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Square Root of Perfect Squares in Vedic Maths



The technique of finding square roots of perfect squares is similar to the technique of finding cube roots of perfect cubes.However, the former has an additional step and hence it is discussed after having dealt cube roots.

What is Square Root? To understand square root, it will be important to understand what are squares. Squaring of number is defined by multiplying a number by itself. When we multiply 5 by 5 means we have squared the number 5.

From the above example 25 is the square of 5 and 5 is the square root of 25.

Method:

First of all you must know the squares of numbers from 1 to 10.

NUMBER SQUARE
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100

In cube root we observed that if the last digit of a cube is 1 the last digit of the cube-root is also 1. If the last digit of the cube is 2 then the last digit of the cube-root is 8 and so on. Thus, for every number there was a unique correspodig number.

But, in square roots we have more than one possibility for every number.In the firs row, we have 1 in the number column and 1 in the square column. Similarly in the nineth row we have 9 in the number column and 1 of (81) in the square column. So if the number ends in 1, the square root ends in 1 or 9.(because 1X1=1 and 9X9=81). To further clarification we are giving the explanations below.

  • Similar to the 1 and 9 relationship, if a number ends in 4 the square root ends in 2 or 8( because 2X2=4 and 8X8=64)
  • If the number ends in 9, the squre root is 3 or 7. Because 3X3=9 and 7X 7 is 49)
  • If a number ends in 6, the square root ends in 4 or 6. (because 4X4 is 16 and 6X6 is 36)
  • If the number ends in 5, the square root ends in 5 because 5X5 is 25.
  • if the number ends in 0, the square root also ends in 0 (Because 10X10 is 100)



  • On the basis of the above observations, we can form a table given below.

The last digit of the square The last digit of the square root
1 1 or 9
4 2 or 8
9 3 or 7
6 4 or 6
5 5
0 0

Whenever we come across a square whose last digit is 9, we can conclude that the last digit of the square will be 3 or 7. Similarly when we come across a square whose last digit is 6, we can conclude that the last digit of the square root will be 4 or 6 and so on.....

Look at the column in the left. It reads "last digit of the square" and the numbers obtained in the column are 1,4,9, 6, 5 and 0. Note that the numbers 2, 3,7 and 8 are absent in the column. That means there is no perfect square which ends with the numbers 2,4,7 or 8. Thus we can deduct a rule:

A perfect square will never end with the digits 2, 3, 7 or 8.

At this point we have well understood how to find the last digit of a square root. However in many cases we have two possibilities and one is correct.. Further we do not know how to find the remaining digits of the square root. So we will solve a few examples and observe the technique used to find the complete square root.

Before proceding ahead with the examples, I have given below a list of the squares of numbers which are the multiples of 10 up to 100. The table below will help us to easily determine the square roots.

NUMBER SQUARE
10 100
20 400
30 900
40 1600
50 2500
60 3600
70 4900
80 6400
90 8100
100 10000

Example:

Find the square root of 7744

Step 1 The number 7744 ends with 4. Therefore the square root ends with 2 or 8. The answer at this stage is ___2 or ___8

Step 2 Next we take the number 7744 lies between 6400(Square of 80) and 8100(Square of 90)

There for the square root lies between 80 to 90.

Step 3 From the first step we know that the square root ends with 2 or 8. From the second step we know that the square root lies between 80 and 90. Of all the numbers between 80 and 90 ( 81, 82, 83, 4, 85, 86, 87, 88, 89) the only numbers ending with 2 or 8 are 82 or 88. Thus out of the 82 and 88 one is the correct answer.Observe the number 7744 ----6400 ----8100

Is it closer to smaller number or closer to bigger number.

If it is closer to smaller number then take the smaller number and if it is close to bigger niumber then take the bigger number.

So the answer is 88


Vandana Sachar

About the Author

"Vandana Sachar, M.A. Eco, teaching Vedic Maths for the last 15 years"

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