What is Cube and Cube Root?

                      

Introduction –  

In geometry, a solid three dimensional object which has equal sides is called cube. It has 6 sides and 12 lines. But here we study about cube in arithmetic.

In number, the term cube is three times multiplication of any number itself. For example take a number 2, cube of 2 means three times multiplication of 2 itself.

                                               2 x 2 x 2 = 8

It is denoted by 2³ as you write in square.  2³ = 8 and said like 2 cube as 2² = 4 is said two square. 

What is Perfect Cube?

A perfect cube is a number that is the cube of an integer like Perfect Square. When you factorize a cube via prime factorization, then each factor should be multiplied thrice itself. If the prime factors are 2 time or four times, then it is not a perfect cube. For example – 81. 125 is a perfect cube since 125 = 5 x 5 x 5 = 5³

How to know the given number is perfect or imperfect cube?

By prime factorization – when factorize any number then the factor should be three time multiplied by itself as we have seen in the above example. Some examples of perfect cubes are :- 



                                             1 = 1 x 1 x 1, 


                                              8 = 2 x 2 x 2, 


                                             27 = 3 x 3 x 3, 


                                             64 = 4 x 4 x 4 x 5, 


                                             125 = 5 x 5 x 5, 


                                              216 = 6 x 6 x 6, 



                                             343 = 7 x 7 x 7, so on.

Imperfect cube – The number is said to be imperfect cube if the prime factor is not multiplied by three times itself. For example, take a number 100, lets factorize it,

                                                     2| 100 |

                                                     2| 50 |

                                                     5| 25 |

                                                     5| 5 |

                                                       | 1 |

               = 2 x 2 x 5 x 5 = 100

This is not perfect cube because both factors are not multiplied three times itself. Hence it is an imperfect cube.

Make Imperfect cube to perfect cube:- You may asked question to make  a imperfect cube into perfect cube by multiplying or dividing. The question could be like it.

Question 1 - What is the least number that make 100 perfect cube by multiplying.

Solution:- As we have seen in the above example - 100 is not perfect cube because its both prime factors 2 and 5 are multiplied only two times.

2 x 2 x 5 x 5, if we added one more 2 and 5 here, it will become perfect cube. It means, we multiply 100 by 10 (2 x 5 ) to make it perfect cube.

2 x 2 x 2 x 5 x 5 x 5 = 2 x 5 = 10

Hence 10 x 10 x 10 = 1000 = 10³ 

Question 2:- Make 392 a perfect cube.

Solution:-  By prime factorization,

                                            2 | 392 |

                                            2 | 196 |

                                            2 | 98 |

                                            7 | 49 |

                                            7 | 7 |

                                               | 1 |

              
                                       = 2 x 2 x 2 x 7 x 7

Here first factor 2 is multiplied thrice while second factor 7 is multiplied twice. If second factor 7 would have been multiplied thrice, then the new number would have been a perfect cube. It means if we multiply number 392 by 7 we get the perfect cube.

2 x 2 x 2 x 7 x 7 x 7 = 2 x 7 = 14; 14³

It means if 392 is multiplied by 7, it will become perfect cube

 = 392 x 7 = 2744 = 14³ = required cube.

Question 3 :- What is least number that divides the given number x to make it perfect cube. For example the given number be 81

Solution – Let us factorize 81

                                                3 | 81 |

                                                3 | 27 |

                                                3 | 9 |

                                                3 | 3 |

                                                   | 1 |

                                                  


                                    = 3 x 3 x 3 x 3

Here the factor is multiplied four times while it should have been multiplied only three times, to be a perfect cube. There is one extra 3. Hence to make it perfect cube, we shall dived the given number 81 by 3.

81 / 3 = 27, the required least number to make perfect cube is 27 = 3³ 

What is Cube Root?

The cube root is just opposite of cubing a number and it is obtained by prime factorization. Since 5³ = 125, the cube root of 125 is 5. The cube root of a perfect cube is an integer.   

Cube root is denoted by mathematical sign ³√.

Finding cube root by prime factorization

Question 1:- Find cube root of 64 or ³√64

Solution:-

                                                4 | 64 |

                                                4 | 16 |

                                                4 | 04 |

                                                   | 01 |

                                         = 4 x 4 x 4 = 4

 Here cube root of 64 is 4.

Question 2:- What is cube root of 216 or ³√216?

Solution:-  

                                                 2 | 216 |

                                                 2 | 108 |

                                                 2 | 54 |

                                                3  | 27 |

                                                 3 | 9 |

                                                 3 | 3 |

                                                    | 1 |   

We can write – 2 x 2 x 2 x 3 x 3 x 3 = 216

Since, 2 and 3 are multiplied three time itself hence the prime factor is 2 and 3.

Therefor, cube root of 216 = 2 x 3 = 6. That means, if you will multiply 3 times (3 x 3 x 3), you’ll get 216.

It is also possible to find the cube root of a negative number. For example, the cube root of – 125 is – 5 or – 216 is – 6.

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