Square Root Of Perfect Squares - The Vedic Mathematics Way

I. Introduction

We will learn a method to calculate the square root of a number which is a perfect square.

The method

II. Finding The Square Root

To use this method you will need to know the last digits of the squares of all single digit numbers, that is $1$ to $9$.

Let us look at the table below:

	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$
Square	$0$	$1$	$4$	$9$	$16$	$25$	$36$	$49$	$64$	$81$
Last Digit	$0$	$1$	$4$	$9$	$6$	$5$	$6$	$9$	$4$	$1$

Once you have familiarized yourself with the above table, we can use the method below to find the square root of fairly large numbers.

Observe from the above table, that except for the digit $5$, every last digit occurs twice. So, other than numbers ending with $5$, the last digit of a perfect square will give two possibilities as the last digit of the square root. We will see below how to select the correct one. Now let us look at the steps first, followed by a few examples.

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$\underline{Steps}$

1. From the last digit of the given number, list the possible last digits of the square root.

2. Strike our the rightmost two digits of the given number. Let us call the remaining number $A$

3. Find the largest number whose square does not exceed $A$. Let us call this number $B$. The square root is $B$ followed by one of the possible last digits. If there is only one last digit, then your answer is already with you. But if you have two possibilities you need the next two steps.

4. Add $B$ to its own square, which gives you $B^2 + B$

5. If the result of step 4 is greater than $A$ then the square root is the smaller of the two choices. If it is less than or equal to $A$ then the square root is the larger of the two choices.

$\underline{Example\ 1:}$

Let us find out the square root of $1849$ using this method.

1. The last digit of the given number is $9$. Therefore, the last digit of the square root will be either $3$ or $7$

2. Now remove the rightmost two digits $18\cancel{49}$ and you are left with $18$

3. The largest number whose square does not exceed $18$ is $4$ whose square is $16$. Therefore, the square root is either $43$ or $49$

4. Add $4$ to the square of $4$, which gives you $16 + 4 = 20$, which is greater than $18$. The square root of the given number is the smaller of the two, that is $43$.

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$\underline{Example\ 2:}$

Let us consider one more number, a little larger number this time. Let's take $13924$.

1. The last digit of the given number is $4$. Therefore the last digit of the square root will be either $2$ or $8$.

2. Remove the rightmost two digits to get $139$.

3. The largest number whose square does not exceed $139$ is $11$. Therefore the square root is either $112$ or $118$

4. Add $11$ to $11^2$, which gives you $121 + 11 = 132$, which is less than the remaining number $139$. Therefore, the square root of the given number is the larger of the two choices, that is $118$

$\underline{Example\ 3:}$

Let us consider a number which ends with $5$, let's say $4225$

1. The last digit of the given number is $5$, therefore the last digit of the square root is also $5$.

2. Eliminating the rightmost two digits of the given number we are left with $42$.

3. The largest number whose square does not exceed $42$ is $6$, whose square is $36$.

4. Since $5$ is the only possibility for the last digit, we get our square root as $65$.

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III. Timed Practice

You can use the widget to do timed practices of this method. We recommend that initially you do a timed practice once every two to three days, subsequently you can reduce it to once a week or even once a fortnight.

Finding Squares -