We have already seen that we can find square roots by many methods. To find the square root of a perfect square number by using the long division method is easy. We have to follow certain steps to find the square root by the long division method.

Step 1: We have to separate the digits, starting from the unit place. For the same, begin by going two places backward. Make pairs of two numbers. The pairs and the remaining numbers are called a period. 

We will divtopicIde the numbers 1369 into two pairs, and the numbers are 13 and 69, that is, 13 69. Here, the numbers 13 and 69 are called a period.

Step 2: Now, you have to find out the largest squared number that is equal to or less than the first pair of numbers (13). In this case, the square of 2^2 is 4, 3^2 = 9, and 4^2 = 16.  

So, the number we will need would be 9 (3^2 = 9). In our case, 9 is the closest number to the first pair 13. 

Next, since 9 is the square of 3, we have found our desired number 3 to proceed forward. 

Now, divtopicIde 13 by 3. Note that to move on to the next process, the quotient and the divisor must be the same. When we divtopicIde 13/3, we get the divisor as 3; the quotient as 3, and the remainder as 4. 

Step 3: Afterward, you need to subtract the product of the divisor and the quotient from the first pair of numbers and bring down the next pair to the right of the remainder. This becomes the new divtopicIdend. 

Step 4: In the next step, our target is to obtain a new divisor. 

For the same, the quotient should be multiplied by 2 or add the same number with the divisor - choose any one step. 

To elaborate, our original divisor was 3. Now, add 3 to divisor 3 again, or multiply 2 by 3. The answer would be the same, which is 6.

Step 5: Further, think about a suitable single-digit or two-digit number, to be placed next to the new divisor 6 (to be placed to the right of the number 6), such that the same number next to the number 6 should appear in the quotient, next to the existing number 3.

We do this so that when the new divisor is multiplied by the new quotient, we will try to get a number closer to or equivalent to the divtopicIdend. Hence, the remainder will be zero.

Step 5: Repeat these steps from Step 2 to Step 4, if more pairs of numbers are present, till they become zero. 

Step 6: Lastly, the quotient so obtained will be the square root of the given number.

Example 1: Find the square root of 5776 with the use of the division method.

Step 1: Initially, we have to separate the digits from the unit place by taking two places backward.

Here, the number is 5776. We will divtopicIde numbers into two pairs and the numbers are 57 and 76

57   76

—-   — 

Step 2: Now, we have to think of the largest number whose square is equal to or less than that of 57. Here, we will take 7 X 7 (since 8x8=64 which will be more than 57. So, we took 7x7= 49)  

Afterward, we will subtract 57- 49 = 08, and we also will pull down the next pair of numbers, i.e. 76. So, the number will be 876.

Step 3: On the divisor part (here ten’s place), the first number will be 14 (first divisor plus the same number as the divisor) so 7 + 7 = 14.

Now for one’s place, we need to write a number from 0 to 9. But this number will be the same as one's place of the quotient. 

How to choose the number: 

141 X 1 = 141

142 X 2 = 284

143 X 3 = 429

144 X 4 = 576

145 X 5 = 725

146 X 6 = 876

We will choose 146 X 6 as its total is 876. It's the same number as our divtopicIdend. So the one's place number for the divisor as well as the quotient will be 6.

Hence √5776 = 76.

Example 2: Find the square root of 15129

Solution: 

 Step 1: We have to group the number in pairs of two from the backstopicIde. 

Here, 1   51   29

         —  —   —-

Step 2: Now, we have to think of the largest number whose square is equal to or less than the first pair of numbers. Here it is 1 X 1= 1.

Then, we will subtract it (1 - 1= 0).

Step 3: We will bring down the digits which are under the next pair. Here, the next set of numbers will be 051 = 51 

Now double the value of the divisor ( 1+ 1 = 2). Thus, 2 will be in the ten’s place. 

We have to select the largest digit in the unit’s place as the divisor such that the new number, when multiplied by the new unit digit, is equal to or less than 51.

Here, it will be 

21 X 1 = 21

22 X 2 = 44

23 X 3 = 69

Since 69 will be greater than 51, we will take 22 X 2 = 44.

Then, we will subtract it, that is, 51 - 44= 07.

Step 4: Now, again we will repeat the full step 3 method.

For the divisor’s place, we will add 22 + 02= 24 

We have to select the largest digit for units in place of the divisor such that the new number, when multiplied by the new unit digit, is equal to or less than 729.

Here it will be 243 X 3 = 729

Then we will subtract 729 - 729 = 000. Since we are not left with any numbers. 

So, √15129 = 123.