Are you supposed to do this by hand or just look it up on your calculator?
You should know that 72 = 49 and 82 = 64 so that sqrt(59) is between 7 & 8.
The Newton formula recurrence relationship is
xn+1 = (1/2)[xn +( N/xn)] where N is the number of which the sqrt is required.
Starting with 8 you get (1/2)[8 + (59/8)] = 123/16 = 7.73 which (as it happens in this case) is correct least to one decimal place. Another iteration would probably get you the answer correct to 3 decimal places.
This not the only way to extract square root, but is quite fast...and the way computers used to do it (there may be quicker algorithms now).
