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Introduction to exponent rules. Created by Sal Khan.
Video transcript
Welcome to the presentation on level one exponent rules. Let's get started with some problems. So if I were to ask you what 2 -- that's a little fatter than I wanted it to be, but let's keep it fat so it doesn't look strange -- 2 the third times -- and dot is another way of saying times -- if I were to ask you what 2 to the third times 2 to the fifth is, how would you figure that out? Actually, let me use a skinnier pen because that does look bad. So, 2 to the third times 2 to the fifth. Well there's one way that I think you do know how to do it. You could figure out that 2 to the third is 8, and that 2 to the fifth is 32. And then you could multiply them. And 8 times 32 is 240, plus it's 256, right? You could do it that way. That's reasonable because it's not that hard to figure out 2 to the third is and what 2 to the fifth is. But if those were much larger numbers this method might become a little difficult. So I'm going to show you using exponent rules you can actually multiply exponentials or exponent numbers without actually having to do as much arithmetic or actually you could handle numbers much larger than your normal math skills might allow you to. So let's just think what 2 to the third times 2 to the fifth means. 2 to the third is 2 times 2 times 2, right? And we're multiplying that times 2 to the fifth. And that's 2 times 2 times 2 times 2 times 2. So what do we have here? We have 2 times 2 times 2, times 2 times 2 times 2 times 2 times 2. Really all we're doing is we're multiplying 2 how many times? Well, one, two, three, four, five, six, seven, eight. So that's the same thing as 2 to the eighth. Interesting. 3 plus 5 is equal to 8. And that makes sense because 2 to the 3 is 2 multiplying by itself three times, to the fifth is 2 multiplying by itself five times, and then we're multiplying the two, so we're going to multiply 2 eight times. I hope I achieved my goal of confusing you just now. Let's do another one. If I said 7 squared times 7 to the fourth. That's a 4. Well, that equals 7 times 7, right, that's 7 squared, times and now let's do 7 to the fourth. 7 times 7 times 7 times 7. Well now we're multiplying 7 by itself six times, so that equal 7 to the sixth. So in general, whenever I'm multiplying exponents of the same base, that's key, I can just add the exponents. So 7 to the hundredth power times 7 to the fiftieth power, and notice this is an example now. It would be very hard without a computer to figure out what 7 to the hundredth power is. And likewise, very hard without a computer to figure out what 7 to the fiftieth power is. But we could say that this is equal to 7 to the 100 plus 50, which is equal to 7 to the 150. Now I just want to give you a little bit of warning, make sure that you're multiplying. Because if I had 7 to the 100 plus 7 to the 50, there's actually very little I could do here. I couldn't simplify this number. But I'll throw out one to you. If I had 2 to the 8 times 2 to the 20, we know we can add these exponents. So that gives you 2 to the 28, right? What if I had 2 to the 8 plus 2 to the 8? This is a bit of a trick question. Well I just said if we're adding, we can't really do anything. We can't really simplify it. But there's a little trick here that we actually have two 2 to the 8, right? There's 2 to the 8 times 1, 2 to the 8 times 2. So this is the same thing as 2 times 2 to the 8, isn't it? 2 times 2 to the 8. That's just 2 to the 8 plus itself. And 2 times to the 8, well that's the same thing as 2 to the first times 2 to the 8. And 2 to the first times 2 to the 8 by the same rule we just did is equal to 2 to the 9. So I thought I would just throw that out to you. And it works even with negative exponents. If I were to say 5 to the negative 100 times 3 to the, say, 100 -- oh sorry, times 5 -- this has to be a 5. I don't know what my brain was doing. 5 to the negative 100 times 5 to the 102, that would equal 5 squared, right? I just take minus 100 plus 102. This is a 5. I'm sorry for that brain malfunction. And of course, that equals 25. So that's the first exponent rule. Now I'm going to show you another one, and it kind of leads from the same thing. If I were to ask you what 2 to the 9 over 2 to the 10 equals, that looks like that could be a little confusing. But it actually turns out to be the same rule, because what's another way of writing this? Well, we know that this is also the same thing as 2 to the 9 times 1 over 2 to the 10, right? And we know 1 over 2 to the 10. Well, you could re-write right this as 2 the 9 times 2 to the negative 10, right? All I did is I took 1 over 2 to the 10 and I flipped it and I made the exponent negative. And I think you know that already from level two exponents. And now, once again, we can just add the exponents. 9 plus negative 10 equals 2 to the negative 1, or we could say that equals 1/2, right? So it's an interesting thing here. Whatever is the bottom exponent, you could put it in the numerator like we did here, but turn it into a negative. So that leads us to the second exponent rule, simplification is we could just say that this equals 2 to the 9 minus 10, which equals 2 to the negative 1. Let's do another problem like that. If I said 10 to the 200 over 10 to the 50, well that just equals 10 to the 200 minus 50, which is 150. Likewise, if I had 7 to the fortieth power over 7 to the negative fifth power, this will equal 7 to the fortieth minus negative 5. So it equals 7 to the forty-fifth. Now I want you to think about that, does that make sense? Well, we could have re-written this equation as 7 to the fortieth times 7 to the fifth, right? We could have taken this 1 over 7 to the negative 5 and turn it into 7 to the fifth, and that would also just be 7 to the forty-five. So the second exponent rule I just taught you actually is no different than that first one. If the exponent is in the denominator, and of course, it has to be the same base and you're dividing, you subtract it from the exponent in the numerator. If they're both in the numerator, as in this case, 7 to the fortieth times 7 to the fifth -- actually there's no numerator, but they're essentially multiplying by each other, and of course, you have to have the same base. Then you add the exponents. I'm going to add one variation of this, and actually this is the same thing but it's a little bit of a trick question. What is 2 to the 9 times 4 to the 100? Actually, maybe I shouldn't teach this to you, you have to wait until I teach you the next rule. But I'll give you a little hint. This is the same thing as 2 the 9 times 2 squared to the 100. And the rule I'm going to teach you now is that when you have something to an exponent and then that number raised to an exponent, you actually multiply these two exponents. So this would be 2 the 9 times 2 to the 200. And by that first rule we learned, this would be 2 to the 209. Now in the next module I'm going to cover this in more detail. I think I might have just confused you. But watch the next video and then after the next video I think you're going to be ready to do level one exponent rules. Have fun.