Answer:
√31 ≈ 5.57
Explanation:
You seem to want the square root of 31, rounded to the nearest hundredth.
Square root
A calculator provides an easy method for finding the square root of a number. The attached shows that ...
√31 ≈ 5.57
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Additional comment
There are a number of ways the square root can be approximated. For integers, one way to approximate the root of n = s² +r is ...
x = √n ≈ s +r/(2s +1)
This approximation can be refined further using a couple of different iterative methods. The "continued fraction method" will give the next approximation as ...
x' = s +r/(s +x) . . . . . for s, r, x defined as above
The "Babylonian method," aka Newton's Method iteration, tells you the next approximation can be ...
x' = (x +n/x)/2
This method converges fairly quickly, doubling the number of accurate decimal places with each iteration.
For the root at hand (s=5, r=6), the first approximation will be ...
√31 ≈ 5 +6/(2·5+1) = 5 6/11 = 5.5454...(repeating)
The next approximation using the Babylonian method will be ...
x' = (61/11 +31/(61/11))/2 = 5 381/671 ≈ 5.5678...
So, √31 ≈ 5.57, as above.
There are also methods that require less reliance upon a calculator for the first few digits of the root. The "long division method" is one of these.