Cube :
The product '2 x 2 x 2' is written in index form as ' 23'.
In other words we say that '8 is a cube of 2'.
Similarly cube of 4 is 4 x 4 x 4 = 64 which can be written as 43 = 64.
The cube of an integer :
(1) 213 = 21 x 21 x 21
= 441 x 21
= 92 61.
(2) (-7)3 = (-7) x (-7) x (- 7)
= 49 x (-7)
= -343
From the above examples we find that
* The cube of a positive number is positive.
* The cube of a negative number is negative.
Perfect cubes:
The cube of any integer is called a perfect cube.
For example : (-1)3 = - 1; 43 = 64; (-7)3 = - 343.
Thus -1, 64, - 343 are perfect cubes.
Cubes of the numbers from 1 to 30 :
Cube roots :
The cube of 2 is 8. Therefore, we say that the cube root of 8 is 2.
The cube of 3 is 27. Therefore, we say that the cube root of 27 is 3.
The cube of - 4 is - 64. Therefore, we say that the cube root of - 64 is - 4.
The symbol 
is used to denote 'the cube root of'
is used to denote 'the cube root of'
'The cube root of 512 is written in symbol form as ' 
'.
'.
Statements given below in symbols and words are shown in table :
Finding the cube root of a perfect cube by using prime factor method:
1) Find the cube root of 216.
Solution : 216 = 2 x 108
= 2 x 2 x 54
= 2 x 2 x 2 x 27
= 2 x 2 x 2 x 3 x 3 x 3
= 2 x 2 x 2 x 3 x 3 x 3
We have grouped the prime factors into pairs of equal prime factors.Taking one number from each of the groups we get the cube root as 2 x 3 = 6. Means 6 is the cube root of 216.
We write this in symbol for as ' 
'.
'.
2) Find the cube root of 1728.
Solution : 1728 = 2 x 864
= 2 x 2 x 432
= 2 x 2 x 2 x 2 x 108
We
have grouped the prime factors into pairs of equal prime factors.Taking
one number from each of the groups we get the cube root by product of them as 2 x 2 x 3 = 12.
Means 12 is the cube root of 1728.
We write this in symbol for as ' 
'.3) Find the cube root of - 1331.
Solution : 1331 = 11 x 121
= 11 x 11 x 11
Therefore the cube root of 1331 is 11.
So
.