Lets find the square root of 5 as an example.
Lets look at the two closest square numbers to 5,
1, 4, 9 (...)
4 and 9 are the closest square numbers. So it looks like our answer is between 2 (22=4) and 3 (32=9). We can now find the average of 2 and 3 (2+3/2 = 2.5)
2.5 squared (2.5*2.5 = 6.25) this is too high so we lower the average a little by 0.3, (2.2*2.2 = 4.84). This should usually be enough to say the answer is approx 2.2, but you can keep going deeper and get a answer of 2.23 (the number is increased 0.3 because 4.84 is a little short of 5.
Alternatively you can simply get a calculator to ease of all this time and effort, calculators have a much more advanced methods and can come up with very precise results.
Hope this helped,
