What algorithm do computers use to compute the square root of a number?
Have a look at the paper
https://www.lirmm.fr/arith18/papers/burgessn-divider.pdf
which describes the division / square root hardware implementation in some ARM processors.
Roughly the idea: For a division unit, you can look up the initial bits of divisor and dividend in a table to get two bits of result. For double precision with a 53 bit mantissa, you repeat that 26 or 27 times and you have the result. For single precision with 24 bit mantissa, you need to repeat only 12 or 13 times. All the other bits, calculating the exponent etc., can be done in parallel.
Now what does square root have to do with division? Simple: sqrt (x) = x / sqrt (x). You get the square root by dividing x by its square root. Now when you start this division, you don't know the square root yet. But as soon as you have say eight bits and you are very, very careful, you can produce two mantissa bits per cycle. And getting the first say eight bits is not hard, since unlike a division x / y, there is only one operand involved.
The paper describes this in much more detail, and importantly describes why this is superior for example to using Newton-Raphson or similar. Note: It is superior for typical floating-point numbers with 24 or 53 bit mantissa. Since that is what is used 99.9999% of the time it is important, but for different situations things are different.