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Square Root by Division Method

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Last updated date: 01st Mar 2024
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Long Division Method

The long division method is a calculation technique for finding the square root of a number. A square root is the product of the numbers that is obtained by multiplying a number to itself. A number can be given in the form of the product of prime numbers. The way of representation of a number in terms of the product of prime numbers is known as the prime factorization method. It is one of the simplest methods of calculation of the square root of a number.


For calculating the square roots of imperfect squares such as 3,5,6,7, 8, etc. this method is useful in order to avoid the use of calculators. Now we are aware that the division method is used to calculate the square root of a number, prime factors of a given number, LCM, and HCF of given numbers. Throughout the article, we will discuss the square root by division method, exemplar problems, and multiple-choice questions, and also a few of the most frequently asked questions will also be answered. 


Square Root of a Number by Long Division Method

Let us understand how to solve square root by division method with the help of an example.


Let us take a number that is 361. Now we will calculate the square root by long division method.

  • Step 1.

The first step is to start from the unit place of the given digit and move to the left by grouping two digits at a time. Then put a bar over pairs of digits that will form a group and the leftover single digit forms another group. 


In the given digit 361 we will follow the above procedure we will make two groups such as \[\bar{3}\] and \[\bar{61}\].


  • Step 2.

Now the process of division process will begin from the highest place value. In 361, the highest place value is 3. The largest square less than or equal to the first group is 1 for  3 then take the square root of 1 as the divisor and quotient, and write this quotient 1 above the first group 3.

   1

1⟌\[\bar{3}\] \[\bar{61}\]


  • Step 3.

Then we will calculate the product of the divisor and quotient. In this example i.e. 1 x 1 = 1. After that the operation of subtraction will be performed between this product from the first group i.e., 3 so after subtracting 3-1 we will get the remainder as 2. Now bring down the next group 61 to the right of the remainder 2. 

     1

  1⟌\[\bar{3}\] \[\bar{61}\]

+1   1

 2⟌ 2

                     

  • Step 4.

The quotient 1 will be doubled i.e. 1+1 = 2. This will be entered in the next divisor column with a blank space on the right for the next digit

     1

1⟌ \[\bar{3}\] \[\bar{61}\]

+1   1

2⟌ 2  61


  • Step 5.

Find the next digit in the quotient. The next important step is to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at the unit’s place, is equal to or less than the dividend (61).

In this case,  29 × 9 = 261. So the new digit is 9.

     19

 1⟌\[\bar{3}\] \[\bar{61}\]  

 +1   1

29⟌2  61


  • Step 6.

Multiply the new digit 9 in the quotient with the second divisor 29 x 9 = 261. At last subtract 261 from 261 which will eventually give the remainder 0 • Therefore, the division process is completed. The quotient obtained by this division process is 19.  So 19 would be the square root of 361. 

      19

  1⟌\[\bar{3}\] \[\bar{61}\]

           +1   1

 29⟌2  61

      2   61

   一一一

      0

Hence the square root of 361 is 19 with the use of the long division method for square root.


Example of Long Division Method

Example: Calculate the root square root by the division method of the number 55225.

Solution: Hence first of all we will do the grouping of numbers from the left then groups would be

 \[\overline{552}\] \[\bar{25}\]. Now let’s perform the rest of the operations.

       2 3 5

  2 ⟌\[\overline{552}\]\[\bar{25}\]

+2   4

  43 ⟌1 52

  +3   1 29

  465 ⟌ 23 25

            23 25

    一一一一

             0

So we get the square root of 55225 through long division method  235.


Finding Square Root of Decimals Through Long Division Method

The long division method is also useful in finding the square root value of the decimal numbers. We follow the same process that we follow in normal numbers. Hence first of all we do a grouping of numbers from left to right into equal groups. After that, it will follow the same steps as that of long division such as division, multiplication, subtraction and bringing down the next group and repeating the process to find the remainder. 

  • First of all, write the division in the standard form of tableau then start making a group from the left to the decimal and from the left side of the decimal to the right.

  • Next is to divide the group of numbers.

  • Bring down the next tenth digit.

  • Now repeat the process of division and bringing down of the other digit in sequence. 

  • Divide until you get 0 in the remainder. 

Let’s take the example of number 338.56 and with the use of the long division method for square root calculate the square root of the given number.    

4 ⟌17.64

     16

42⟌ 164

       164

   一一

        0                      

Hence the square root of decimal number 17.64 would be 4.2


Fun Facts

  • When the number 123454321 is divided by 11111. It gives a quotient of 11111 and the remainder of 0.

  • We can check our answer with the help of the formula Dividend = (Divisor × Quotient) + Remainder. This formula is useful in checking the quotient and the remainder of the division.

The square root by long division is useful when we are dividing a large number such as a number that consists of three digits or more by a two-digit or more number. It is an easy method for calculating square root values of numbers. By memorising few basic steps we can calculate square root by division method of large and small numbers.


Conclusion

The square root by long division is useful when we are dividing a large number such as a number that consists of three digits or more by a two-digit or more number. It is an easy method for calculating square root values of numbers. By memorising a few basic steps we can calculate square root by division method of large and small numbers. Students can visit the official website of Vedantu where we also have other topics related to square roots and long division methods.

FAQs on Square Root by Division Method

1. What are the major five steps to find the square root by long division method?

Sometimes the calculation of finding the square root of a number becomes complex when the given number is large. Hence in order to find the square root of large numbers we use the long division method. The long division method to find square root is a very popular and easy method to calculate the accurate roots of numbers. In this method, we observe five main steps such as divide, multiply, subtract, bring down and repeat. We should always remember these five basic steps to find the square root by long division method.

2. How does the long division method work to find the square root by long division method?

With the help of basic mathematics operation we can calculate the square root by division method to find square root of numbers. A tableau is drawn to divide two numbers by this method. The divisor is written outside the right of the tableau and the dividend is placed within. The quotient is placed above the overbar on top of the dividend. In order to find we follow five main steps such as divide, multiply, subtract, bring down and repeat. First of all, we do a grouping of two numbers from left to right and then we perform division, multiplication and then bring down the next group and repeat the process again.

3. How can we find the square root of a decimal number by the division method? 

The square root of a decimal number can be easily calculated by using the long division method. The long division method is one of the easiest methods to find the square root of any number. Whether it's a decimal number, a perfect square, or a non-perfect square we can find the square root by the long division method. In the case of decimal numbers, we have to make pairs of the whole number parts and the fractional parts and then follow the same process of long division as mentioned throughout the article.