Video Transcript: Exponent Properties

Exponent properties with products 

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. 

Exponent properties with parentheses 

And now I want to go over some of the other core exponent properties. But they really just fall  out of what we already know about exponents. Let's say I have two numbers, a and b. And I'm  going to raise it to-- I could do it in the abstract. I could raise it to the c power. But I'll do it a  little bit more concrete. Let's raise it to the fourth power. What is that going to be equal to?  Well that's going to be equal to-- I could write it like this. Copy and paste this, copy and paste.  That's going to be equal to ab times ab times ab times ab times ab. But what is that equal to?  Well when you just multiply a bunch of numbers like this it doesn't matter what order you're  going to multiply it in. This right over here is going to be equivalent to a times a times a times a  times-- We have four b's as well that we're multiplying together. Times b times b times b times  b. And what is that equal to? Well this right over here is a to the fourth power. And this right  over here is b to the fourth power. And so you see, if you take the product of two numbers and  you raise them to some exponent, that's equivalent to taking each of the numbers to that  exponent. And then taking their product. And here I just used the example with 4, but you  could do this really with any arbitrary-- actually any exponent. This property holds. And you  could satisfy yourself by trying different values, and using the same logic right over here. But 

this is a general property. That-- let me write it this way-- that if I have a to the b, to the c  power, that this is going to be equal to a to the c times b to the c power. And we'll use this to  throughout actually mathematics, when we try to simplify things or rewrite an expression in a  different way. Now let me introduce you to another core idea here. And this is the idea of  raising something to some power. And I'll just use example of 3. And then raising that to some  power. What could this be simplified as? Well let's think about it. This is the same thing as a to  the third-- let me copy and paste that-- as a to the third times a to the third. And what is a to  the third times-- So this is equal to a to the third times a to the third. And that's going to be  equal to a to the 3 plus 3 power. We have the same base, so we would add and they're being  multiplied. They're being raised to these two exponents. So it's going to be the sum of the  exponents, which of course is going to be equal to a-- that's a different color a-- it's going to be  a to the sixth power. So what just happened over here? Well I took two a to the thirds. And I  multiplied them together. So I took these two 3s and added them together. So this essentially  right over here, you could view this as 2 times 3. That's how we got the 6. When I raise  something to one exponent, and then raised it to another, that's the equivalent to raising the  base to the product of those two exponents. I just did it with this example right over here. But I  encourage you try other numbers to see how this works. And I could to do this in general. I  could say a to the b power. And then-- let me copy and paste that-- and then I'm going to raise  that to the c power. Well what is that going to give me? Well I'm essentially going to have to  take c of these, so one, two, three. I don't know how large of a number c is, so I'll just do the  dot, dot dot. So dot, dot, dot. I have c of these, right over here. So what is that going to be  equal to? Well that is going to be equal to a to the-- well for each of these c, I'm going to have a  b that I'm going to add together. So let me write this. So I'm going to have a b plus b plus b plus  dot, dot, dot plus b. And now I have c of these b's, so I have c b's right over here. Or you could  view this as a, this is equal to a to the c times b power. c or a, you could do a to the cb power. So very useful. So if someone were to say what is 35 to the third power, and then that raised to the  seventh power? Well this is going to obviously be a huge number. But we could at least simplify  the expression. This is going to be equal to 35 to the product of these two exponents. It's going  to be 35 to the 3 times 7, or 35 to the 21, or to the 21st power. 

Exponent properties with quotients 

Let's do some exponent examples that involve division. Let's say I were to ask you what 5 to the sixth power divided by 5 to the second power is? Well, we can just go to the basic definition of  what an exponent represents and say 5 to the sixth power, that's going to be 5 times 5 times 5  times 5 times 5-- one more 5-- times 5. 5 times itself six times. And 5 squared, that's just 5 times itself two times, so it's just going to be 5 times 5. Well, we know how to simplify a fraction or a  rational expression like this. We can divide the numerator and the denominator by one 5, and  then these will cancel out, and then we can do it by another 5, or this 5 and this 5 will cancel  out. And what are we going to be left with? 5 times 5 times 5 times 5 over 1, or you could say  that this is just 5 to the fourth power. Now, notice what happens. Essentially we started with six in the numerator, six 5's multiplied by themselves in the numerator, and then we subtracted  out. We were able to cancel out the 2 in the denominator. So this really was equal to 5 to the  sixth power minus 2. So we were able to subtract the exponent in the denominator from the 

exponent in the numerator. Let's remember how this relates to multiplication. If I had 5 to the-- let me do this in a different color. 5 to the sixth times 5 to the second power, we saw in the last  video that this is equal to 5 to the 6 plus-- I'm trying to make it color coded for you-- 6 plus 2  power. Now, we see a new property. And in the next video, we're going see that these aren't  really different properties. They're really kind of same sides of the same coin when we learn  about negative exponents. But now in this video, we just saw that 5 to the sixth power divided  by 5 to the second power-- let me do it in a different color-- is going to be equal to 5 to the-- it's  time consuming to make it color coded for you-- 6 minus 2 power or 5 to the fourth power.  Here it's going to be 5 to the eighth. So when you multiply exponents with the same base, you  add the exponents. When you divide with the same base, you subtract the denominator  exponent from the numerator exponent. Let's do a bunch more of these examples right here.  What is 6 to the seventh power divided by 6 to the third power? Well, once again, we can just  use this property. This going to be 6 to the 7 minus 3 power, which is equal to 6 to the fourth  power. And you can multiply it out this way like we did in the first problem and verify that it  indeed will be 6 to the fourth power. Now let's try something interesting. This will be a good  segue into the next video. Let's say we have 3 to the fourth power divided by 3 to the tenth  power. Well, if we just go from basic principles, this would be 3 times 3 times 3 times 3, all of  that over 3 times 3-- we're going to have ten of these-- 3 times 3 times 3 times 3 times 3 times 3. How many is that? One, two, three, four, five, six, seven, eight, nine, ten. Well, if we do what we did in the last video, this 3 cancels with that 3. Those 3's cancel. Those 3's cancel. Those 3's  cancel. And we're left with 1 over-- one, two, three, four, five, six 3's. So 1 over 3 to the sixth  power, right? We have 1 over all of these 3's down here. But that property that I just told you,  would have told you that this should also be equal to 3 to the 4 minus 10 power. Well. What's 4  minus 10? Well, you're going to get a negative number. This is 3 to the negative sixth power. So  using the property we just saw, you'd get 3 to the negative sixth power. Just multiplying them  out, you get 1 over 3 to the sixth power. And the fun part about all of this is these are the same  quantity. So now you're learning a little bit about what it means to take a negative exponent. 3  to the negative sixth power is equal to 1 over 3 to the sixth power. And I'm going do many,  many more examples of this in the next video. But if you take anything to the negative power,  so a to the negative b power is equal to 1 over a to the b. That's one thing that we just  established just now. And earlier in this video, we saw that if I have a to the b over a to the c,  that this is equal to a to the b minus c. That's the other property we've been using. Now, using  what we've just learned and what we learned in the last video, let's do some more complicated  problems. Let's say I have a to the third, b to the fourth power over a squared b, and all of that  to the third power. Well, we can use the property we just learned to simplify the inside. This is  going to be equal to-- a to the third divided by a squared. That's a to the 3 minus 2 power, right? So this would simplify to just an a. And you could imagine, this is a times a times a divided by a  times a. You'll just have an a on top. And then the b, b to the fourth divided by b, well, that's just going to be b to the third, right? This is b to the first power. 4 minus 1 is 3, and then all of that in  parentheses to the third power. We don't want to forget about this third power out here. This  third power is this one. Let me color code it. That third power is that one right there, and then  this a in orange is that a right there. I think we understand what maps to what. And now we can use the property that when we multiply something and take it to the third power, this is equal 

to a to the third power times b to the third to the third power. And then this is going to be equal to a to the third power. times b to the 3 times 3 power, times b to the ninth. And we would have  simplified this about as far as you can go. Let's do one more of these. I think they're good  practice and super-valuable experience later on. Let's say I have 25xy to the sixth over 20y to  the fifth x squared. So once again, we can rearrange the numerators and the denominators. So  this you could rewrite as 25 over 20 times x over x squared, right? We could have made this  bottom 20x squared y to the fifth-- it doesn't matter the order we do it in-- times y to the sixth  over y to the fifth. And let's use our newly learned exponent properties in actually just simplify  fractions. 25 over 20, if you divide them both by 5, this is equal to 5 over 4. x divided by x  squared-- well, there's two ways you could think about it. That you could view as x to the  negative 1. You have a first power here. 1 minus 2 is negative 1. So this right here is equal to x to the negative 1 power. Or it could also be equal to 1 over x. These are equivalent. So let's say that this is equal into 1 over x, just like that. And it would be. x over x times x. One of those sets of x's would cancel out and you're just left with 1 over x. And then finally, y to the sixth over y to the  fifth, that's y to the 6 minus 5 power, which is just y to the first power, or just y, so times y. So if  you want to write it all out as just one combined rational expression, you have 5 times 1 times y, which would be 5y, all of that over 4 times x, right? This is y over 1, so 4 times x times 1, all of  that over 4x, and we have successfully simplified it.