There is a pseudo long division process. Hopefully you can follow my explanation which I've merged with an example.
Separate the digits of the number into pairs working left and right of the decimal point. The lead ‘pair’ may only be a single digit. e.g 985.96 becomes 9 85. 96 Set out the display as for long division but as yet there is no divisor.
Find the square root equal to or less than the 1st pair. E.g. √9 = 3. Put 3 as the start of the quotient (above the 9).
Multiply 3 by 20 = 60 and use this as the divisor for the next stage. Bring down the next pair of digits (85). From now on when ‘dividing’ you also have to allow for the square of the new part quotient to be accommodated. So, 85 ‘divides’ by 60 ONE time but the remainder is not 25. We need to accommodate 1² = 1; the remainder is 25 – 1 = 24 which is carried over to the next stage. Put 1 as the next digit of the quotient (above 85). The quotient now reads 31.
Multiply 31 by 20 = 620 and use this as the divisor for the next stage. Bring down the next pair of digits (96) but as there is a carry over, you are now ‘dividing’ 2496 by 620. This divides FOUR times and the remainder of 16 accommodates 4² = 16. So there is no final remainder. No further pairs of digits to bring down and when 4 is put as the next digit of the quotient (above 96) then answer reads 31.4.
This is a very basic example and hopefully you can follow my explanation. It’s a great arithmetic challenge but becomes number crunching with a vengeance when the square root of numbers containing seven or more digits is required.