Set the upper and lower limits of your search. These limits will change as the search continues and you narrow in on the square root. If your five-digit number is "abcde," a reasonable lower limit is 100 multiplied by "x," where "x" is the largest digit such that "x" squared is less than or equal to the "a" in number "abcde." The upper limit should be 100 plus the lower limit. If you are looking for the square root of 56789, the lower limit would be 200 because 2 is the largest number whose square is less than or equal to 5 -- 2^2 = 4 -- and the upper limit would be 300. Note that 200^2 = 40000, which is less than 56789, and 300^2 = 90000, which is greater than 56789.
Make the first estimate equal to the lower limit + (upper limit - lower limit)/2. Check the estimate by squaring it. If the square of the estimate is greater than the number, the estimate becomes the new upper limit. If the square of the estimate is less than the number, the estimate becomes the new lower limit. When taking the square root of 56789, if 200 + (300 - 200)/2 = 250; and 250^2 = 62500, then the new upper limit becomes 250. The limits are now 200 and 250 -- we are narrowing in on the square root.
Complete the formula from Step 2 with the new limits. The next estimate, 225, becomes the new lower limit because 225 squared is 50625, which is less than 56789.
Stop the process if the upper and lower limits close around a single digit. If the upper and lower limits fall between two digits, the square root is irrational and can never be expressed exactly. At any step in the process, the error between our current estimate and the root will be less than the result of (upper limit - lower limit)/2. When this error is low enough we can stop the process.