# Cube and Cube Roots

## Hardy - Ramanujan Numbers

Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be
expressed as sum of two cubes in two different ways.
1729 = 1728 + 1 = 123 + 13
1729 = 1000 + 729 = 103 + 93
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

## Cube Number

Numbers obtained when a number is multiplied by itself three times are known as cube numbers
Example
1=13
8=23
27=33
Or   if a natural number m can be expressed as n3 where n is also a natural number, then m is a cube number
The numbers 1, 8, 27, 125 ... are cube numbers. These numbers are also called perfect cubes.

## Prime Factorization of Cubes

When we perform the prime factorization of cubes number, we find one special property
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.

## Smallest multiple that is a perfect cube

Here we find the prime factorization of the number. Then we find the prime factor required to make all of them in triplet.
Example
Find the smallest multiple that will make 392 perfect cube
Solution:
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

## Cube Root

Cube root of a number is the number whose cube is given number
So we know that
27=33
Cube root of 27

Cube root is denoted by expression

## How to Find Cube root

### Finding cube root through prime factorization

This method, we find the prime factorization of the number.
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)

Example
5832
Solution
5832= (2 × 2 × 2) × (3 × 3× 3) × (3 × 3× 3)

### Finding cube root by estimation method

This can be well explained with the example
The given number is 17576.
Step 1 Form groups of three starting from the rightmost digit of 17576.
17 576. In this case one group i.e., 576 has three digits whereas 17 has only two digits.
Step 2 Take 576.
The digit 6 is at its one’s place.
We take the one’s place of the required cube root as 6.
Step 3 Take the other group, i.e., 17.
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,

Extra Zing
1) for any Positive integer m, m3 > m2 i.e cube is greater than square
2) For any negative integer m, m3 < m2 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 = 13

3 + 5 = 8 = 23

7 + 9 + 11 = 27 = 33

13 + 15 + 17 + 19 = 64 = 43

21 + 23 + 25 + 27 + 29 = 125 = 53