how do you do division exponents with positive exponents

Okay, that’s all the math we have time for today. You can find answers and explanation in my post this week on The Quick and Dirty blog. Your goal is to use the quotient rule to simplify the following problems: Practice Problemsīut before we finish up for today, here are a few practice problems to help you get comfortable with dividing numbers that have exponents. 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. 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. 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. If you’re really on the ball, you may have noticed something about the quotient rule…something that’s a little concerning.

If they’re not, you can’t solve the problem using the quotient rule! What Do Negative Exponents Mean? And Zero? And that is that the bases of the numbers you’re dividing must be the same in order for you to subtract the exponents. Warning: The Bases Must Be the Same!Īs with last week’s product rule, there’s one word of warning I’d like to emphasize about the quotient rule. Of course, the base number doesn’t have to be 2-the rule works for any other base number as well. 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. The letters m and n here are placeholders that can represent any number. This rule, called the quotient rule for exponents, says that: Just as we did last week with the product rule, we can write a rule to summarize how division with exponents works. Or, equivalently, of subtracting 1 from the exponent in the numerator. 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. And, as you can check, the other problems work the same way. 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. 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. To see why this is important, let’s summarize the three problems we’ve looked at:ĭo you see a pattern here? Remember that for multiplication problems we added exponents. Okay, but so what? Why are we doing this? A Rule For Dividing Numbers With Exponents

And just because it’s so much fun, let’s try 2^3 / 2^2. So, what’s 2^2 / 2^1? Well, that’s just 4 / 2 = 2-which, of course, is the same thing as 2^1. Just for fun, let’s now take the answer to this problem, 2^2, and divide it by 2. 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. If we go ahead and multiply out the numerator, 2 x 2 x 2, we get 8.

What does this look like if we expand out all those 2s? Well, it’s just (2 x 2 x 2) / 2. It also means that the problem 2^3 / 2 is the same as the problem 2^3 / 2^1. 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. Easy enough! What Happens When We Divide Numbers with Exponents?īut what about division with exponents? How does that work? Well, let’s start by looking at the problem 2^3 / 2.

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.

How do you do division exponents with positive exponents