Cube and cube root

Ramanujan, the great mathematician India ever had. He loved numbers experimented with numbers to a vast extent. Once G.H Hardy came to meet him in India. The taxi which he booked has a number 1729. When he met S. Ramanujan he said that this is a very dull number. Ramanujan replied quickly that this is not a dull number but its interesting number. Then also explained about cube and cube root.

He said that 1729 is the smallest number which can be expressed as the sum of two cubes in two different ways.
1729 = 1728 + 1 = 12³ + 1³
1729 = 1000 + 729 = 10³ + 9³

After that day 1729 is known as Hardy-Ramanujan number.

Perfect cubes or cube number:

When any number is obtained when a number is multiplied by itself three times is perfect cube or cube number.

or,
If in the prime factorisation of any number each factor appears three times, then the number is a perfect cube. Thus necessary for studying cube and cube root.
Example: 10 x 10 x 10 =1000, this number is perfect cube number.

Properties of perfect cube number:

1. Cube of even number is always even.
   Example: 2³ = 8, 4³ = 64 etc.
2. Cube of even number is always even.
   Example: 3³ = 27, 5³ = 125
3. We get a perfect cube by adding consecutive odd numbers in an amazing pattern.

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

The number of odd numbers in cube of any number is same as the number whose cube is obtained.

4. When prime factorization of a cube of a number then each prime factor appears 3 times.

Cube roots:

Cube root is simply the converse of cube of any number.
The symbol for representation of cube root is . it is special symbol as it has same symbol as square root only the redical attached with it.
Example: 8 = 2 x 2 x 2,  27 = 3 x 3 x 3 etc.

Example related to cube and cube root:

1. Find the value of (8)³.

Solution:
For getting cube of any number multiplies it by itself 3 times.

Thus, (8)³ = 8 x 8 x 8 = 512

2. Find the value of (13)³.

Solution:
we can write 13 as 10 + 3
thus, (13)³ = (10 + 3)³
= (10)³ + (3)³ + 3 x 10 x 3(10 + 3)
= 1000 + 9 + 90 x 13
= 1009 +1170
= 2179

3. Find the cube root of 1000.
   Solution:
   Directly we can assume the cube root as seeing it.
   In this case there are 3 zeroes on number than it is confirmed that the number must contain a zero.
   So we put that 10 x 10 x 10 = 1000, so cube root of 1000 is 10.