How do you extract square root manually with examples?

Method 1: You experiment with different numbers. For example, to get the square root of 2: 1 squared = 1, and 2 squared = 4, therefore the root of 2 must be somewhere between 1 and 2. If you continue experimenting, you'll eventually find that 1.42 = 1.96, while 1.52 = 2.25, so the square root is somewhere between 1.4 and 1.5. Continue experimenting to find more and more digits.

Method 2 (much faster): Start with a preliminary estimate for the root, for example, estimate the square root of 2 as 1. Divide by the preliminary root, 2 / 1, the result is 2. That means that 2 x 1 = 2, the real root must be somewhere between both. Take the average, 1.5, as the next estimate for the square root.

Continue by dividing 2 / 1.5 = 1.3333. Take the average of 1.5 and 1.3333, which is 1.416.

Continue by dividing 2 / 1.416, etc.

The number of correct digits in the result tends to double with each step, making it much faster than the first method.