Write the number you want to calculate the surd of. Write it under a radical sign and group the digits in pairs in both directions starting at the decimal point. You can add as many pairs of zeros as you want to the right of the decimal point. When the algorithm stops, the answer will be on top of the radical sign. There will be one digit for each pair inside the radical sign. To take find the square root of 547 to the nearest hundredth, you would write inside the radical 5 47.00 00.
Find the largest X such that X^2 is less than or equal to the leftmost group -- which may be a pair or may be a single digit. If you are taking the square root of 547, the leftmost group is 5 and 2 is the largest digit that can be squared and still be less than or equal to 5. Write a 2 on top of the radical sign -- right over the 5, and write 2 X 2 = 4 under the 5. Draw a line and subtract. Then bring down the next group. You should have 1 47 under the line.
Multiply the new digit that was just written on top of the radical sign by 20 to produce the "base candidate" and write this to the left of the new number under the last line drawn. Find a digit N such that N X (base candidate + N) is less than or equal to the number under the line. For the example of taking the square root of 547, N would be 3 because 3 X 43 = 129. Write 3 above the radical directly over the 47. Write 43 under the 40 and 1 29 under the 1 47, draw a line and subtract.
Repeat the last step -- getting different numbers each time -- until you have as many decimal points as you want.