Most numbers you square root are indefinitely long, so define how many digits you want the answer to be.
How long does it take you to do square root of imperfect number in your mind, without calculator, pen nor paper?
Can any one do a square root of any number up to 4 or 5 s.f. within 1 or 2 minutes, in your mind?
Just wondering what is the quickest way to do square root with mind alone
I can't explain it well, so take a look at the third method here. It works similar to long division, but with important differences. I learned it in High School and, though there is no real need for it any more (cell phones can do square roots now!) it's a neat trick to learn. The other two methods work quickly but if you need the square root of 127 the third one is likely the one you'll need.
-Dan