Square Roots: How to

Square Roots: How to
By Madison Nef

What is a square root?
The square root of a number is the number that, when multiplied by itself, produces the number you are trying to find the square root of. For example, the square root of 100 would be 10, since 10*10=100. Finding perfect square roots is easy- the square root of 16 is 4 and so on, because there are no decimal places involved. However, without a calculator, it is much harder to find the square root of a number like 10- and to get to an accurate answer, simplification is needed. Otherwise, you could be doing math for hours just to get to your conclusion!

So: how DO you find the square root of a number with an imperfect square?
Start off by finding two perfect squares that your number is in between. For example, the square root of the number 5 would be somewhere between the square root of 4 (2) and the square root of 9 (3). Since 5 is a number closer to 4, we can figure that the square root of 5 is going to be closer to 2 than it is to 3. This gets rid of all the numbers above 2.5, and cuts down the size of the work we need to do.

Next, you want to take the number you are trying to find the square root of and divide it by the number closest to it (in this case, we would divide 5 by 2). If needed, round off your answer. The next step is to get the average of the answer you’ve come up with: so for the problem we are using, (2.5+2)/2… which would equal 2.25. Repeat the second step of the process, and divide 5 by 2.25. This leaves us with 2.2222222… which can be rounded off to 2.22. Finally, average out 2.22 and 2.25. Take the number you get and square it. If the number you come up with is accurate enough for you, then you have your answer: but to get the EXACT square root would be a more complicated process of repeating the same step over and over again; and sometimes the decimals go on for 15-20 places before coming to the EXACT number.

In conclusion, here is the written out problem that we solved:
1) Find two numbers with perfect square roots that 5 is between. In this case, the numbers are 4 and 9.
2) Since 5 is closer to 4 than to 9, we can assume that the square root of 5 is closer to the square root of 4 (2). 
3) Divide 5 by 2. Simplify your answer if necessary.
4) Average the answer you got from the previous step with 5. 
5) Repeat step 5 with the number you got.
6) Round, and you have your answer.

5/2 = 2.25
2.25 + 2 = 4.25 / 2 = 2.22 (simplified)
2.22 + 2.25 = 4.47 / 2 = 2.235
2.235^2 = 4.995225 (round to 5)

* This technique can be used to find the square root of any number.