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What are Exponents?

Learn what exponents are and why there’s an exponent controversy!

By
Jason Marshall, PhD
May 16, 2011
Episode #062

In the last article, we talked about the rule of 72 and how you can use it to figure out how long it will take for the value of an investment to double. At the end of the article, I left you to ponder where the rule of 72 comes from and why it works. But to really understand the answers to these questions, we need to learn about something called compound interest. And to really understand compound interest, we need to learn about a fundamental piece of math called exponents. Which is precisely what we’re going to talk about today.

What Is Exponentiation?

So what is exponentiation? Well, one answer—which comes with a bit of controversy that we’ll talk about in a minute—is that exponentiation is repeated multiplication. In other words, just as we can do something called multiplication to repeatedly add a number to itself a bunch of times, we can also do something called exponentiation to repeatedly multiply a number by itself a bunch of times.

For example, we can find the answer to the problem 2 x 2 x 2 x 2 x 2—that is, multiplying 5 copies of the number 2 together—by finding what’s called the 5th power of 2. We’ll talk more about what all this lingo means in a minute, but for now all you need to focus on is the fact that exponentiation is the idea of multiplying a number by itself some number of times.

The Exponentiation Controversy

However, some people would contest this description of exponentiation—this is the controversy—and claim that while it is certainly true that exponentiation can be used as a convenient way to solve the problem of multiplying a number by itself some number of times, it is fundamentally something different.

You can get a flavor for why this is by thinking about what it means to raise a number to a fractional power and asking yourself how you can multiply a number by itself a fractional number of times? What does that even mean? In truth though, the details of this controversy aren’t really important for us to discuss right now, but I do think that it’s interesting to at least point out that it exists. For our purposes, we can just leave it at that and go about our merry lives learning how to deal with exponents!

What Are Bases and Exponents?

In order to properly talk about exponentiation, we need to understand the words used to describe it. First of all, when we talk about exponentiation we often say that we’re “raising a number to a power.” For example, in the problem 2 x 2 x 2 x 2 x 2 where we multiply 5 copies of the number 2 together, we say that we are raising the number 2 to the 5th power. The number 2 here is called the “base,” and the number 5 is called the “exponent” or “power.” The base tells you what number to multiply by itself, and the power tells you how many times.

To write this, you simply write the base number 2 followed by the exponent as a superscript, like this: 2^5 = 2 x 2 x 2 x 2 x 2. Sometimes you’ll see exponents on a computer written using a caret symbol, like this: 2^5…with the caret used to indicate that the number coming after it is an exponent. In fact, this is how you can enter exponents to do calculations right in the Google search box or on a site like WolframAlpha.

What Do Squared and Cubed Mean?

And that’s pretty much all there is to understanding the basics of exponents. But there are two more very commonly used words that you should know: “squared” and “cubed.” Here’s the quick and dirty tip: Squaring a number is multiplying two copies of itself together, and cubing a number is multiplying three copies of itself together. So “two squared” is written 2^2 = 2 x 2 = 4, and “two cubed” is written 2^3 = to 2 x 2 x 2 = 8. You might be wondering if there are any other fancy names like “squared” and “cubed” for raising numbers to powers besides 2 and 3? And, for that matter, you might be wondering where these names come from in the first place? I’m going to let you think about those questions on your own for a bit, and we’ll talk about the answers in next week’s article.

Handy Number of the Week

[[AdMiddle]As I mentioned in the podcast version of last week’s article, I’ve recently started a new daily feature on the Math Dude’s Facebook page where every weekday I feature a different number that’s either particularly interesting or useful to know in your daily life. And each week I’m also going to feature one of the five numbers from the previous week in the new Math Dude article.

So, without further ado, the inaugural number of the week is 12. But why 12? Well, it’s because that’s the approximate number of standard Manhattan city blocks that are in one kilometer. Which means that the next time you hear that something is such-and-such a number of kilometers away, you can use this number to get a good intuitive feeling for just how far away that something really is in terms of easy-to-picture city blocks! And just in case you’re wondering what this number is in terms of miles instead of kilometers, it turns out there are about 20 standard Manhattan city blocks in a mile.

Wrap Up

If you think those numbers are handy, be sure to become a fan of the Math Dude on Facebook where you’ll learn a new number that’s just as handy as these every single weekday. And if you’re on Twitter, please follow me there as well to get updates about the podcast, the numbers of the day, daily math puzzles, and lots of other great links to stories about math and science. Finally, remember to email any math questions that you may have to 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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