How do you evalutate square roots?

Answer 1

Since the calculations are rather involved, this is usually done with a calculator.

Otherwise, you can use trial-and-error - squaring different numbers until you get close to the target. For example, to get the square root of 2: 1 squared is 1, 2 squared is 4, so the square root is between those two values. 1.5 squared is 2.25 (too much), 1.3 squared is 1.69 (too little), 1.4 squared is 1.96 (too little), so now, we know that the square root of 2 is between 1.4 and 1.5. Continue experimenting to get the second digit.

A faster method is as follows. Take a first estimate, for example, the estimated square root of 2 is 1. Divide 2 by 1. The result is 2. So we have 2 as a product of two numbers: 2 = 1 x 2. The square root is between those two values. Take the average, and divide again: 2 / 1.5 = 1.33, or 2 = 1.5 x 1.33. Take the average again; I'll round this to 1.4. If you divide, you get the factors 2 = 1.4 x 1.428. The average is 1.414, which is already quite close. Roughly, with each step you double the amount of significant digits in the answer.