Learn how to divide numbers that have exponents.
Understanding how to work with exponents is incredibly important since they show up all the time in both the abstract world of math and in the real world of science and engineering. With this goal in mind, last week we learned how to multiply numbers with exponents, which naturally leads to this week’s complimentary topic: how to divide numbers with exponents.
Review: The Product Rule for Exponents
Before we talk about division, let’s quickly review how multiplying numbers with exponents works. As you’ll recall, a problem like 2^2 x 2^3 just says to multiply two copies of 2 with three additional copies of 2, like this: 2 x 2 x 2 x 2 x 2. Of course, that’s really just five copies of 2 multiplied together. In other words, it’s the same thing as 2^5. Which, if you think about it, naturally leads to the rule for multiplying exponents we learned last time—called the product rule—which states that:
2^m x 2^n = 2^m+n
This says that some base number (it could be anything, but in this case it’s 2) raised to some power (in this case we call it m) multiplied by another number with the same base raised to a different power (in this case represented by the letter n) is just equal to the base raised to the sum of the exponents. Easy enough!
What Happens When We Divide Numbers with Exponents?
But what about division with exponents? How does that work? Well, let’s start by looking at the problem 2^3 / 2. As we learned last time, any number raised to the first power is just equal to that number—the exponent of 1 doesn’t really do anything. Which means that 10^1 = 10, 5^1 = 5, 2^1 = 2, and so on. It also means that the problem 2^3 / 2 is the same as the problem 2^3 / 2^1. What does this look like if we expand out all those 2s? Well, it’s just (2 x 2 x 2) / 2. If we go ahead and multiply out the numerator, 2 x 2 x 2, we get 8. If we then divide this by the denominator, 2, we see that 2^3 / 2^1 = 8 / 2 = 4—which is also equal to 2^2.
Just for fun, let’s now take the answer to this problem, 2^2, and divide it by 2. So, what’s 2^2 / 2^1? Well, that’s just 4 / 2 = 2—which, of course, is the same thing as 2^1. And just because it’s so much fun, let’s try 2^3 / 2^2. Well, that’s just 8 / 4 = 2—aka, 2^1. Okay, but so what? Why are we doing this?
A Rule For Dividing Numbers With Exponents
To see why this is important, let’s summarize the three problems we’ve looked at:
2^3 / 2^1 = 2^2
2^2 / 2^1 = 2^1
2^3 / 2^2 = 2^1
Do you see a pattern here? Remember that for multiplication problems we added exponents. But, if you look closely at these three problems, you’ll see that with division we don’t need to add exponents, we need to subtract them. In other words, in the problem 2^3 / 2^1 = 2^2, all you have to do to find the exponent of the answer is subtract the exponent in the denominator (which is 1) from the exponent in the numerator (which is 3) to get 3 – 1 = 2. And, as you can check, the other problems work the same way. In truth, this isn’t really surprising since dividing by a base number has the effect of canceling out one of the base numbers in the numerator. Or, equivalently, of subtracting 1 from the exponent in the numerator.
Just as we did last week with the product rule, we can write a rule to summarize how division with exponents works. This rule, called the quotient rule for exponents, says that:
2^m / 2^n = 2^m-n
The letters m and n here are placeholders that can represent any number. In one of our earlier problems, m was 3, n was 1, and the division problem 2^3 / 2^1 was equal to 2^(3-1) = 2^2. Of course, the base number doesn’t have to be 2—the rule works for any other base number as well.
Warning: The Bases Must Be the Same!
As with last week’s product rule, there’s one word of warning I’d like to emphasize about the quotient rule. And that is that the bases of the numbers you’re dividing must be the same in order for you to subtract the exponents. If they’re not, you can’t solve the problem using the quotient rule!
What Do Negative Exponents Mean? And Zero?
If you’re really on the ball, you may have noticed something about the quotient rule…something that’s a little concerning. While using the quotient rule to solve a problem like 2^5 / 2^3 = 2^(5-3) = 2^2 makes perfect sense, the answer you get for a problem in which the exponent in the numerator is less than the exponent in the denominator is a little strange. For example, if we use the quotient rule to solve the problem 2^3 / 2^5, we get the answer 2^(3-5) = 2^-2. A negative exponent…does that even make sense? Is the quotient rule broken? No, the quotient rule is not broken, negative exponents (and even an exponent of zero) make perfect sense, and everything is just fine…as we’ll talk about next time.
But before we finish up for today, here are a few practice problems to help you get comfortable with dividing numbers that have exponents. Your goal is to use the quotient rule to simplify the following problems:
5^5 / 5^2 = … ?
4^2 x 3^3 / 4^2 = … ?
2^6 / 4^2 = … ?
You can find answers and explanation in my post this week on The Quick and Dirty blog.
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 firstname.lastname@example.org.
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!