How do you find the square root of 200 without a calculator?

Well, the easiest way is to note that √200 = √2 x √100 = 10√2. You can look up √2 in a book, and then just shift the decimal one place.

Alternatively, there is a procedure for computing the square root of a number with pencil and paper. It looks very much like old-fashioned "long division," but with some differences in the details.

1. First, write the number you are finding the square root of, grouping the digits in pairs on either side of the decimal point. Thus, 200 becomes 2 00 . 00 00 00 00
2. The first group of digits is just the number 2. Find the largest square that is less than that number. In this case, that's 1. The square root of 1 is also 1, and that is the first digit of the answer.
3. Subtract the square from the previous step, leaving a remainder of 1, and bring down the next pair of digits, leaving you with 100.
4. You now take the answer so far, double it, and put a blank next to it. In this case, that's 2_.
5. Fill in the blank with the largest number that gives you a multiple 2_ * _ that is less than your working remainder. 21 * 1 = 21, 22 * 2 = 44, 23 * 3 = 69, 24 * 4 = 96, 25 * 5 = 125, so the largest multiple that fits our requirements is 24 * 4. Write the 4 as the next digit of the answer, and subtract 96 from the working remainder of 100, and bring down the next pair of digits.
6. Repeat steps 4 and 5 until you reach the precision you require.

Here's what it looks like:

1 4 . 1 4 2 1 3 5 6 2 √ 2 00 . 00 00 00 00 00 00 00 00 1 * 1 = 1 - 1 1 00 24 * 4 = 96 - 96 4 00 281 * 1 = 281 - 2 81 1 19 00 2824 * 4 = 11296 - 1 12 96 6 04 00 28282 * 2 = 56564 - 5 65 64 38 36 00 282841 * 1 = 282841 - 28 28 41 10 07 59 00 2828423 * 3 = 8485269 - 8 48 52 69 1 59 06 31 00 28284265 * 5 = 141421325 - 1 41 42 13 25 17 64 17 75 00 282842706 * 6 = 1697056236 - 16 97 05 62 36 67 12 12 64 00 2828427122 * 2 = 5656854244 - 56 56 85 42 44 10 55 27 21 56