Calculate Cube Roots with Vedic Mathematics

Basic Rules for extraction of Cube Roots. The given  number is first arranged in three-digit groups  from right to left. A single digit if any left over at the left hand  is counted as a simple group itself The number of  digits in the cube root will be the same as  the number of digit-groups  in the given  number itself.

• 125 will count as one group 

• 1000 will count as 2 groups 

• 15625 as two groups 

If the cube root contains 'n' digits ,  the cube must contain 3n or 3n-1 digits. If the given number has 'n' digits the cube root will have n/3 or (n+1)/3 digits. The first digit of the Cube root will always be obvious from the first group in the cube . For example a cube number with first group as 226 , the first digit of the cube root will be 6 since 6³ is 216 which is a perfect cube closer to 226. The Cubes  of the first nine natural numbers

    
1 = 1                2³ = 8              3³ =27                  4³ = 64            5³ = 125                
6³ = 216          7³ = 343         8³ =512               9³ = 729

This means, the last digit of the cube root of an exact cube is

• Cube ends in  1 , the Cube Root ends in 1

• Cube ends in  2 , the Cube Root ends in 8  

• Cube ends in  3 , the Cube Root ends in 7 

• Cube ends in  4 , the Cube Root ends in 4 

• Cube ends in  5 , the Cube Root ends in 5  

• Cube ends in  6 , the Cube Root ends in 6  

• Cube ends in  7 , the Cube Root ends in 3  

• Cube ends in  8 , the Cube Root ends in 2 

• Cube ends in  9 , the Cube Root ends in 9 

We can see that

• 1,4,5,6,9,0 repeat themselves in the cube ending

• 2,3,7 and 8 have their complements from 10, in the cube ending

Example 1: Find the cube root of 417 to 3 decimal places
 Arrange the number as follows groups of 3 digits starting from right.      

Step 1
         417  :   0    0    0 

By inspection write down 7 and 74 as the first Q and R . Since 343 is the perfect cube close to 417  and the reminder from 417 is 74

Step 2
                 417    :      0    0  0   0
       147  :           :   74
                :    7     :
The dividend is found by multiplying the Quotient Squared by 3  ,  72 X 3 =  147

Step 3
                 417    :      0       0     0      0
       147  :           :   74   152
                :    7    :      4
The second gross dividend is 740 , Do not subtract anything from this, divide it by 147 and put down 4 as Quotient and 152 as Remainder.

Step 4
                 417    :      0       0       0      0
       147  :          :   74   152   155
                :    7   :      4    7

The third gross dividend is 1520 ,  subtract 3ab2  ,  3 x 7 x 42  = 336 . The third actual working Dividend is 1520 - 336 = 1184 . Divide 1184  by 147 and put down 7 as Quotient and 155 as Remainder.

Step 5
                 417    :      0       0       0        0
       147  :          :   74   152   155   163
                :    7   :      4    7      1

The 4th gross dividend is 1550 ,  subtract 6abc + b3 ,  6 x 7 x 4 x 7 + 43  = 1176 + 64 = 1240. 

The 4th actual working Dividend is  1550 - 1240 = 310. Divide 310  by 147 and put down 1 as Quotient and 163 as Remainder.

Step 6
                 417    :      0       0       0        0
       147  :          :   74   152   155   163   118
                :    7   :      4    7      1      1 

The 5th gross dividend is 1630 ,  subtract 3ac2 + 3b2c ,  3 x 7 x 12 + 3 x 42 x 7   =  1029  + 336  = 1365.  The 5th actual working Dividend is  1630  - 1365 = 265.Divide 265 by 147 and put down 1 as Quotient and 118 as Remainder.

The number of digits in the cube root will be 1 , so the cube root = 7.4711