## CUBES AND CUBE ROOTS- ESSENTIAL POINTS

• Cube number: Number obtained when a it is multiplied by itself three times.
• 33 = 3 x 3 x 3 = 27, 43 = 4 x 4 x 4 = 64 etc.
• Numbers that can be expressed as sum of two cubes in two different ways are known as Ramanujan numbers. Example, 1729, 4104, 13832 etc.
• Numbers obtained when a number is multiplied by itself three times are known as Cube Numbers. For example, 1, 8, 27, ... etc.
• If in prime factorization of any number each factor appears three times, then the number is a Perfect Cube.
• The symbol denotes cube root. Example, = 4.
• Perfect Cube: A natural number is said to be a perfect cube if it is the cube of some natural number. Example: 8 is perfect cube, because there is a natural number 2 such that 3 8 = 2, 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:
1. Cubes of even number are even.
2. Cubes of odd numbers are odd.
3. The sum of the cubes of first n natural numbers is equal to the square of their sum.
4. 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.
5. Cube of the number ending in 2 ends in 8 and cube of the number ending in 8 ends in 2.
6. Cube of the number ending in 3 ends in 7 and cube of the number ending in 7 ends in 3.
7. Cube of n digit number can have max 3n digits.