 In this video, I'm going to talk about the properties of exponents. I'm actually going to do about five examples of the different properties of exponents just to give because with exponents There's a lot of different variability that we can have so I'm just going to go through a few examples This should be pretty quick This should be a pretty quick video But hopefully I'll be able to show you enough examples so we have a better understanding of what's going on Okay, so my first example something pretty basic What if I have three to the negative second okay now if the if I have a product if this is part of a problem I might be asked to simplify this I might be asked to evaluate this something to that effect Okay, so what I the first thing that I notice is that this is a negative exponent remember negative exponents make fractions Negative make exponents make fractions So what I'm going to do is I want to make a fraction out of this So what I'm going to do is I'm going to change this position and put it on the bottom of the fraction It's right now is on top of the fraction if you need to put a bar right there to see that it's on top Just take this and put it to the bottom when you do that we go from a negative two exponent to a positive two x point Okay, and then from there once I have changed the position once I have made the exponent positive I now need to evaluate this One on top three squared on bottom is nine. Okay, so there's a simple example Just with negative exponents. Okay, so let's do another quick example What if I have a fraction? Fractions can sometimes be a little bit a little bit more difficult There's a couple of different ways to look at this one. I will try to show you the simplest way So what I'm going to do since I see this negative exponent This negative exponent is being applied to both of these numbers So that's what I'm going to do I'm going to take this negative exponent and apply it to the two negative two and apply it to the three So it looks a little bit like this two to the negative two over three to the negative two Okay, now this problem is a little bit confusing because notice now we have a negative exponent in the bottom of the fraction So what I'm going to do is just like the top example I'm going to change its position so that We have positive exponents So on this first example up here on this first example up here It was on top and I brought it to the bottom which was fine But now here some are on top some are on bottom. So all I need to do now is just simply change its position So this is two to the negative second. It's on top. I'm going to take it to the bottom This three here three the negative second. It's on the bottom. I'm going to change its position and bring it to the top Okay, once I change its position the exponents actually become positive And once I change position the exponents become positive now after I have now that I have positive exponents now I can actually evaluate the Exponent this will be three squared is nine two squared is four. So that would be simply nine fourths Okay, that's one way to look at it another way to look at it if I can do this rather quickly You'd also think of it this way if I take the fraction and flip it initially if I take the fraction and just flip it initially That actually will change the exponent outside to be go from a negative two to a positive two This is another way to do this and then all you simply need to do is take this Exponent of two and apply it to the numbers inside. So three squared is nine and two squared is four That's another way to look at it. That's one of the really quickly go over that Okay, so those are two examples of Using exponents. Let's go through a couple more Here's another one. This is kind of long-winded one three z to the seventh times negative four Z to the second Okay, now I got numbers. I got variables. I got exponents. I got all sorts of stuff in here My my advice to you is just look at one thing at a time So the first thing I want to look at is I'm gonna look at the three and a negative four Those are numbers. These are constants. They're just numbers. So actually I'm gonna figure that out first So three times negative four is just going to be a negative 12 Okay, so I've evaluated the number portion of this first Okay Now what I have here is z to the seventh and z squared these z's are multiplying times one another so go look at your Property your power first you need your product property if I multiplying like bases I need to add the exponents multiplying light bases add the exponents So this is going to be z to the ninth power seven plus two gives me nine Okay, so that's what that would simplify to again when you look at something like this don't don't get Overwhelmed by everything that's going on just take one thing at a time All right, especially for this next one if I have y z to the third over Z to the fifth quantity third Okay, so this is a different type of example I get a lot of stuff going on But again take one thing at a time and you'll do just fine So as I look at this one thing that I noticed is that inside the parentheses I see a z on bottom and a z on top I'm going to evaluate that first and then I'm going to worry about the three exponent outside first are next Okay, so why is I only see one why so that doesn't is an effect and I see Z on top See how about I see three Z's on top five Z's on bottom So if I'm dividing like bases subtract the exponents, this is going to be y to the This is one way to explain now. I'm actually going to change this up here in just a second Okay, so if I take three minus five I get a negative two But there's kind of another way to explain this it might be a little bit easier I like to think of it this way. I have three Z's on top five Z's on bottom They're going to cancel so that I only have two left on the bottom Now notice this is z to the negative second if I have a negative exponent what I need to do with this is take this And put that on the bottom So notice z to the third z to the fifth three on top five on bottom They're going to cancel so I have two left on the bottom I think that's kind of a simpler way to look at it Yes, you can have Z to the negative second in there, but I would prefer that we have this way All right now that we've evaluated with the Z now let's take a look at the three I'm going to take that three and apply it to everything inside y to the third Z squared to the third Z squared to the third Squared to the third this is a power of a power Property that I'm going to use here if I'm taking the power of a power power of a power. I need to multiply The exponents so that two and three I actually multiply together to get six and that was a little bit tough there's a lot of properties that are intermixed into Into a problem like that. There's a lot of different ways to do it The only the only real good way of understanding these problems is to practice them over and over again Practice and a little bit of experience will get you to understand these a little bit better. Okay last example That I want to go with is having to use scientific notation 4.5 times 10 to the negative fifth over 1.5 times 10 to the sixth okay now in mathematics One place that we that is that this is that this is used in the real world It's an astronomy or even chemistry of where you use very very large numbers in astronomy You use very very large numbers light years Distance of the moon is from the earth. These are very very large numbers And chemistry you use very very small numbers the size of an atom Things like that. So what I'm going to show you just real quick using scientific notation using property of exponents We evaluate something like this Okay, so now this looks kind of confusing and you might try to plug this into a calculator But it's really very messy if you try to do that. So we're going to evaluate this using our property of exponents Now this right here 4.5 over 1.5 I can actually divide that and then these over here notice base of 10 base of 10 This is just going to be like the other problems that we've been working with okay But first I'm going to look at 4.5 divided by 1.5. Actually I'm going to look at that as 45 and 15 15 goes into 45 three times if I put a decimal here decimal here 1.5 goes into 45 three times Okay, so 4.5 divided by 1.5 is actually just three simply enough Okay, so now I'm going to look at this part here And I'm going to use the same rules For exponents if I have a base of 10 and a base of 10 if I'm dividing if I'm dividing like bases I need to subtract the exponents negative 5 Minus 6 so this is times 10 to be negative 11 negative 5 minus 6 got me that negative 11 Okay, so there's just an example one example using scientific notation It's not really that difficult to use scientific notation But it does help just to have an example so that you've seen one before and those are five about five examples For all the properties of exponents