# Square root of decimals

In this chapter we will learn to calculate decimal square root using long division method.

We have already discussed the long division method of square root calculation for whole numbers in previous chapter. Click the red link to read about the same.

## How to find square root of decimal numbers ?

We will understand the process with the help of examples.

Let the given decimal is 56.25

To calculate the square root, follow the below steps;

(a) Form pair of two’s

⟹ for number before decimal point start form pair from the right

⟹ for number after decimal point form pair also from the right

For numbers before and after decimals, the pair formation should be done separately.

(b) Select the pair from left.

Here number 56 is the first pair.

(c) Calculating first divisor and quotient

Select the number which when multiplied by itself gives value less then or equal to 56.

Using hit and trial method.

Let digit be 8.
Multiply by itself 8 x 8 = 64.

Since 64 > 56, we reject the digit 8.

Let digit be 7.
Multiply by itself 7 x7 = 49.

Since 49 < 56, we select the digit 7.

Write number 7 in the place of both divisor and quotient.

Subtract the number 49 from 56.

Include the next pair 25 in the divided area.

Since number 25 lies after decimal point, we will include decimal point in the quotient.

(b) Calculating second divisor and quotient
.

Next divisor calculation
Here the second divisor calculation follows two steps;

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

Previous divisor = 7
Last digit of quotient = 7

Adding = 7 + 7 = 14

Include number 14 in the divisor area.

(b) Insert digit after 14.

The digit placed should be such that if we multiplied the number formed with the same digit we get the value less or equal than 725.

Using hit and trial method.

Let the digit is 6.
Number formed 146.
Multiply 146 x 6 = 876.

Since 876 > 725, we reject the digit 6.

Let the digit is 5.
Divisor formed 145.
Multiply 145 x 5 = 725.

Since 725 = 725, we accept the digit 5.

Now place digit 5 in both divisor and quotient.

Also subtract 725 – 725, in the dividend area.

Here we got the dividend equals zero.

Hence, 7.5 is the square root of number 56.25

Example 02
Find the square root of decimal 5.29

Solution
To find the square root, follow the below steps;

(a) Form pair of two separately for numbers before and after decimal place.

(b) Select the pair from the left.

Here we have selected the single number 5.

(c) First divisor and quotient.

Calculating first divisor
Select the number which when multiplied by itself gives number less than or equal to 5.

Using hit and trial.

Let the digit be 3.
Multiply by itself 3 x 3 = 9
Since 9 > 5, the digit 3 is rejected.

Let the digit be 2.
Multiply by itself 2 x 2 = 4.
Since 4 < 5, we accept digit 2.

Put number 2 in both divisor and quotient.

Also put number 2x 2 = 4, in the divisor area.

Now include next pair, number 29, in the dividend area.

Since number 29 is after decimal point, we will include decimal point in the quotient.

(d) Second divisor and quotient

Calculating second divisor.
This process involves two steps;

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

Previous divisor = 2
Last digit of quotient = 2

Adding = 2 + 2 = 4

Insert 4 in place of next divisor.

(ii) Insert digit after divisor 4.

The digit should be such that the number formed if multiplied by same digit produce value less or equal than 129.

Using hit and trial method.

Let the digit be 4.
Divisor formed 44.
Multiply 44 x 4 = 176

Since 176 > 129, we reject the digit 4.

Let digit be 3.
Divisor formed 43.
Multiply 43 x 3 = 129

Since 129 is equal to dividend 129, we accept the digit 3.

Put number 3 in place of divisor and quotient.

Also include 129 in dividend place for subtraction.

Note that after subtraction, the dividend equals 0.

Hence, 2.3 is the square root of number 5.29.

Example 03
Find the square root of 1.296 up to two decimal places.

Solution
Follow the below steps;

(a) Form pair of two’s separately for numbers before and after decimals.

Since the three numbers after decimal cannot form pair of two’s, we shall include digit 0 at the end.

Now the number becomes 1.2960

(b) Select the first pair from left side.

Here we will select the single digit 1.

(c) Finding first divisor and quotient

First divisor
Select the number which when multiplied by itself produce number less than dividend 1.

Selecting digit 1.
Because when we multiply it by itself we get 1 x 1 = 1, which is equal to dividend value.

Put 1 in divisor and quotient.

Now include next pair of dividend number 29.

Since the number is after decimal point, we will include decimal point at the quotient.

(d) Finding second divisor and quotient

Calculating second divisor.
Calculation of second divisor involves two steps;

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

Previous divisor = 1
Last digit of quotient = 1
Adding = 1 + 1 = 2

Insert number 2 on the divisor area.

(ii) Insert digit after divisor 2

Insert digit such that the number formed when multiplied by that digit gives value less than or equal to 29.

The digit is 1.
Divisor formed is 21.
Multiply 21 x 1 = 21.

Since 21 < 29, we select the digit 1.

Put digit 1 in divisor and quotient area.
Also subtract 29 – 21 in dividend area.

Now insert the next pair of dividend 60 in the main calculation.

(e) Finding third divisor and quotient

Third divisor calculation
The process involves two steps;

(i) Add previous divisor and last digit of quotient

Previous divisor = 21
Last digit of quotient = 1
Adding = 21 + 1 = 22

Insert 22 in divisor area.

(ii) Insert digit after divisor 22.

The digit should be such that the number formed when multiplied by same digit is less than or equal to dividend 860.

Using hit & trial.

Let the digit be 4.
Divisor becomes 224.
Multiply 224 x 4 = 896

Since 896 > 860, we reject the digit 4.

Let the digit be 3.
Divisor becomes 223
Multiply 223 x 3 = 669

Since 669 < 860, we accept the number.

Inserting 3 in both divisor and quotient.
Also subtract 860 – 669.

Here we got the remainder 191.

The process will go on and on.

Since the question asked square root up-to two decimal places. We got 1.13 as the solution.

Note that 1.13 is just an approximate square root of number 1.296

Example 04
Find square root of 1.44

Solution
Follow the below steps;

(a) Form pair of two separately for both number before and after the decimal.

(b) Select the pair from left side.

From the left side we have single digit 1.

Hence, selecting 1 as a dividend.

(c) Finding first divisor and quotient

First divisor
Select the highest number which when multiplied by itself produce number less or equal than 1.

Here we will select number 1.
Because when we multiply 1 by itself 1 x 1 = 1.

After multiplication we get 1 which is equal to dividend.

Insert 1 in both divisor and quotient.

And in dividend, subtract 1 – 1 = 0

Now insert the next pair of dividend 44.
Since pair 44 is after decimal point, we will insert decimal point in quotient also.

(c) Finding second divisor and quotient

Calculating second divisor
This process involves two steps;

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

Last divisor = 1
Last digit of quotient = 1
Adding = 1 + 1 = 2

Insert 2 on the divisor area.

(ii) Insert digit after number 2 in divisor.

The digit should be such that the number formed if multiplied with same digit is less or equal than 44.

Using hit and trial.

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

Since 69 > 44, we reject the digit 3.

Let digit be 2.
Divisor becomes 22.
Multiply 22 x 2 = 44

Since 44 is equal to dividend 44, we accept digit 2.

Now insert 2 on divisor and quotient.
And in dividend, subtract 44-44 = 0.

Note that dividend has become 0.

Hence, 1.2 is the square root of 1.44

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