How do you find the square root of any number?

The simplest way is to make a series of educated guesses until you get as close as you need to be. What is the highest number you can think of that, when squared, is still less than 80? What is the lowest number that, when squared, will be higher than 80? 8 squared is 64, and 9 squared is 81. You now know the answer is between 8 and 9. Now for the bad news: it won't be a rational number, and to get closer, it gets messy.

We could try 8.5 squared, which is 72.25. Still low, so now we know it's between 8.5 and 9.

How about 8.8? That squared is 77.44. Still low. Must be between 8.8 and 9.

Eventually you will find two numbers close enough that when they are squared, the result, rounded to the nearest ___ (depends upon how close you need to come), will be 80. For example, 8.94 squared is 79.9236, which rounded to the nearest one is 80, as is 8.95 squared (which is 80.1025).

If you need it closer than that, pick a number between 8.94 and 8.95 and continue from there in the same way.

## Comments

Brilliantly simple. Whether or not you can leave it as something like:

4 * sq rt (5)

depends upon why you want to find the square root in the first place. But it generally helps if you can simplify first. Thanks for pointing this out.