Learn what negative exponents and an exponent of zero mean. Try your new skills with a few patented Math Dude practice problems.

As we’ve been learning, exponents are used all the time in both the abstract world of math that you see in subjects like algebra and in the real world of problems that you see in science, engineering, finance, and lots of other areas too. So far we’ve learned how to multiply numbers with exponents and how to divide numbers with exponents. But for each of these cases we’ve only dealt with exponents that are positive integers. However the world of math is more complex—it’s full of negative integer exponents and even exponents of zero! What do these all mean? Keep on reading to find out.

## Review: The Quotient Rule for Exponents

To figure out what negative exponents mean, we need to start with the quotient rule for exponents that we learned earlier. As you’ll recall, a problem like 2^3 / 2^2 just says to divide three copies of 2 (the numerator) by two copies of 2 (the denominator)—as in (2 x 2 x 2) / (2 x 2). Since every copy of 2 in the denominator basically “cancels out” a copy of 2 in the numerator, 2^3 / 2^2 is equal to 2^1…or just 2. As we learned last time, we can generalize this result into what’s called the quotient rule for exponents:

2^m / 2^n = 2^(m–n)

In other words, some base number (here it’s 2) raised to some power (which we’re calling m) divided by another number with the same base raised to a different power (which we’re calling n) is equal to the base number raised to the difference of the powers. For example, in the problem 2^3 / 2^2, m is equal to 3 and n is equal to 2. So 2^3 / 2^2 = 2^(3–2) = 2^1…or just 2—exactly as we found before.

## What Happens When We Keep Dividing Numbers with Exponents?

But this problem, and all of the other problems we’ve looked at so far, have all dealt exclusively with exponents that are positive integers—1, 2, 3, and so on. According to the quotient rule, as long as m (the exponent in the numerator) is greater than n (the exponent in the denominator), the answer will always have an exponent that’s a positive number (since m–n must be a positive number if m is greater than n). But what if the exponent m is less than n? For example, what about 2^2 / 2^3? If we use the quotient rule, we find that this must be equal to 2^(2–3) = 2^–1! What does that mean?

To see, let’s start with the problem 2^2 / 2. As we know, this is the same as the problem 2^2 / 2^1 which the quotient rule tells us is equal to 2^(2–1) = 2^1…or just 2. Now, what if we divide this number by 2? Well, that’s really easy since 2 / 2 is just equal to 1. But let’s also look at this in terms of a problem with exponents. Remember, we started with the problem 2^2 / 2 (which we found is equal to 2), and then we divided this by 2. That’s really the same thing as the problem 2^2 / 2^2. Using the quotient rule here, we find that 2^2 / 2^2 is equal to 2^(2–2) = 2^0. But we just saw that the answer to this problem has to be 1, so what’s going on here?

## What Does an Exponent of Zero Mean?

What we’ve discovered is that any number raised to the 0th power must be equal to 1. Well, perhaps not quite any number: It’s debatable whether or not to say that 0^0 is equal to 1. You’ll often see 0^0 called “indeterminate” or “undefined”. Although this is an interesting point, it’s not one that we need to worry about for now. The important thing to remember is that 2^0, –2^0, 1000^0, or anything else (except perhaps 0) raised to the 0th power is always equal to 1. And it’s helpful to remember that you can always figure this out just by looking at the quotient rule for exponents.

## What Do Negative Exponents Mean?

But we don’t have to stop with an exponent of zero. In fact, let’s continue with the original problem of dividing 2^2 by larger an larger powers of 2 and see what we can learn. So far we’ve seen that 2^2 divided by two powers of 2—that is 2^2 / 2^2—is equal to 1. What happens if we now divide this answer of 1 by 2? Of course, 1 divided by 2 is just the fraction 1/2. And if we look at this problem in terms of exponents, we see that this is the answer to the problem 2^2 / 2^3. If we use the quotient rule, we see that this is equal to 2^(2–3) = 2^–1…a negative exponent! So 2^–1 is equal to the fraction 1/2.

I’ll let you work out the details yourself, but if we once again divide by 2, you’ll find that 2^–2 is equal to the fraction 1/(2^2)…aka, 1/4. What does this all mean? Well these examples show us that any number raised to a negative integer power is equal to the reciprocal of that number raised to the absolute value of that power. In other words, the negative part of an exponent tells you that the number is actually a fraction with a 1 in its numerator and the base number (possibly raised to a power) in its denominator. For example, 5^–2 is equal to 1/(5^2) = 1/25 and 2^–3 = 1 / (2^3) = 1/8.

## Practice Problems

Before we finish up for today, here are a few practice problems to help you get more comfortable with negative exponents. Your goal is to use everything we’ve learned so far to simplify these problems:

1. 5^5 / (5^2 x 5^4) = … ?

2. 4^-2 x 3^3 / 4^-2 = … ?

3. 2^–6 x 2^2 = … ?

You can find answers and explanations in my post this week on The Quick and Dirty.

## 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!