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Home >> Square Root >> Square Root by Long Division Method >>

Square Root by Long Division Method

Square Root by Repeated Subtraction Method Square Root by Prime Factorization Method Square Root by Long Division Method

Before you understand How to calculate Square Root by Long Division Method, you are advice to read:

What is Square Root ?
What is Square Number ?
Fundamental Operations: Divisor, Dividend, Quotient, Remainder

Steps of finding Square root by division method:

  • Step 1: Make pair of digits of given number starting with digit at one’s place.
    For ease of calculation; put bar on each pair. If the number of digits of the given number is odd, then place the bar on this remaining digit also.
  • Step 2: Find the divisor, such that its square is either equal or less than the number under first bar from the left. Here divisor and first digit of quotient will be same. Now do the division and get the remainder
  • Step 3: Now bring down the next digits, under second bar and place them to the right of remainder. This will become new dividend.
  • Step 4: Starting digit(s) of new divisor is equal to twice of first digit of quotient (calculated in step 2)
  • Step 5: Now the digit at one’s place of new divisor is equal to next digit of quotient.
    And it is to be selected in such a way that, when new divisor is multiplied by this next digit of quotient, the product will be equal or less than the new dividend (as calculated in step 3)
  • Step 6: Divide and get the new remainder.
  • Step 7: This process is to be repeated till all numbers are used under the bars.

    Example 1: Find square root of 225 using division method
    Solution: Step are as follows:

  • Step 1: Make pair of digits of given number starting with digit at one’s place.
    For ease of calculation; put bar on each pair. If the number of digits of the given number is odd, then place the bar on this remaining digit also.
    Make pair of 25 (under bar red) and as we are left with only single digit i.e. 2, so put it under green bar (as shown below)



  • Step 2: Find the divisor, such that its square is either equal or less than the number under first bar from the left. Here divisor and first digit of quotient will be same. Now do the division and get the remainder
    Number under green bar i.e. 2 is the dividend. Calculated divisor is 1 and first digit of quotient is also 1. After division we get 1 as remainder. (as shown below.)



  • Step 3: Now bring down the next digits, under second bar and place them to the right of remainder. This will become new dividend.
    Bring 25 down (under red bar), place it to the right of remainder 1 (calculate in step 2) and new dividend is 125 (as shown below)



  • Step 4: Starting digit(s) of new divisor is equal to twice of first digit of quotient (as in step 2)
    First digit of quotient calculated in step2 is 1 and twice of 1 is 2,
    so 2 is starting digit of new divisor. (as shown below)



  • Step 5: Now the digit at one’s place of new divisor is equal to next digit of quotient.
    And it is to be selected in such a way that, when new divisor is multiplied by this next digit of quotient, the product will be equal or less than the new dividend (as calculated in step 3)
    5 should be the digit at one’s place of new divisor because when 25 is multiplied by 5 we get 125, which is equal to our new dividend.
    So new divisor is 25 and next digit of quotient is 5. (as shown below):



  • Step 6: Divide and get the new remainder. After division we get remainder Zero (as shown below)



  • Step 7: Since all the numbers under the bars are used, so this puts end of this method and we get:
    15 is the square root of 225
    Or we can write it as
      225   = 15



    Example 2: Find square root of 2401 using division method
    Solution: Step are as follows:

  • Step 1: Make pair of digits of given number starting with digit at one’s place.
    For ease of calculation; put bar on each pair.
    Make pair of 24 (under bar green) and pair of 01 (under bar red);as shown below:



  • Step 2: Find the divisor, such that its square is either equal or less than the number under first bar from the left. Here divisor and first digit of quotient will be same. Now do the division and get the remainder
    Number under green bar i.e. 24 is the dividend. Calculated divisor is 4 and first digit of quotient is also 4. After division we get 8 as remainder. (as shown below.)



  • Step 3: Now bring down the next digits, under second bar and place them to the right of remainder. This will become new dividend.
    Bring 01 down (under red bar), place it to the right of remainder 8 (calculate in step 2) and new dividend is 801 (as shown below)



  • Step 4: Starting digit(s) of new divisor is equal to twice of first digit of quotient (as in step 2)
    Starting digits of quotient calculated in step2 is 4 and twice of 4 is 8,
    so 8 is starting digit of new divisor. (as shown below)



  • Step 5: Now the digit at one’s place of new divisor is equal to next digit of quotient.
    And it is to be selected in such a way that, when new divisor is multiplied by this next digit of quotient, the product will be equal or less than the new dividend (as calculated in step 3)
    9 should be the digit at one’s place of new divisor because when 89 is multiplied by 9 we get 801, which is equal to our new dividend.
    So new divisor is 89 and next digit of quotient is 9. (as shown below):



  • Step 6: Divide and get the new remainder.
    After division we get remainder Zero (as shown below)



  • Step 7: Since all the numbers under bars are used, so this puts end of this method and we get:
    49 is the square root of 2401

    Or we can write it as
      2401   = 49

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