Just as a reminder, so far we’ve been dividing numbers, and we discussed in the last video that when you take a number and divide it by itself you cancel and are left with ones on the top and the bottom and 1 divided by 1 is simply 1. So any number divided by itself is 1. If you take a number with an exponent: let’s say we have an 8 to the 6th power and divide it by 8 to the 6th power that is absolutely the same thing. You’re taking your numerator and you’re dividing it by itself, which is the same number in the denominator. And in expanded notations, what this looks like is each of your digits cancelling out and dividing into each other and leaving you with 1 over 1 which is the same as 1.

If we apply our properties’ rule, which we learned a little bit ago, we can take our exponents here and when we’re dividing bases you can subtract your exponents. So we’re dividing 8 to the 6th by 8 to the 6th so we can subtract our exponents. So now we have 8 to the 6 minus 6 which 8 to the zero power. So in this case 8 to the zero power is the same as 1.

If you take a look at another example: Let’s say we just pick a random number here. 12 to the 5th power divided by 12 to the 5th power. 5 minus 5 in this case is zero so I have 12 to the zero power, and if I wanted to, I could write this in expanded form just to prove that this is indeed going to cancel out the way it’s supposed to, and when it does you’re left with 1 over 1. So if I translate that into another property, I now can make the connection that 12 to the 5th divided by 12 to the 5th is the same as the 12 to the 5 minus 5, which is 12 to the zero, and therefore 12 to the zero is going to equal 1. That is another property and that is called the “Zero Property.”