Answer:
about 63.261
Explanation:
You want the square root of 4002 by the division method.
Division method
The division method of finding a square root makes use of the relation ...
N = (x +a)² = x² +2ax +a²
That is, we start by approximating the root of N by x. The next step in the process is to subtract x² from N. This leaves the difference ...
N -x² = (x +a)² -x² = 2xa +a² = (2x +a)·a
The divisor for the remainder from the subtraction looks like double the current value of the root, multiplied by 10 to leave room for the next digit 'a'.
Root of 4002
The first digit of the root (6) is the integer portion of the square root of the first pair of digits. You can find this based on your knowledge of multiplication tables. (Digits are marked off in pairs in either direction from the decimal point.)
The second row of the attachment shows the divisor 12_, where 12 = 2×6, twice the root to that point. The largest digit 'a' that can fill the blank is 3, so the divisor used is 123, and the next subtraction is of (2·6·10 +3)·3 = 369.
When the difference after the subtraction is zero, the process ends. Unless the number being rooted is a perfect square, the root is irrational, so will have infinitely many digits.
The approximate square root of 4002 is 63.261.
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Additional comment
In order to properly provide a rounded value, a digit beyond is required. That is, we do not know if 63.261 is properly rounded or not. We know that 63.26 would be a properly rounded root to 2 decimal places.
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