# Finding square root using Long division method

In this chapter, we will learn method to find square root of perfect square by long division method.

The method is somewhat complicated, so I urge you to take out pen and paper and start solving the problem as you move ahead with the chapter.

## How to find square root using long division method ?

Given below are the steps to find the square root. We will understand the process with the help of an example.

Let’s find the square root of number 3844.

(a) First form pair of two from the unit place of the given number.

(b) Select the pair from left side.

The first pair is 38.

(c) Finding First divisor and quotient.

First Divisor
Select highest number whose value when multiplied by itself is less than 38.

Using hit and trial method for calculation.

Digit 7
7 x 7 = 49
NO, as 49 > 38.

Digit 6
6 x 6 = 36
YES, as 36 < 38.

Write number 6 in place of both divisor and quotient.

Subtract number 36 from 38.

Include the next pair 44 in the main dividend.

(c) 2nd Divisor and quotient

Getting next Divisor .
The step to find next divisor is divided into two parts.

(i) Add current divisor 6 with last quotient digit (number 6)

Current divisor = 6
Last quotient digit = 6
Adding the numbers = 6 + 6 =12

Insert 12 in the place of next divisor.

(ii) Insert digit after 12.

Now we have to insert another digit after 12 such that when the completed number is again multiplied by digit, we get value equal / smaller than dividend 244.

Here we have to use hit and trial method.

Let digit = 4.
Then divisor will become 124.
Multiply divisor with 4, we get 124 x 4 = 496.

Since 496 > 244, the digit 4 is rejected.

Let digit = 3
New divisor = 123.
Multiply 123 x 3 = 369.

Since 369 > 244, digit 3 is also rejected.

Let digit = 2
New divisor = 122
Multiply 122 x 2 = 244
Since 244 = 244, digit 2 is accepted.

Hence, digit 2 is added both on divisor and quotient.

Note that subtracting dividend gets 0.

So the final quotient 62 is the square root of number 3844.

Example 02
Find the square root of 19044

Solution

(a) Form pair of two from unit place.

Select the first pair from the left side.

Here we get the single number 1.

(b) Finding first divisor and quotient.

First divisor
Select the highest number which when multiplied by itself is less /equal than first number 1.

Using hit & trial method.

Let digit = 2
Multiply by itself we get 2 * 2 = 4
Since 4 > 1, this digit is rejected.

Let digit = 1
Multiply by itself we get 1 * 1 = 1
Since 1 = 1, this one is selected.

Put 1 in the divisor & quotient.

Include the next pair 90 in main divisor.

(c) Finding second divisor and quotient

Next divisor
Finding next divisor involve two steps;

(i) Add previous divisor and last digit of quotient.

Previous divisor = 1
Last digit of quotient = 1

Add both the digits = 1 + 1 = 2.

Insert number 2 on divisor area.

Now we have to insert another digit on the right side of current divisor.

(ii) Insert digit after 2 such that the number formed when multiplied with digit will produce value less than or equal to 90.

Using hit & trial method;

Let digit = 4
Divisor becomes 24.
Multiply 24 x 4 = 96.

Since 96 > 90, the digit 4 is rejected.

Let digit = 3
Divisor becomes 23.
Multiply 23 x 3 = 69

Since 69 < 90, digit 3 is selected.

So, put digit 3 on divisor and quotient.

Include next pair of numbers in main divisor.

(d) Finding third divisor and quotient

Next divisor
Again the calculation involves two steps;

(i) Add past divisor and last digit of quotient

Past divisor = 23
Last quotient digit = 3

Adding both numbers = 23 + 3 = 26

Insert number 26 on divisor area.

(ii) Insert digit after 26 such that on multiplication of number with same digit we get value less or equal than 2144.

Using Hit & Trial method.

Let digit = 9
Divisor becomes 269
Multiply 269 x 9 = 2421.

Since 2421 > 2144, the digit 9 is rejected.

Let digit = 8
Divisor becomes 268
Multiply 268 x 8 = 2144.

Here 2144 = 2144, hence digit 8 is accepted.

Insert number 8 on quotient and divisor.

After subtraction, the dividend is 0.

Hence, 138 is the square root of number 19044.

Example 03
Find the square root of 2809

Solution

(a) Form pair of two starting form unit place.

Select the first pair from left side, we get number 28.

(b) Finding first divisor and quotient

First divisor
Select the highest number which when multiplied by itself has value less/equal than 28.

Using hit and trial method.

Let digit be 6.
Multiply by itself 6 x 6 = 36
Since, 36 > 28, we reject the digit 6.

Let digit be 5.
Multiply 5 x 5 = 25
Since, 25 < 28, we select the number.

Now insert number 5 on divisor and quotient.

Noe get the next pair 09 on the main dividend.

(c) Finding second divisor and quotient.

Second Divisor;
Finding the next divisor involve two steps;

(i) Add previous divisor and last digit of quotient.

Previous divisor = 5
Last digit of quotient = 5.

Adding both digits = 5 + 5 = 10

Insert 10 on divisor area.

Now, we have to insert another digit on right side of divisor 10.

(ii) Insert digit after 10 such that on multiplying that number with the digit produce value less than or equal to 309.

Let the digit be 4.
Divisor becomes 104.
Multiply 104 x 4 = 408.

Since 408 > 309, we reject the digit 4.

Let the digit be 3.
Divisor becomes 103.
Multiply 103 x 3 = 309

Since 309 is equal to divisor 309, we select digit 3.

Insert digit 3 on divisor and quotient.

Note that after calculation, we got the divisor = 0.

Hence, 53 is the square root of number 2809.

Example 04
Find square root of 11025

Solution

(a) Form pair of two from units place.

Select the first pair from the right side.

Here we will get single digit 1 as it is not forming pair with other digit.

(b) First Divisor and quotient

First divisor
Select the number which multiplied by itself is less than divided 1.

Digit 1 is only number that fits the above specification. So put number 1 in divisor and quotient.

Now insert next pair of dividend in main area.

(b) Finding second divisor and quotient.

Second divisor
This process involves two steps;

(i) Add previous divisor and last digit of quotient.

Previous divisor = 1
Last digit of quotient = 1

Adding the numbers = 1 + 1 = 2

Putting digit 2 on divisor area.

Now we have to insert another digit on right side of current divisor.

(ii) Insert digit after 2 such that if we multiply the complete divisor with that digit we get value less than 10.

Using hit and trial.

Let digit = 1;
Divisor becomes 21.
Multiply 21 x 1 = 21

Since 21> 10, we reject digit 1.

Let digit = 0
Divisor becomes 20.
Multiply 20 x 0 = 0

Since 0 < 10, we accept digit 0.

Insert digit 0 on divisor and quotient.

Now insert next pair 25 in the main dividend calculation.

(d) Finding third divisor and quotient

Calculating next divisor
The process involves two steps;

(i) Add previous divisor and last digit of quotient.

Previous divisor = 20
Last digit of quotient = 0

Adding the numbers = 20 + 0 = 20

Hence, insert 20 on the divisor area.

We insert another digit after 20 to complete the divisor.

(ii) Insert digit after 20 such that if we multiply the whole divisor with the digit, it’s value is less or equal to 1025.

Using hit and trial method.

Let digit be 5.
Divisor becomes 205.
Multiply 205 x 5 = 1025.

Since 1025 is equal to divisor 1025, we accept the digit 5.

Insert digit 5 in divisor and quotient and do the calculation of dividend.

Since dividend is reduced to zero, the number 105 is the square root of number 11025.