I use this trick quite often while computing square roots. It is extremely simple and utilizes the binomial expansion. It assumes that you know a couple of common squares and their roots. For example, 7 is , 11 is etc.
Let’s take a number 86.
Get the closest square. In this case 81 it’s square root is 9.
Divide 86 by the square root, 9 we get 9.555
Take the average of 9 and 9.555 = 9.277.
Well the exact answer is 9.273, pretty close isn’t it 😉
Let’s try one more example. What is the square root of 420 ?
Closest square is 400, square root is 20. 420/20 = 21. Average of 20 and 21 is20.5. Well, the exact answer is 20.494 😉
Estimating Cube Roots