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 Skip Navigation LinksMath Help > Procedures > Square Root

This is a "pencil and paper" procedure for finding square roots.  The method is similar to long division.  It is illustrated using an example.

As an example, let's take the square root of 819531.8784.

1. To set up the problem, put the number of which you want to find the square root inside the square root symbol.  Then, starting at the decimal point, group the digits in pairs.  Pad any leftover single digits with zeros.

   ---------------
  V 81 95 31.87 84

2. Determine the highest perfect square that is less than or equal to the first pair of digits.  Put its square root above the pair of digits, where the "quotient" would be in a long-division problem.

     9
   ---------------
  V 81 95 31.87 84

3. Multiply this digit by itself, and subtract from the first pair of digits.  Then bring down the next pair of digits.

     9
   ---------------
  V 81 95 31.87 84
   -81
   ---
     0 95

4. Double the partial square root that you know so far, write it down, and then append a "mystery digit".  Multiply this number by that mystery digit, then write this product below the numbers you've written so far.  It should look like this:

       9
     ---------------
    V 81 95 31.87 84
     -81
     ---
18_    0 95
* _ = 

5. Select the "mystery digit" so that the number is as large as possible, but not larger than the numbers above it.  In this case the largest digit that fits is 0.

       9
     ---------------
    V 81 95 31.87 84
     -81
     ---
180    0 95
* 0 =     0

6. Write the mystery digit, 0, at the top, then subtract, and bring down the next pair of digits:

       9  0
     ---------------
    V 81 95 31.87 84
     -81
     ---
180    0 95
* 0 =     0
       ----
         95 31

Now repeat steps 4 through 6.

       9  0  5
     ---------------
    V 81 95 31.87 84
     -81
     ---
180    0 95
* 0 =     0
       ----
 1805    95 31
 5 = -90 25
         -----
           506 87

And again, two more times.  Remember to place the decimal point in the square root just above the decimal point in the original number.

       9  0  5. 2  8
     ---------------
    V 81 95 31.87 84
     -81
     ---
180    0 95
* 0 =     0
       ----
 1805    95 31
 5 = -90 25
         -----
  18102    506 87
   2 = -362 04
          -------
   181048   14483 84
    *   8 =-14483 84
           ---------
                   0

The final answer, then is:

     ------------
    V 819531.8784  =  905.28

Internet references

algorithm to find the square root of a six-digit number 

Related pages in this website

Procedures

Back to Word Problems

 

The webmaster and author of this Math Help site is Graeme McRae.