Topic A: Exponential notation and properties of integer exponents
In this video, I want to do a bunch of examples involving exponent properties. But, before I even do that, let's have a little bit of a review of what an exponent even is. So let's say I had 2 to the third power. You might be tempted to say, oh is that 6? And I would say no, it is not 6. This means 2 times itself, three times. So this is going to be equal to 2 times 2 times 2, which is equal to 2 times 2 is 4. 4 times 2 is equal to 8. If I were to ask you what 3 to the second power is, or 3 squared, this is equal to 3 times itself two times. This is equal to 3 times 3. Which is equal to 9. Let's do one more of these. I think you're getting the general sense, if you've never seen these before. Let's say I have 5 to the seventh power. That's equal to 5 times itself, seven times. 5 times 5 times 5 times 5 times 5 times 5 times 5. That's seven, right? One, two, three, four, five, six, seven. This is going to be a really, really, really, really, large number and I'm not going to calculate it right now. If you want to do it by hand, feel free to do so. Or use a calculator, but this is a really, really, really, large number. So one thing that you might appreciate very quickly is that exponents increase very rapidly. 5 to the 17th would be even a way, way more massive number. But anyway, that's a review of exponents. Let's get a little bit steeped in algebra, using exponents. So what would 3x-- let me do this in a different color-- what would 3x times 3x times 3x be? Well, one thing you need to remember about multiplication is, it doesn't matter what order you do the multiplication in. So this is going to be the same thing as 3 times 3 times 3 times x times x times x. And just based on what we reviewed just here, that part right there, 3 times 3, three times, that's 3 to the third power. And this right here, x times itself three times. that's x to the third power. So this whole thing can be rewritten as 3 to the third times x to the third. Or if you know what 3 to the third is, this is 9 times 3, which is 27. This is 27 x to the third power. Now you might have said, hey, wasn't 3x times 3x times 3x. Wasn't that 3x to the third power? Right? You're multiplying 3x times itself three times. And I would say, yes it is. So this, right here, you could interpret that as 3x to the third power. And just like that, we stumbled on one of our exponent properties. Notice this. When I have something times something, and the whole thing is to the third power, that equals each of those things to the third power times each other. So 3x to the third is the same thing is 3 to the third times x to the third, which is 27 to the third power. Let's do a couple more examples. What if I were to ask you what 6 to the third times 6 to the sixth power is? And this is going to be a really huge number, but I want to write it as a power of 6. Let me write the 6 to the sixth in a different color. 6 to the third times 6 to the sixth power, what is this going to be equal to? Well, 6 to the third, we know that's 6 times itself three times. So it's 6 times 6 times 6. And then that's going to be times-- the times here is in green, so I'll do it in green. Maybe I'll make both of them in orange. That is going to be times 6 to the sixth power. Well, what's 6 to the sixth power? That's 6 times itself six times. So, it's 6 times 6 times 6 times 6 times 6. Then you get one more, times 6. So what is this whole number going to be? Well, this whole thing-- we're multiplying 6 times itself-- how many times? One, two, three, four, five, six, seven, eight, nine times, right? Three times here and then another six times here. So we're multiplying 6 times itself nine times. 3 plus 6. So this is equal to 6 to the 3 plus 6 power or 6 to the ninth power. And just like that, we/ve stumbled on another exponent property. When we take exponents, in this case, 6 to the third, the number 6 is the base. We're taking the base to the exponent of 3. When you have the same base, and you're multiplying two exponents with the same base, you can add the exponents. Let me do several more examples of this. Let's do it in magenta. Let's say I had 2 squared times 2 to the fourth times 2 to the sixth. Well, I have the same base in all of these, so I can add the exponents. This is going to be equal to 2 to the 2 plus 4 plus 6, which is equal to 2 to the 12th power. And hopefully that makes sense, because this is going to be 2 times itself two times, 2 times itself four times, 2 times itself six times. When you multiply them all out, it's going to be 2 times itself, 12 times or 2 to the 12th power. Let's do it in a little bit more abstract way, using some variables, but it's the same exact idea. What is x to the squared or x squared times x to the fourth? Well, we could use the property we just learned. We have the exact same base, x. So it's going to be x to the 2 plus 4 power. It's going to be x to the sixth power. And if you don't believe me, what is x squared? x squared is equal to x times x. And if you were going to multiply that times x to the fourth, you're multiplying it by x times itself four times. x times x times x times x. So how many times are you now multiplying x by itself? Well, one, two, three, four, five, six times. x to the sixth power. Let's do another one of these. The more examples you see, I figure, the better. So let's do the other property, just to mix and match it. Let's say I have a to the third to the fourth power. So I'll tell you the property here, and I'll show you why it makes sense. When you add something to an exponent, and then you raise that to an exponent, you can multiply the exponents. So this is going to be a to the 3 times 4 power or a to the 12th power. And why does that make sense? Well this right here is a to the third times itself four times. So this is equal to a to the third times a to the third times a to the third times a to the third. Well, we have the same base, so we can add the exponents. So there's going to be a to the 3 times 4, right? This is equal to a to the 3 plus 3 plus 3 plus 3 power, which is the same thing is a the 3 times 4 power or a to the 12th power. So just to review the properties we've learned so far in this video, besides just a review of what an exponent is, if I have x to the a power times x to the b power, this is going to be equal to x to the a plus b power. We saw that right here. x squared times x to the fourth is equal to x to the sixth, 2 plus 4. We also saw that if I have x times y to the a power, this is the same thing is x to the a power times y to the a power. We saw that early on in this video. We saw that over here. 3x to the third is the same thing as 3 to the third times x to the third. That's what this is saying right here. 3x to the third is the same thing is 3 to the third times x to the third. And then the last property, which we just stumbled upon is, if you have x to the a and then you raise that to the bth power, that's equal to x to the a times b. And we saw that right there. a to the third and then raise that to the fourth power is the same thing is a to the 3 times 4 or a to the 12th power. So let's use these properties to do a handful of more complex problems. Let's say we have 2xy squared times negative x squared y squared times three x squared y squared. And we wanted to simplify this. This you can view as negative 1 times x squared times y squared. So if we take this whole thing to the squared power, this is like raising each of these to the second power. So this part right here could be simplified as negative 1 squared times x squared squared, times y squared. And then if we were to simplify that, negative 1 squared is just 1, x squared squared-- remember you can just multiply the exponents-- so that's going to be x to the fourth y squared. That's what this middle part simplifies to. And let's see if we can merge it with the other parts. The other parts, just to remember, were 2 xy squared, and then 3x squared y squared. Well now we're just going ahead and just straight up multiplying everything. And we learned in multiplication that it doesn't matter which order you multiply things in. So I can just rearrange. We're just going and multiplying 2 times x times y squared times x to the fourth times y squared times 3 times x squared times y squared. So I can rearrange this, and I will rearrange it so that it's in a way that's easy to simplify. So I can multiply 2 times 3, and then I can worry about the x terms. Let me do it in this color. Then I have times x times x to the fourth times x squared. And then I have to worry about the y terms, times y squared times another y squared times another y squared. And now what are these equal to? Well, 2 times 3. You knew how to do that. That's equal to 6. And what is x times x to the fourth times x squared. Well, one thing to remember is x is the same thing as x to the first power. Anything to the first power is just that number. So you know, 2 to the first power is just 2. 3 to the first power is just 3. So what is this going to be equal to? This is going to be equal to-- we have the same base, x. We can add the exponents, x to the 1 plus 4 plus 2 power, and I'll add it in the next step. And then on the y's, this is times y to the 2 plus 2 plus 2 power. And what does that give us? That gives us 6 x to the seventh power, y to the sixth power. And I'll just leave you with some thing that you might already know, but it's pretty interesting. And that's the question of what happens when you take something to the zeroth power? So if I say 7 to the zeroth power, What does that equal? And I'll tell you right now-- and this might seem very counterintuitive-- this is equal to 1, or 1 to the zeroth power is also equal to 1. Anything that the zeroth power, any non-zero number to the zero power is going to be equal to 1. And just to give you a little bit of intuition on why that is. Think about it this way. 3 to the first power-- let me write the powers-- 3 to the first, second, third. We'll just do it the with the number 3. So 3 to the first power is 3. I think that makes sense. 3 to the second power is 9. 3 to the third power is 27. And of course, we're trying to figure out what should 3 to the zeroth power be? Well, think about it. Every time you decrement the exponent. Every time you take the exponent down by 1, you are dividing by 3. To go from 27 to 9, you divide by 3. To go from 9 to 3, you divide by 3. So to go from this exponent to that exponent, maybe we should divide by 3 again. And that's why, anything to the zeroth power, in this case, 3 to the zeroth power is 1. See you in the next video.