} =3 $

We will get same prime number occurring in pair for perfect square number. Square root will be given by multiplication of prime factor occurring in pair

Consider

1331

1331= (11×11×11)

$\sqrt[3]{1331} =11 $

5832

5832= (2 × 2 × 2) × (3 × 3× 3) × (3 × 3× 3)

$\sqrt[3]{5832} =2 \times 3 \times 3=18 $

The given number is 17576.

17 576. In this case one group i.e., 576 has three digits whereas 17 has only two digits.

The digit 6 is at its one’s place.

We take the one’s place of the required cube root as 6.

Cube of 2 is 8 and cube of 3 is 27. 17 lies between 8 and 27.

The smaller number among 2 and 3 is 2.

The one’s place of 2 is 2 itself. Take 2 as ten’s place of the cube root of

17576.

Thus, $\sqrt[3]{17576} =26$

1) for any Positive integer m, $m^3 > m^2$ i.e cube is greater than square

2) For any negative integer m, $m^3 < m^2$ i.e cube is less than square , as the cube is always negative number and square is positive number

3) Cubes can be written as Addition consecutive odd numbers

$1 = 1 = 1^3$

$3 + 5 = 8 = 2^3$

$7 + 9 + 11 = 27 = 3^3$

$13 + 15 + 17 + 19 = 64 = 4^3$

$21 + 23 + 25 + 27 + 29 = 125 = 5^3$

Cube root of a number is the number when multiplied itself by thrice gives the original value Example Cube root of 8 is 2 as when 2 is multipled by itself thrice, it gives the Original value $2 \times 2 \times 2=8$

Cube root of a number can be found using prime factorization method $729= 3 \times 3 \times 3 \times 3 \times 3 \times 3$ $\sqrt[3]{729}=3 \times 3 =9$

Cube root of a number can be found using prime factorization method $10000= 10 \times 10 \times 10 \times 10$ It is clear 10000 is not a perfect cube

Cube root of a number can be found using prime factorization method $512= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 $ $\sqrt[3]{729}=2 \times 2 \times 2=8$

Cube of a number is given by $n \times \times n =n^3$ Example Cube root of 2 is $2 \times 2 \times 2=8$ Cube root in inverse of Cube and its formula is $\sqrt[3]{n}$

Properties of Cube of Number:
(i) Cubes of even number are even whereas Cubes of odd numbers are odd.
(ii) The sum of the cubes of first n natural numbers is equal to the square of their sum.
(iii) The below properties will help in finding the Unit digits of the cubes

**Notes**-
**Assignments & NCERT Solutions**

Class 8 Maths Class 8 Science