If you are a good guesser and can do simple arithmetic, you can do square root.

Make a guess, divide the guess into the number, use the average of the guess and the quotient as the next guess. I will illustrate by giving an example.

Suppose you did not know that the square root of 100 is 10, and wanted to calculate it. You are a bad guesser and decide that 20 is a reasonable guess.

100/20 = 5
Average(5, 20) = 12.5

100/12.5 = 8
Average(8, 12.5) = 10.25 (We are getting closer)

100/10.25 = 9.756 097 561
Average(9.756 097 561, 10.25) = 10.003 048 751

100/10.003 048 751 = 9.996 952 178
Average(9.996 952 178, 10.003 048 751) = 10.000 000 465

Keep going until you think you are close enough.

Note that as you get closer, the guesses do not change much. This is how you know you are close.

There is a method which is a bit like long division, but I find the above easier. I make good first guesses.