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The Process of Finding the Square Root

The "square" in "square root" is the result of multiplying a number by itself. For example, the square of 2 is 4, because 2 x 2 = 4. Similarly, the square of 7 is 49, because 7 x 7 = 49. Finding the square root, or the 2 and the 7, respectively, in the previous examples, is somewhat harder.

Guessing
- Guessing is helpful, especially when trying to find the square root of a larger number. For most manual methods of finding a square root, you will need a number to start the calculation with, which is another reason guessing is helpful. If trying to find the square root of 54,321, first go to 100, since 100 x 100 = 10,000. Next, go to 1,000, since 1,000 x 1,000 = 1,000,000. You can see that 54,321 is far closer to 10,000 than it is to 1,000,000, so you might guess 150, and 150 x 150 = 22,500. That is not very close, but close enough to start the process.
High/Low
- An effective process, if not a quick process, for finding the square root of a number is to use high/low guesses to slowly narrow down the possible answers. This is really just a continuation of the guessing method, but with a little intelligence. If you are given the number 105, for example, then you might guess 10. 10 x 10 = 100, so you would then try 11. 11 x 11 = 121. Since the number we are looking for is 105, a good second guess would be 10.25. 10.25 x 10.25 = 105.0625. 10.25 was just a little bit too high, so you might try 10.24 and see what that gets you. Keep guessing from there until you have enough decimal points.
Babylonian Method
- A slightly more complicated process to find the square root of a number is the Babylonian method. The equation looks like this: .5(X+S/X), where X is your guess and S is the number you are trying to find the square root of. If the number is 105 and the guess is 8, then the formula would look like .5(8+105/8), which comes out to 10.5625, which is quite close. You would then redo the equation, with 105,625 as your guess. Do the equation 6 times and for an answer.
Calculator
- The easiest process for finding a square root, however, is to let your calculator do the work for you. Most calculators will have a square root function key, but you may need to resort to a scientific calculator if yours lacks one. The key you are looking has this symbol on it: √. Type in the number to find the square root of, press the "√" key, and you will be instantly told the square root of the number.

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