Square Root and Cube Root Procedure

Square Roots

To extract the square root of a positive number by hand, use the following procedure:

1. Group the digits to the left of the decimal in pairs with possibly a first incomplete pair.
2. Similarly pair digits to the right of the decimal point. Complete any last incomplete pair with a zero.
3. Append additional zero pairs so that the total number of pairs is equal to the desired number of digits in the answer.
4. Estimate the (1 digit) square root of the leftmost pair. This is the first digit of the result.
5. Form the first partial. (The partials are listed along the left margin in the example below.)
6. Double the answer digits obtained so far: these are the most significant digits of the the next partial. The least significant digit is obtained by estimation. (It is shown with an underscore.)
7. The estimated digit is the largest such that its product with the completed partial is less than or equal to the current radicand.
8. The largest such product is the next subtrahend. It is written underneath the current radicand.
9. Subtracting the subtrahend produces the next radicand.
10. Repeat until the desired number of digits are generated.

Example

            1. 4  1
           ___________________________
          /
       \/   2.00 00   <------------- radicand
     1       
     _      1
            -------
	    1 00      <------------- next radicand
    24        96      <------------- subtrahend      
     -      ____
	       4 00
   281         2 81
     -  
            -------
    ^
    |
 Partials (Estimated digit is underscored)

Cube Roots

One begins by grouping the radicand digits to the left of the decimal point in threes (with the most significant digits possibly forming a partial group of 1 or 2.) The mechanics are similar in outline to those of the square root algorithm (see above) except for the formation of the partials. Each (total) partial is the sum of a major and a minor partial. The major is gotten by multiplying the square of the current result digits by 300. The minor partial is the product of the next result digit and the minor factor. The minor factor is gotten by multiplying the current result digits by 30, where the trailing zero is to be replaced by the next result digit. Finally, the next result digit is the largest possible digit such that its product with the resulting total partial will not exceed the current radicand.

             2.  1   5   4
           ___________________________
        3 /
       \/   10.000 000   <------------- radicand
             8
	    ----------
       	     2 000       <------------- second radicand
 1200 
  (61) 
    -
   61
   --
 1261        1 261       <------------ subtrahend
            ------
 132300        739 000
   (635)
      -                 ( Underscores denote digits filled in after
   3175                    estimation. )
   ----
 135500        677 500 
               -------
 13867500               <-------------- major partial
    (6454)              <-------------- (minor factor)
        -
    25816               <-------------- minor partial
    -----
 13893370               <-------------- total partial