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What are Square Roots?

Learn what square roots are, how they’re related to perfect squares, and what pitfalls to watch out for when calculating them. Then try your hand at a few practice problems to make sure you understand everything.

By
Jason Marshall, PhD,
Episode #103

What are Square Roots?

by Jason Marshall

We’ve spent a lot of time recently investigating the very cool properties of the numbers known as perfect squares. Perfect squares, you’ll remember, are all the numbers you get when you multiply some number by itself—as in 1 (since 1^2 = 1), 4 (since 2^2 = 4), 9 (since 3^2 = 9), and so on. I’m sure you’ll agree with me that those were some pretty good times we had with perfect squares, and I’ve got great news because those good times aren’t over yet. In fact, I’ve got another question for us to think about that’s related to perfect squares. Namely, if I give you a perfect square, can you figure out what number was squared to get that perfect square? By the end of this article, you will. And you’ll also be prepared to deal with some of the more common obstacles that you might encounter along the way.

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What are Square Roots?

Imagine I give you a number and tell you that it’s a perfect square. For example, let’s say I give you the perfect square 49. We already know that since 49 is a perfect square, there must be some number that when multiplied by itself gives 49. This is a pretty simple example, so you probably already know the answer. But the real question that we need to think about is how can you figure out what this unknown number is for any perfect square?

As it turns out, the answer to this question is pretty simple…and it’s probably something you’re already familiar with. We can find the unknown number we’re looking for by taking what’s called the squareroot of the perfect square. Using our example, the square root of the perfect square 49 is written using the very impressive looking mathematical symbol that looks kind of like a “check-mark” known as a radical (obviously because it’s pretty radical). As in: √49. (Note: we usually draw a line over the “49” to show that the whole thing is included within the radical; but unfortunately that’s really hard to do on the web so we’ll go with this). As we learned long ago when we first memorized the multiplication table, 7^2 = 49. Which means that √49 = 7. Pretty easy, right? Sure, but is that really all there is to it?

Can Square Roots Be Negative?

If you’re really paying attention, you might be thinking: Isn’t it also true that (–7)^2 = –7 x –7 = 49? And doesn’t that mean that –7 is an equally valid answer? In other words, shouldn’t we actually say that the square root of 49 is equal to either 7 or –7? My answer is: Not really. Although you’ll often see people make the claim that the square root of a number like 49 can be equal to –7 as well as 7, I think it’s less confusing to think of square roots in terms of what’s more technically known as principal square roots—which are just the positive square roots we’ve been talking about.

This discrepancy stems from the fact that in an equation like x^2 = 49, the variable x is a solution if it is equal to either the square root of 49 (which is 7) or –1 times the square root of 49 (which is –7). As you can check by plugging in these two values for x, either one will make the left and right sides of the equation the same. But saying that this equation has two possible values of x that make it true is not the same thing as saying that the square root of 49 itself is equal to two different numbers. So while an equation like this may have multiple solutions, it’s best to think of the square root of a number as the single positive number that, for future reference, is also known as the principal square root.

How to Calculate Square Roots

Your best bet for calculating square roots is usually to use a calculator or computer.

Which means that the square root of 49 is just equal to 7. That was pretty easy to figure out, right? Perhaps you’re thinking that square roots aren’t so tough to calculate after all. Well, it’s true that it’s not too difficult to find the square root of a perfect square. But don’t get too excited because life isn’t always so simple. In particular, it’s not nearly as easy to find the square root of non-perfect squares. For example, what’s the square root of 60? That’s a lot harder to figure out because there is no whole number that you can multiply by itself to get 60. So what can we do?

We’ll talk next time about some quick and dirty techniques that you can use to estimate square roots (since it’s always nice to have tricks like that in your tool belt), but the truth is that your best bet for calculating square roots—especially when high precision is needed—is usually to use a calculator or computer. After all, methods for calculating square roots by hand were developed hundreds of years ago before machines that could do the job existed. But now those machines do exist, and they’re a lot faster and much more accurate than you are. So doesn’t it make sense to use them? After all, we’ve got better things to do with our brains!

Do Negative Numbers Have Square Roots?

Before we finish up, and before you get so excited that you run off to calculate square roots, I should warn you that not every number has one. Which are those? Well, let’s think about it. When you square a positive number, the answer is always a positive number. And when you square a negative number, the answer is once again always a positive number. Which means that the square of any number is always a positive number. So, since there are no numbers that we can square to get back a negative number (at least none that we know of yet), we can conclude that negative numbers don’t have square roots!

Practice Problems

With that, it’s now time to test your square root taking skills. Try to solve 1/3 of these problems exactly, another 1/3 of them approximately using a calculator, and, finally, don’t solve 1/3 of them at all! You can choose which method you want to use for each problem, but choose wisely!

  1. √16 = ____ ?

  2. √–1 = ____ ?

  3. √π = ____ ?

  4. √81 = ____ ?

  5. √–42 = ____ ?

  6. The square root of 55 = ____ ?

You can find the answers to these practice problems next week on The Math Dude’s Facebook page.

Wrap Up

Okay, that’s all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new featured number or math puzzle posted every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

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