Answer:
3.5
Explanation:
As a first approximation of the square root, you can look at the integers relating the given number to the next lower square number. The next lower square number is 3^2 = 9.
12 = n^2 +r = 3^2 +3 . . . . . n = 3, r = 3
The approximate square root is ...
n +r/(2n+1) = 3 +3/7 ≈ 3.429
This value tends to be a little low. Alternatively, the root can be estimated as ...
n +r/(2n) = 3 +3/6 = 3.5
This value tends to be a little high.
A reasonable estimate of √12 is 3.5.
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Of course, your calculator can tell you instantly that the root is about 3.464102, close to 3.5.
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Additional comment
The relation we used above simplifies the work of linear interpolation between the points (9, 3) and (16, 4) on the square root curve. For some n between 9 and 16, the interpolated root value r would be found from ...
(n -9)/(16 -9) = (r -3)/(4 -3)
(n -9)/7 = (r -3)/1
r = 3 +(n -9)/7 = 3 +(12 -9)/7 = 3 +3/7 . . . . as above
Note that "r" is used for "root" in this discussion. In the above answer, "r" is used for "remainder."