How do you work out square roots?

Answer 1

The process of trial and error is sometimes sufficient if you know squares of common numbers.

For example, the square root of 14400 is 120, because you know that 12 x 12 is 144 and 10 x 10 is 100.

The actual (tedious) process involves multiple steps in a process akin to long division, where you locate quotients for each number of the root.

Fortunately, there are now calculators that provide this information instantly.

Answer 2

There are several methods; you might just use trial and error, that is, try the squares of different numbers, getting closer and closer to the actual square root.

Another method consists of starting with an initial estimate, then dividing the number by the estimate. Then calculate the average between the initial estimate and the result, and use this for your next estimate. For example, you want to calculate the square root of 2, and use 1 as your initial estimate. 2 / 1 = 2, so you take the average of 1 and 2 for your next estimate. That would be 1.5. Next step: 2 / 1.5 = 1.33; average between 1.5 and 1.33 is 1.42. This is already fairly close to the square root of 2, but you can continue this process to get more significant digits. You double the amount of significant digits in each step, more or less.