Long Division Formula

Long Division

The square root of a number can be calculated either through the prime factorization method or by the long division method. Calculating a square root by prime factorization method can sometimes be hectic and time-consuming.To overcome this problem, the long division method can be used to find out the square root of a large number. Long Division method can be used to calculate square roots of both perfect squares and not a perfect square.

How to find Square Roots by Long Division Method?

Steps to calculate square root by long division method is explained with the help of an example.

Step 1. Take the number whose square root is to be calculated. Let's take it 484. Place a bar and arrange the number in pairs starting from the unit digit. If the number is given in an odd digit then the number placed on the left side will also have a bar.

Step 2. Now take the largest number and consider it as a divisor whose square should be less or equal to the number placed on the left side. The number placed on the left side will now be considered as a dividend. Divide the number and write the quotient value. Here the quotient will be 2 and the remainder will be Zero.

Step 3.  Bring down the next number and place it to the right side of the remainder. In this value, we will bring down 84 and this will be considered as our new dividend.

Step 4.  Now take the square of quotient value and place it on the right side.


Step 5. The next step is to find out the largest digit to be placed in unit place and the new number which will get multiplied by the new digit at a unit place should be equal to or less than the dividend 84. In this case, the new digit will be 2 as when we multiply 42 by 2, we will get 84.


Step 6. Now finally we got the remainder 0 and nothing  is left for further multiplication

 Hence, the square root of 484 is 22

How to Find The Square Root of a Decimal Number by the Long Division Method?

The square root of a decimal number using the long division method will be calculated as similar to the square root of a number without decimal. The only difference you will find in placing the bars before and after the decimal point. The bar should be placed after each pair of the number starting from the unit digit (before the decimal point) and even the bar should even be placed after each pair of a number starting from the left side of a number after the decimal point. For example, if you want to find out the square root of 12.25. Then in such a case, the bar will be placed over 12 as one pair and another bar will be placed on another pair i.e. on 25.

What is a Square Root?

The square root of a given number gives the same original numbers when multiplied by itself. For example-If you want to find out the square root of number 9 then you will get the answer 3. Here you can see that when number 3 is multiplying by 3, it is giving back original number 9.

Square Root Symbol

The square root symbol is known as a radical symbol. It is denoted with the symbol √. The square root of 2 is denoted as √2. The digit inside the radical symbol is called a radicand. For example, the square root of 2 is also represented as radical of 2, both the values represent the same.

Properties of a Square Root

  •  The perfect square will only exist with the perfect square

  • The square of an even perfect square will be even and the square of an odd perfect square will be odd.

  • The square root of a negative number cannot be defined.

  • There will always be a perfect square root if the number ends with 1,4,5,6, or 9.

  • The perfect square root will not be possible if the number will end with 2, 3, 7 or 8.

  •  There will be no square root if the number ends with the odd number of zeros.

Applications of a Square Root

The square root is the most important application in Mathematics as square root values can be used to simplify mathematical problems. Computation of square roots can be done using a calculator, prime factorization method or long division method.

Solved Example

1.  Find square root of 1750 and 445.21

Hence, the square root of 1750 is 41


2. √445.21

Hence, the square root of 445. 21 is 21.1

Fun Quiz

1.  Find √0.9

a)     0.3

b)     0.03

c)      0.33

d)     0.94

2. Find the least number which must be added to 6072 to make it a perfect square?

a)     1

b)     10

c)      12

d)     16

FAQ (Frequently Asked Questions)

1. What is a Perfect Square?

Any natural numbers which are the squares of other natural numbers are known as a perfect square.

A perfect square is basically what you are left with when you will multiply two equal integers with each other. For example-1,4,9,16,25 are perfect squares because we know when we will multiply the number with itself, we will get the original answer.







Any number ends with 2, 3, 7, 8 or odd number of zeros will never be a perfect square

A number will only be a perfect square if it will end with 1,4,5,6, or 9

2. Is the Number 0 Considered as a Perfect Square?

The digit zero is considered as a perfect square because we know the perfect square refers to a number whose roots are determined as a rational number. And we know 0 is a rational number and it can be expressed as 0/1. Therefore, we can say the digit 0 is a perfect square.

 The digit with the odd number of zeros will never be considered as a perfect square and the digit with the even number of zeros may or may not be considered as a perfect square. For example-400 is a perfect square but 600 is not a perfect square.