Sunday, May 24, 2020

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 factorizationwhen 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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