First let's learn the definition of squaring. To square a number means to multiply it by itself. For example, the square of 4 is 16. We could say that four squared is 16. We could also write that as 4² = 16. The little 2 is called an exponent. It doesn't mean to multiply 4 times 2, which students sometimes think. It means to multiply 4 times itself, in this case, giving us 16. A few more examples are 4² = 16, 9² = 81, and 53² = 2809. Notice how quickly the results of squaring become very large. We refer to numbers like 16 and 81 as perfect squares, simply because they are the result of having squared a number.
For practice, list the perfect squares through 144. For example, start with 1², then 2², then 3², etc. It is absolutely essential that you commit this list to memory, ideally taking it a bit further than 12². In later math you will need to utilize these perfect squares very frequently, and you will need to quickly identify what is and is not a perfect square.
Before we move onto square roots, there is one other point to make about squaring which is a huge source of confusion for many students. If we square a negative number, we get a positive result. This is because a negative times a negative equals a positive. For example, (-4)² = 16 (positive). (-10)² = 100 (positive). In the next steps you will see why squaring negative numbers is important.
Taking the square root of a number is a mathematical operation in the same way that squaring, adding, and multiplying are also operations. It means that we need to do something in particular with the given number. In this article the notation sqrt(16) is used to mean "the square root of 16," as an example, but we typically use the symbol shown at left. The symbol is often referred to as a "radical," and we would say, "radical 16" or "square root of 16."
To find the square root of a number, we must ask ourselves what number needs to be squared in order to obtain the number that we are trying to find the square root of. That sounds quite complicated, but all you need to realize is that squaring and "square rooting" are inverse operations. They are opposites. For example, sqrt(16) = 4, since 4² = 16. The question asked us what number we would need to square in order to get 16. The answer is 4. Be very careful. The answer to sqrt(16) is plain 4. The answer is not 4² nor is it sqrt(4).
Let's look at a few more examples. Sqrt(81) = 9. Sqrt (100) = 10. Sqrt(4) = 2. This is quite easy if you have memorized your list of perfect squares. Note that we cannot easily compute the value of something like sqrt(17) since 17 is not a perfect square. You could do it on the calculator, and you'd get an answer that is slightly higher than 4, as you would probably expect. That is outside the scope of this article, though, since we will just deal with perfect squares for now.
There is one last important thing to understand about finding the square root of a number. Recall we said that sqrt(16) equals 4. While that is correct, there is another answer that is also correct. Remember we saw that (-4)² also equals 16. What that means is that -4 is another correct answer to sqrt(16). If we square either 4 or -4, we get 16.
We refer to the answer of 4 as the principal square root. Usually we will just give that as our answer. Certainly if we're dealing with a geometry problem involving lengths, a negative answer would be meaningless. However, -4 is most definitely also a correct answer to the problem, and sometimes we are tested to see if we know that. We can use the notation above (±4) to show that there are two answers to the problem. We would read the answer as "plus or minus 4."
Make sure you remember that squaring and square rooting are inverse operations. Also remember that when we give the answer to a square root problem, we must give an actual number (sometimes both the positive and negative versions). We don't give our answer as a squaring or square root. It is important that you fully master this topic or you will be completely lost when it comes up again and again in later math.