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**

Follow the below steps;**(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 quotien**t**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**

Follow the below process.**(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**

Follow the below steps;**(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.**