Question

Division method to calculate sqaure root of large numbers

Answer 1

First off, know that 7600 squared is 57,760,000 and that 7700 squared is 59,290,000.

The trick is that:
sqrt(x) = (a + b)
Where x is the value you want to get the root of, and a is the closest perfect square to the large number.
In this case, x = 58,271,391, and a = 7600, since 7600 squared is 57760000, which is the closest square to 58,271,391.
x^2 = a^2 + 2ab + b^2
Hopefully you a^2 value is close enough to x^2 so that the b^2 value is negligible. Assuming b^2 is negligible:
x^2 = a^2 + 2ab
b = (x^2 -a^2) / (2a)
Since you want a + b:
a + b = a + (x^2 -a^2) / (2a)
x^2 will be 58,271,391
a^2 = 57,760,000

By mental math, you will have to be able to compute 58,271,391 - 57,760,000 = 511,391
Then, by mental math, you will need to compute 2 * 7600 = 15200

Finally, you will need to approximate 511,391 / 15,200 pretty accurately.
This value will turn out to be around 34, thus, a + b = 7600 + 34 = 7634.

Checking on a calculator, the value is actually 7633.57, thus, using this technique, you should be able to estimate the root of a number like that to within the nearest whole number.