Set up the number you want to take the square root of by writing the number under a radical sign. The digits of the number should be written in groups of two starting at the decimal point and going both directions. For some numbers the leftmost group will have a single digit. For example, if you are taking the square root of 123, the digits under the radical would be written 1 23.00 00 00 00. The solution will appear as one digit over each group.
Determine the first digit in the answer by figuring the largest number that can be squared to be equal to or less than the leftmost group. For taking the square root of 123, this first digit will be 1.
Write the first digit above the fist group and write the square of the first digit under the first group. Draw a line and subtract. Then bring down the next group. The number under the line is called the residue. For taking the square root of 123, the first residue will be 23.
Multiply the number above the radical sign -- the root so far -- by 20 and write this number to the left of the residue. This is the candidate base. Now find a digit d that is the largest digit so that d(candidate base + d) is less than or equal to the residue. Write d above the next group, and write (candidate base + d) below the candidate base -- this is the actual candidate. Determine the product of d multiplied by the actual candidate and write this below the residue. Draw a line and subtract to find the next residue. For taking the square root of 123, the second digit in the answer is 1, and the new residue is 200.
Repeat the previous step until you have as many decimals as you need -- until your approximation is as close to the square root as you want.