Vedic Math - Square roots

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Earlier we discussed "Squaring numbers near base" and "General Squaring through Duplex Process" and now we will find out how to calculate the square root of numbers. To understand this, let us first learn basic rules for finding the square root.

(1). The given number is first arranged in two-digit groups from right to left. If on left hand side, a single digit is left, that will also be counted as a group.
(2). The number of digits in the square root will be the same as the number of groups derived from the number. Examples are:
  • 25 will be having one group as '25', hence square root should be of one digit.
  • 144 will be having two groups as '44' and '1', hence the square root should be of two digits.
  • 1024 will be having two groups as '24' and '10', hence the square root should be of two digits.
(3). If the given number has 'n' digits then the square root will have n/2 or (n+1)/2 digits
(4). The squares of the first nine natural numbers are 1,4,9,16,25,36,49,64, and 81. All of these squares end with 1, 4, 5, 6, 9, 0. This means
  • An exact square never ends in 2, 3, 7 or 8
  • If a number ends in 2, 3, 7 or 8, its square root will always be an irrational number
  • If an exact square ends in 1, its square root ends in 1 or 9
  • If an exact square ends in 4, its square root ends in 2 or 8
  • If an exact square ends in 5, its square root ends in 5
  • If an exact square ends in 6, its square root ends in 4 or 6
  • If an exact square ends in 9, its square root ends in 3 or 7
(5). If a perfect square is an odd number, the square root is also an odd number
(6). If a perfect square is an even number, the square root is also an even number
(7). A whole number, which ends with an odd numbers of 0's, can never be the square of a whole number
(8). An exact square never ends in a 6 if the penultimate digit(digit that is next to the last digit) is even (eg. exact squares can not end in 26, 46, 86, etc.)
(9).An exact square never has an odd penultimate digit unless the final digit is a 6 (thus, exact squares can not end in 39,71, etc.)
(10).An exact square never ends with an even number when the last two digits taken together are not divisible by 4 (thus, no exact square can end in 22, 34 and other non-multiples of 4 if the last digit is even)

Firstly, we use "The First by the First and the Last by the Last" technique to solve the square root.

(1). 6889
     There are two groups of figures, '68' and '89'. So we expect 2-digit answer.
     Now see since 68 is greater than 64(82) and less than 81(92), the first figure must be 8.

     So, 6889 is between 6400 and 8100, that means, between 802 and 902.
     Now look at the last figure of 6889, which is 9.
     Squaring of numbers 3 and 7 ends with 9.
     So, either the answer is 83 or 87.
     There are two easy ways of deciding. One is to use the digit sums.
     If 872 = 6889
     Then converting to digit sums
     (L.H.S. is 8+7 = 15 -> 1+5 -> 6 and R.H.S. is 6+8+8+9 -> 31 -> 3+1 -> 4)
     We get 62 -> 4, which is not correct.
     But 832 = 6889 becomes 22 -> 4, so the answer must be 83.
     The other method is to recall that since 852 = 7225 and 6889 is below this. 6889 must be below 85. So it must be 83.

Note: To find the square root of a perfect 4-digit square number we find the first figure by looking at the first figures and we find two possible last figures by looking at the last figure. We then decide which is correct either by considering the digit sums or by considering the square of their mean.

(2). 5776
     The first 2-digit(i.e. 57) at the beginning is between 49 and 64, so the first figure must be 7.
     The last digit (i.e. 6) at the end tells us the square root ends in 4 or 6.
     So the answer is 74 or 76.
     742 = 5776 becomes 22 -> 7 which is not true in terms of digit sums, so 74 is not the answer.
     762 = 5776 becomes 42 > 16 -> 7, which is true, so 76 is the answer.
     Alternatively to choose between 74 and 76 we note that 752 = 5625 and 5776 is greater than this so the square root must be greater than 75. So it must be 76.

Second technique is useful for bigger numbers and in this method, we use "Duplex". In the next article, we shall continue to discuss this second technique. Until then, good luck and happy computing!!

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Anish's Blog said...

Thank you for the trick. Can you explain me how can i do with large number for eg. 151321

Mohit Gupta said...

Please refer to the example given in

Anonymous said...

Take first 4 digit of no that is 1513
Search for the least square that is in this case 38 .
Now see the last digit of no I.e 1
Since. 1 is having perfect square of 1 or 9

After that combine the no I.e
381 or 389
Make square of 381 if it yields our answer than OK if not than make the square of 389 we will get desire result I.e 151321

abhilash said...

Kindly help me in finding sq root for 5256

Purnaabh said...

How can we find the square root of numbers which are not perfect squares?
Thank You

Pooja Gupta said...

Please refer
For Imperfect square root, procedure is same. Difference will be that we shall not get zeroes after decimal, while calculating the square root.

: 73 : 2 1 0 8
16 : :9 12 16 14 15
: 8 : 5 5 . 6 3 3

Here, square root of 732108 is 855.633...

amit said...

The calculation is applicable for perfect square only. Here the answer is nearly 72.5

Unknown said...

By the first method can you tell me how to find the perfect square of 11664..since it has three groups I am a bit confused

Furkan said...

If new dividend find is negative is negative then what can i do

Pooja Gupta said...

Please refer Case2 under complicated cases

Unknown said...

Extremely helpful for competitive exams.

Sarthak Mishra said...

How can we find square root of 15, 35, 125 and so on by this method.

NAS said...

Hw can we find square of 1764. Using abov method I am getting as 48 but d square is 42

Unknown said...

Can you explain the square root for √82356?

Jai Shiv Enterprises said...


goodinfo said...

Very good compilation of maths square root tricks.....i really loved it

Raji said...

This technique is not working for 1009. Can someone explain

Unknown said...

Square root of 7744 not possible by this method

Mudunuri Lalitha Sowmya said...

389 38*38=1444,39*39=1521

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