Key Points - Cubes and Cube Root

- Cube number: Number obtained when a number is multiplied by itself three times.
23 = 2 × 2 × 2 = 8 , 33 = 3 × 3 × 3 = 27 , etc.

- Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be expressed as sum of two cubes in two different ways.
- Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1, 8, 27, ... etc.

- If in the prime factorisation of any number each factor appears three times, then the number is a perfect cube.
27 = 3
- The symbol
- Perfect Cube: A natural number is said to be a perfect cube if it is the cube of some natural
3
 denotes cube root. For example
3
 number. Example: 8 is perfect cube, because there is a natural number 2 such that 8 = 23 , but
18 is not a perfect cube, because there is no natural number whose cube is 18.
- The cube of a negative number is always negative.

- Properties of Cube of Number: 
(i) Cubes of even number are even.
(ii) Cubes of odd numbers are odd.
(iii) The sum of the cubes of first n natural numbers is equal to the square of their sum.
(iv) Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9 end with digits 0, 1, 4, 5, 6 and 9 respectively.
(v) Cube of the number ending in 2 ends in 8 and cube of the number ending in 8 ends in 2.
(vi) Cube of the number ending in 3 ends in 7 and cube of the number ending in 7 ends in 3.