Article: To understand square roots, it's best to start with squares. Squares are easy — taking the square of a number is just multiplying it by itself. For instance, 3 squared is the same as 3 × 3 = 9 and 9 squared is the same as 9 × 9 = 81. Squares are written by marking a small "2" above and to the right of the number being squared — like this: 32, 92, 1002, and so on. Try squaring a few more numbers on your own to test this concept out. Remember, squaring a number is just multiplying it by itself. You can even do this for negative numbers. If you do, the answer will always be positive. For example, -82 = -8 × -8 = 64. The square root symbol (√, also called a "radical" symbol) means basically the "opposite" of the 2 symbol. When you see a radical, you want to ask yourself, "what number can multiply by itself to give the number under the radical?" For instance, if you see √(9), you want to find the number that can be squared to make nine. In this case, the answer is three, because 32 = 9.  As another example,  let's find the square root of 25 (√(25)). This means we want to find the number that squares to make 25. Since 52 = 5 × 5 = 25, we can say that √(25) = 5. You can also think of this as "undoing" a square. For example, if we want to find √(64), the square root of 64, let's start by thinking of 64 as 82. Since a square root symbol basically "cancels out" a square, we can say that √(64) = √(82) = 8. Up until now, the answers to our square root problems have been nice, round numbers. This isn't always the case — in fact, square root problems can sometimes have answers that are very long, inconvenient decimals. Numbers that have square roots that are whole numbers (in other words, numbers that aren't fractions or decimals) are called perfect squares. All of the examples listed above (9, 25, and 64) are perfect squares because when we take their square roots, we get whole numbers (3, 5, and 8). On the other hand, numbers that don't give whole numbers when you take their square roots are called imperfect squares. When you take one of these numbers' square roots, you usually get a decimal or fraction. Sometimes, the decimals involved can be quite messy. For instance, √(13) = 3.605551275464... As you've probably noticed, taking the square root of perfect squares can be quite easy! Because these problems are so simple, it's worth your time to memorize the square roots of the first dozen or so perfect squares. You'll come across these numbers a lot, so taking the time to learn them early can save you lots of time in the long run. The first 12 perfect squares are:  12 = 1 × 1 = 1  22 = 2 × 2 = 4  32 = 3 × 3 = 9  42 = 4 × 4 = 16  52 = 5 × 5 = 25  62 = 6 × 6 = 36  72 = 7 × 7 = 49  82 = 8 × 8 = 64  92 = 9 ×  9  = 81  102 = 10 × 10 = 100  112 = 11 × 11 = 121  122 = 12 × 12 = 144 Finding the square roots of imperfect squares can sometimes be a bit of a pain — especially if you're not using a calculator (in the sections below, you'll find tricks for making this process easier). However, it's often possible to simplify the numbers in square roots to make them easier to work with. To do this, you simply need to separate the number under the radical into its factors, then take the square root of any factors that are perfect squares and write the answer outside the radical. This is easier than it sounds — read on for more information!  Let's say that we want to find the square root of 900. At first glance, this looks very difficult! However, it's not hard if we separate 900 into its factors. Factors are the numbers that can multiply together to make another number. For instance, since you can make 6 by multiplying 1 × 6 and 2 × 3, the factors of 6 are 1, 2, 3, and 6. Instead of working with the number 900, which is somewhat awkward, let's instead write 900 as 9 × 100. Now, since 9, which is a perfect square, is separated from 100, we can take its square root on its own. √(9 × 100) = √(9) × √(100) = 3 × √(100). In other words, √(900) = 3√(100). We can even simplify this two steps further by dividing 100 into the factors 25 and 4. √(100) = √(25 × 4) = √(25) × √(4) = 5 × 2 = 10. So, we can say that √(900) = 3(10) = 30. Think — what number times itself equals -16? It's not 4 or -4 — squaring either of these gives positive 16. Give up? In fact, there isn't a way to write the square root of -16 or any other negative number with ordinary numbers. In these cases, we have to substitute imaginary numbers (usually in the form of letters or symbols) to take the place of the negative number's square root. For example, the variable "i" is usually used for the square root of -1. As a general rule, the square root of a negative number will always be an imaginary number (or include one). Note that although imaginary numbers can't be represented with ordinary digits, they can still be treated like ordinary numbers in many ways. For instance, the square roots of negative numbers can be squared to give those negative numbers, just like any other square root. For example, i2 = -1
What is a summary of what this article is about?
Square a number by multiplying it by itself. For square roots, find the "reverse" of a square. Know the difference between perfect and imperfect squares. Memorize the first 10-12 perfect squares. Simplify square roots by removing perfect squares when possible. Use imaginary numbers for the square roots of negative numbers.