Vedic Math - Cube Roots of more than 6-Digit Number - Part I

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Back after a long break. In previous article, we learn how to find the cube root of 4 or 5 or 6 digits perfect cubes. Let us continue it further and discuss how to find the cube root of perfect or imperfect cubes.

In this article, we shall learn to find the cube roots for:
1. Cube root of perfect cubes, for any number of digits.
2. Cube root for all the cubes, whether perfect cubes or not.

To summarize what we have learned till now for cube root:

Arrange the given number in three-digit groups, starting from right to left. A single digit, if any left over at the left hand side, 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.

• 169 will count as 1 group 
• 1 258 will count as 2 groups 
• 43 781 will count as 2 groups
• 2 154 890 will count as 3 groups

If the given number has 'n' digits, its cube root will be having n/3 or (n+1)/3 digits. Also remember few other points from previous article:

The Cubes  of the first nine natural numbers    
1³ = 1        2³ = 8        3³ =27        4³ = 64        5³ = 125        6³ = 216        7³ = 343        8³ = 512         9³ = 729        10³ = 1000
From it, we understand that

• 1,4,5,6,9,0 numbers repeat themselves in the ending of their cubes
• 2,3,7 and 8 have their complements from 10, at the end of their cube

Let us start with actual technique now. Any number can be written in an algebraic expression. For example, if arithmetical number is 'dcba', it can be written in algebraic form as:
Algebraic Expression is: a + 10b + l00c + 1000d.

Now if we need to find the cube of a number 'cba', algebraically we can expand it like  (a+10b+10²c)³. Let us expand it:
(a+10b+10²c) ³ =  a³ + 10 (3a²b) + 10² (3ab²+3a²c) + 10³ (b³+6abc) + 10⁴ (3ac²+3b²c) + 10⁵ (3bc²) + 10⁶ (c³)

Now removing the powers of ten and putting the result in algebraic form, it tells us the formation of cube as:
(1) The units' place is determined by a³.
(2) The tens' place is contributed by 3 a²b
(3) The hundreds' place is contributed to by 3ab² + 3a²c
(4) The thousands' place is formed by b³ + 6abc
(5) The ten thousands' place is given by 3ac² + 3b²c
(6) The hundred thousands' (lakhs') place is constituted of 3bc² ; and
(7) The millions' place is formed by c³.

The number of zeroes in the various coefficients of the expanded Algebraic Expression are the basis of the formula / analysis.

Case1 :  Cube root of perfect cubes for any number of digits

Suppose we have a cube number n of any number of digits. To find its cube root, find following:
- The number of groups (N) in cube (as we discussed above to make the sets of 3 digits)
- First digit of cube root denoted as 'F' (Nearest cube root of first group from left)
- Last digit of cube root denoted as 'L' (Cube root of last group from left)
- Middle numbers of cube root(i.e. 'M' or 'H' or 'J'....), we shall find using the procedure.

Following are the steps for the procedure:
(i) From the units' place of given number, subtract the L³ (i.e. a³, refer to algebraic expression above); and that eliminates the last digit of the number.
(ii) From the ten's place, we subtract 3L²M (i.e. 3a²b) and thus eliminate the second last digit (penultimate digit).
(iii) From the hundreds' place, we subtract 3LM² + 3L²F (i.e. 3ab² + 3a²c) and hence eliminate the pre-penultimate digit.
and so on