First split the number into pairs of digits starting from the right (that is, left of the decimal point):
31 | 36.
After the decimal point you can add zeroes in pairs if necessary.
Look at the first pair of digits and find the nearest square lower than 31: answer is 25=5^2.
5 | 6
31 | 36
106 ) 6 36
Subtract 25 from 31 to leave 6, and bring down the next pair of digits.
We double the 5 in our result to get 10. We now form a "divisor" made up of this result and another digit, denoted by X, so that we have the number 10X. We choose X such that 10X times X is as close as possible to 636. If we picked X=5 we would get the product 105*5=525. Too small, maybe. Try 6: 106*6=636. Exactly right! so the next digit in the answer is 6 and when we subtract 636 from 636 we get zero, so the square root of 3136=56.
If there had been a remainder we would have had to continue the process by doubling the last digit of the "divisor" and then selecting an X value as before.