Write your square root as a radical, leaving space above the radical to write your answer, as you would do in a long division calculation. Either leave a space or otherwise separate the digits of the number you are taking the square root of in pairs, moving from right to left. If there are an odd number of digits (ignoring trailing zeroes), the first (leftmost) digit will be unpaired. As an example, the square root of 4.5369 would be written "4. 53 69."
Guess and check to determine the largest whole number whose square is equal to or less than the number you are taking the root of, and write it above your radical's first digit pair (or single). In the sample problem, the square root of 4.5369, the largest whole number whose square does not exceed this value is 2. This will be written directly above the 4.
Take the square of that number and subtract it from the number you are extracting the root of. Write the difference below, bringing down the next pair with it. In the example, the difference will yield 0.53.
Write double the value of what you have above the radical so far on the same line as the difference you've calculated -- but clearly separated. Leave space for one more digit after this value. For the example you would take the double of 2, which is 4, and leave space for another digit.
Determine, through guess and check, a digit such that the product of this digit with the doubled number followed by the same digit will be the largest number not exceeding the difference you calculated. For the example, 42 x 2 = 84, which is larger than 53. Therefore, you have to go with 41 x 1 = 41. This last digit will be written above the next digit pair of the radical.
Take the aforementioned product and subtract it from the difference on your current line. Write the new difference below and bring down the next digit pair. In this case, the difference will be 12, and you will bring down our next digit pair of 69, giving you 1,269 as the new bottom line.
Repeat Steps 4 through 6 until you have calculated your square root to the desired number of decimal places, or until your difference is equal to zero. In the latter case, you will have calculated the value of the radical exactly. In the example, you would double the 21 from above the radical to get 42( ), leaving space for an unknown digit. The product of 423 x 3 = 1,269, which is the remainder exactly, yielding 3 as the next, and last, digit of the calculation. So the exact square root of 4.5369 is 2.13.