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Square Roots and Cube Roots (Vedic Mathematics)

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1. Square Root of a Number

The method of finding the square root of a number is based on the application of “Dvandvayoga” or Duplex. Calculation of “Dvandvayoga” (Duplex) is very important in calculating the square root of any number.
Watch the following video to understand the method of calculating the Duplex of a number.
Solved Examples
Example 1:
Calculate the Duplex of 2852
Solution:
Pair the digits of the number as shown below

Example 2:
Calculate the Duplex of 93242
Solution:
Pair the digits of the number as shown below

In order to calculate the square root of a number, the following points should be kept in mind:
  • An exact square cannot end in 2, 3, 7, and 8.
  • A perfect square ending in 1 must have either 1 or 9 (compliment of 1 from base 10) as the last digit of the square root.
  • If a perfect square ends in 4, then the square root can only end in either 2 or 8 (complimentary to each other).
  • If a perfect square ends in either 5 or 0, then its square root also ends in 5 and 0 respectively.
  • If a perfect square ends in 6, then its square root ends in either 4 or 6.
  • If a perfect square ends in 9, then its square root ends in either 3 or 7.
Last digit of perfect square
Last digit of square root
1
1, 9
4
2, 8
5
5
6
4, 6
9
3, 7
0
0
Few facts
  • If the number of digits of a number is even, say “n”, then the number of digits in its square root is , i.e., half of the number of digits of the given number. Also, the first digit of the square root of a number is equal to the square root of the closest perfect square, which is less than the first two digits of the
    number.
  • If the number of digits of a number is odd, say “n”, then the number of digits in its square root is. Also, the first digit of the square root of a number is equal to the square root of the closest perfect square, which is less than the first
    digit of the number.
Note:
The above analysis is only valid for whole numbers and not for decimal numbers.
Now, watch the following video to learn how to calculate the square root of a given number.
Solved Examples
Example 1:
Calculate
Solution:
Number of digits = 4
Number of digits in its square root
Since the number of digits in 3249 is even, the first two digits (32) are taken on the left side for square root calculation.

52 = 25 < 32 < 62 = 36

⇒ Divisor = 5 × 2 = 10
Step I:
32 − 52 = 32 − 25 = 7
Step II:
74 ÷ 10 gives Q = 7, R = 4
Step III:
Gross dividend = 49
Actual dividend = 49 − Duplex (7) = 49 − 72 = 49 − 49 = 0
0 ÷ 10 gives Q = 0, R = 0

Also, 3249 is a perfect square.
Example 2:
Calculate
Solution:
Number of digits = 5
Number of digits in its square root
Since the number of digits in 16384 is odd, the first digit (1) is taken on the left side for square root calculation.

02 = 0 < 1 < 22 = 4

Note: In case 0 is taken, then the divisor will be 2 × 0 = 0, which is not possible. Hence, in such a case, always take 1 instead of
0.
⇒ Divisor = 1 × 2 = 2
Step I:
1 − 12 = 1 − 1 = 0
Step II:
06 ÷ 2 gives Q = 3, R = 0
However, if R = 0, then in the next step, the actual dividend comes out to be negative.
Hence, Q = 2, R = 2 is taken.
Step III:
Gross dividend = 23
Actual dividend = 23 − Duplex (2) = 23 − 22 = 23 − 4 = 19
19 ÷ 2 gives Q = 9, R = 1
However, if R = 0, then in the next step, the actual dividend comes out to be negative.
Hence, Q = 8, R = 3 is taken.
Step IV:
Gross dividend = 38
Actual dividend = 38 − Duplex (28) = 38 − 2 × 2 × 8 = 38 − 32 = 6
6 ÷ 2 gives Q = 3, R = 0
However, if R = 0, then in the next step, the actual dividend comes out to be negative.
Hence, Q = 0, R = 6 is taken.
Step V:
Gross dividend = 64
Actual dividend = 64 − Duplex (280) = 64 − (2 × 2 × 0 + 82) = 64 − (0 + 64) = 64 − 64 = 0
0 ÷ 2 gives Q = 0, R = 0

Also, 16384 is a perfect square.

2. Cube Root of a Number

This lesson will describe the method to calculate the cube root of a number, which is not described in high school textbooks.
If a number comprises n digits, then its cube root has digits, where represents the least integer more than or equal to it.
To understand the method, go through the following video.
Solved Example
Example 1:
Calculate
Solution:
Number of digits in 15625 = 5

∴Number of digits in cube root of 15625 = 2

23 = 8 < 15 < 33 = 27

⇒ a = 2
Also, divisor = 3 × 22 = 3 × 4 = 12
Step I:
15 − a3 = 15 − 23 = 15 − 8 = 7
Step II:
76 ÷ 12 gives Q = 6, R = 4
However, in that case, the actual dividend comes out to be negative in the next step.
Hence, Q = 5, R = 16 is taken.
⇒ b = 5
Step III:
Gross dividend = 162
Actual dividend = 162 − 3ab2 = 162 − 3 × 2 × 52 = 162 − 150 = 12
12 ÷ 12 gives Q = 1, R = 0
However, in that case, the actual dividend comes out to be negative in the next step.
Hence, Q = 0, R = 12 is taken.
⇒ c = 0
Step IV:
Gross dividend = 125
Actual dividend = 125 − (6abc + b3) = 125 − (6 × 2 × 5 × 0 + 53) = 125 − 125 = 0
0 ÷ 12 gives Q = 0, R = 0

Also, 15625 is a perfect cube
Points to remember:
  • First number selection: The first number that is used for calculating the cube root of a number will consist of the first n digits of the number from left to right, where n is the remainder left after dividing the number of digits of the number by 3.
If R = 0, then n = 3
  • Divisor: 3a2
  • For Actual Dividend:
  • The 2nd gross dividend remains the same.
  • From the 3rd gross dividend, subtract 3ab2
  • From the 4th gross dividend, subtract 6abc + b3
  • From the 5th gross dividend, subtract 3ac2 + 3b2c
  • From the 6th gross dividend, subtract 3bc2
  • From the 7th gross dividend, subtract c3
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