If a number is multiplied three times, the product or the resultant number is known as the cube of the given number. For example, 2 * 2 * 2 = 8, here two is multiplied three times, and the resultant answer eight will be a cube of two. The cube is written as x^3 (^ means raise to the number). 

The cube root on the other hand is the reverse process of the cube of a number. It is denoted by 3√. 

For example 3√64  = √4 *4*4 = 4 

Here is the answer for 3√64= 4 . So the cube root is 4. Cube root is the inverse of the cube of the number. 

Perfect cubes: 

A perfect cube is an integer or a number that can be expressed as a product of three equal integers or numbers. For example: 7 * 7 * 7 = 343. So, it is a perfect square.  If I take 

2 * 2 *3 =12, so 12 is not a perfect square. 

Let’s see the following table of perfect cube roots: 

Number / Cube root  Perfect Cube 

1 1*1*1=1

2 2*2*2=8

3 3*3*3= 27

4 4*4*4= 64 

5 5*5*5=125

6 6*6*6=216

7 7*7*7= 343

8 8*8*8= 512

9 9*9*9= 729

10 10*10*10=1000

11 11*11*11=1331

12 12*12*12=1728

13 13*13*13= 2197

14 14*14*14=2744

15 15*15*15=3375

16 16*16*16=4096

17 17*17*17=4913

18 18*18*18= 5832

19    19*19*19= 6859

20     20*20*20=8000

Method to find the cube root of a number: 

In order to find out the cube root of a number, we have to first start with the prime factorization method and then we have to find the factors of the number. 

Afterward, we have to divtopicIde the factors obtained into groups containing three similar factors. Then, remove that cube root symbol. 

If there is any factor left and which cannot be divtopicIded equally into a group of three, that means the given number is not a perfect cube. We can understand more by the following example:

Example 1: Find a cube root of 3√27000.

     3 27000

     3 9000

     3 3000

     2 1000

     2 500

     2 250

     5 125

    5 25

    5 5

        1

3√27000= 3 * 3 * 3 * 2 *2 *2 * 5 * 5 *5

            ————   ———-  ————

          =3^3   *   2^3     *   5^3 

          = 3 * 2 * 5

          = 30 

The cube root of 27,000 is 30.

Example 2: Is the number 3√17576 a perfect cube? 

   2 17576

   2 8788

   2 4394

  13 2197

  13 169

  13 13

1

3√17576= 2 * 2 * 2 * 13 *13 *13 

                ————-   ————-

              = 2^3  * 13^3

              = 2  *  13 

              = 26 

Since 3√17576 = 26 , it is a perfect cube. 

Example 3:  Is 243 a perfect cube? 

243= 3 *3 *3* 3 *3 

         ———

      = 3^3  *3^2

In the above factorization, one group of three is Cube but in the other group, there are only two threes. Therefore the number 243 is not a perfect cube. 

Definition: Cube of a number: 

Whenever we multiply a number three times by itself the product is known as the cube of the number. For example, if 4* 4 * 4 = 64, here four is multiplied three times and the resultant is 64. So 64 will be a cube of four. 

Cube of Negative Numbers - is it negative or positive?

Whenever we multiply three same negative numbers, the product obtained is the cube of negative numbers. Here, the resultant will always be negative.

For example: (-3) * (-3) *(-3) = (-27)

Hence, we can say that the cube of -3 is -27. Remember the resultant of a cube of negative numbers will always be negative.

Cube of a fraction - Types of a cube: 

There are two types of cube of fractions number. a) Positive cube of fraction  b) Negative cube of fraction 

Positive cube of Fractions: 

Whenever we multiply three same fractions, the product obtained is the cube of a fraction.

For example: 5/7 * 5/7  * 5/ 7 = 5*5*5/7*7*7 = 125 / 343.

The resultant of a positive cube of a fraction will always be a positive fraction. 

Negative cube of Fraction: 

Whenever we multiply three same negative fractions the product octane is the cube of a fraction. The result will always be negative.

For example: (-1/7) * (-1/7) * (-1/7) = -1* -1 *-1 / 7*7*7 = (-1/343).