It is basically a brute force, trial-and-error method. You insert a number into a formula, compare the formula result to the desired result; then keep changing the original number until the formula result matches the desired result within the level of tolerance.
Which translated into English means, rather than solve for the square root, you instead take x * x and compare the result to your starting number:
Suppose you want the square root of 144.
(a) start with x = 10, you get 10 x 10 = 100 which is < 144 so increase 10.
(b) Try 11, you get 11 x 11 = 121 < 144, so increase 11.
(c) Try 13 you get 13 x 13 = 169 > 144, so decrease 13
.
.
. etc etc etc until eventually you get to x = 12 and you get 12 x 12 = 144 = 144 and so you stop.
It is ugly and inelegant and it works by using raw processing power.
It's ironic, when I used to use a slide rule I would spend more time figuring out where to put the decimal place than I did in solving for the number itself. I probably still have my table of logarithms book in storage somewhere. When the EMP hits I'll be scrambling to find it while I unearth the silver coins from their hiding spot.