Answer:
see below
Step-by-step explanation:
The attached picture shows the calculation of the first several digits of the square root of 50.
To start with, you group digits of the number in pairs. Here, there is one pair of digits left of the decimal point, so the integer portion of the root will have one digit.
For that first digit, you want the largest integer whose square is smaller than the leftmost digit pair. Here that pair is 50, so the largest square less than this is 7² = 49 and the first root digit is 7. We subtract this square and append the next digit pair to the result.
At this stage, and each succeeding stage, we double the root value to use as the divisor of the "dividend" we just found. The least-significant digit of this "divisor" will be the next root digit. We will subtract the product of that root digit and the "divisor" from the "dividend" to form the left digits of the next dividend. I have shown the appended root digit in red.
You can see that as the number of root digits grows, so does the number of digits in the divisor. This makes it be somewhat tedious to continue the process beyond about 8 or 9 digits.
Each step adds one digit to the root.
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<em>Comment on an alternate root-finding method</em>
You can compare this to the Babylonian method, where the next root guess is the average of the previous one and the square divided by that previous guess. For a square of 50 and a first guess of 7, the root estimates are ...
(50/7 +7)/2 = 7 1/14
(50/(7 1/14) +7 1/14)/2 = 7 197/2772 ≈ 7.07106782
This value is correct to 8 significant digits and took a lot less work. Each step will double the number of accurate significant digits.
