We’re going to try to solve the example 5 to the 6th power divided by 5 to the 2nd power. I’m going to show you how this looks in expanded notation. That means I’m going to multiply my numerator out so that I have six 5’s so I am multiplying 5 six times and my denominator two times. So I have 5 times 5 times 5 times 5 times 5 times 5 on the top and 5 times 5 on the bottom. At this point we know how to divide these. These are treated just like typical fractions that you might divide or numbers that you might divide in fraction format and you’ll see that we have cancelled two 5’s on the top with two 5’s on the bottom. I am left with 5 times 5 times 5 times 5 on the top, which is the same as 5 to the 4th power and I’m left with 1 on the bottom. Whenever you divide a number by 1 you don’t even need to write it down; you can just ignore it or leave it off. But if you take a look and compare that to the original problem, do you see a pattern here? 5 to the 6th over 5 to the 2nd and we end up with 5 to the 4th as an answer.

Let’s try another example and we’ll write it out in expanded format and see if you can come up with what that pattern is. And the next example we’re going to try is 8 to the 9th power and we’re going to divide by 8 to the 3rd power. Now notice my bases are the same. My 8 is the base in the numerator and the denominator. So I can go ahead and divide these easily, but I’m going to write them out...1, 2, 3, 4, 5, 6, 7, 8, 9, and the reason I say we’re going to divide these easily is I’ll be able to cancel just like we did in that first example. My 8’s cancel one at a time. So for every one on the top I get to divide by one on the bottom. So it actually works out pretty nicely here. I’m left with 1, 2, 3, 4, 5, 6 on the top, and then, of course, because I cancelled I have a 1 on the bottom which is just 8 to the 6th power. So notice here I have 8 to the 9th divided by 8 to the 3rd and my answer is 8 to the 6th. Did you guess what that pattern is? When you divide numbers that have the same base with different exponents, the shortcut to that is that you can take your exponents and subtract them and you don’t have to write all of this in expanded format. You can just skip that step and jump right to using your property, and that property is something that you definitely want to put into your notebook or into the graphic organizer in class. It’s called “The Quotient of Powers Property.”