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So a few videos ago, I told you that anything to the zeroth power is equal to 1. So x to the zeroth power is equal to 1. And I gave you one argument why this is the case. I used the example of, if we have 3 to the first power, that is equal to 3. 3 to the second power is equal to 9. 3 to the third power is equal to 27. So every time we decrease by a power, we're dividing by 3. 27 divided by 3 is 9. 9 divided by 3 is 3. Then 3 divided by 3 is 1. And that should be what 3 to the zeroth power is. So that's one way to think about it. The other way to think about it is that we need this for the exponent properties to work. For example, I told you that a to the b times a to the c is equal to a to the b plus c. Now, what happens if c is 0? What happens if we have a to the b times a to the 0? Well, by this property, this needs to be equal to a to the b plus 0, which is equal to a to the b. So a to the b times a to the 0 must be equal to a to the b. If you divide both sides of this times a-- let me rewrite this-- a to the b times a to the 0, if we use this property up here, must be equal to a to the b, right? b plus 0 is b. If you divide both sides by a to the b, what do you get? On the left-hand side, you're left with just a to the 0, right? These cancel out. a to the 0 is equal to 1. And you can use a similar argument in pretty much all of the exponent properties, that we need anything to the zeroth power to be equal to 1. And it also makes sense as we divide by 3, each step as we decrement our exponent. It keeps working. When you take 3 to the negative 1 power, we saw on the last video that that's equal to 1 over 3 to the first power, or 1/3 . So once again, from 3 to the 0 to 1/3, you're dividing by 3 again. So it really makes sense on some level that 3 to the zeroth power is equal to 1. But that leaves a little bit of a gap. What about 0 to the zeroth power? This is a very strange notion. 0 multiplied by itself 0 times. And it depends what context you're using. Sometimes people will say that this is undefined, but many more times, at least in my experience, this'll be defined to be 1. And the reason why-- even though this is completely not intuitive, and you could type in 0 to the zeroth power in Google, and it'll give you 1. Even though this is completely not intuitive, the reason why this is defined to be this way is that makes a lot of formulas work. One in particular, the binomial formula works for your binomial coefficients, which I'm not going to go over right here, when 0 to the zeroth power is equal to 1. So that's an interesting thing for you to think about, what that might even mean. So let's talk about some of the other properties. And then we can put them all together with a couple of example problems. I told you in the last video what it means to raise to a negative power. a to the negative 1 power, or maybe I should say a to the negative b power is equal to 1 over a to the b power. So just to do that with a couple of concrete examples, 3 to the negative 3 power is equal to 1 over 3 to the third power, which is equal to 1 over 3 times 3, times 3, which is equal to 1 over 27. If I were to ask you what 1/3 to the negative 2 power is-- well, this is going to be equal to 1 over 1/3 to the second power. You get rid of the negative and you inverse it. So this is going to be equal to 1 over-- what's 1/3 times 1/3? 1/9. Which is equal to-- this is 1 divided by 1/9 is the same thing is 1 times 9, so this is equal to 9. And this makes complete sense, because 1/3, remember, 1/3 is the same thing as 3 to the negative 1 power, right? 3 to the negative 1 is equal to 1 over 3 to the 1 power, which is the same thing is 1/3. So if we replace 1/3 with 3 to the negative 1, this is 3 to the negative 1 to the negative 2 power. These two things are equivalent statements. And if we use one of the properties we learned in the first video, we can take the product of these two exponents. So this is equal to 3 to the negative 1, times negative 2, which is just positive 2, which is equal to 9. So it's really neat how all of these exponent properties really fit together in a nice, neat puzzle, that they don't contradict each other. And it doesn't matter which property you use, you'll get the right answer in the end, as long as you don't do something crazy. Now, the last thing I want to define is the notion of a fractional exponent. So if I have something to a fractional power-- so let's say I have a to the 1 over b power. I'm going to define this. This is going to be equal to the bth root of a. So let me be very clear here. Let me make it with some numbers here. If I said 4 to the 1/2 power right there, this means this is equivalent to the square root of 4. Which is equal to, if we're taking the principal root, this is equal to 2. So if I were to take, let's be clear, 8 to the 1/3 power, this is taking the cube root of 8. And this is, on some level, one of the most sometimes confusing things in exponents. Here I'm saying, what number times itself 3 times is equal to 8 ? So if I said that x is equal to 8 to the 1/3 power, this is the exact same thing as saying x to the third power is equal to 8. And how do I know that these are equivalent statements? Well, I could take both sides of this equation to the third power. If I take the left-hand side of the third power and the right-hand side of the third power, what do I get? On the left-hand side, I get x to the third. On the right-hand side, I get 8 to the 1/3 times 3, which is just 3 over 3, which is just 1. So if x is equal to 8 to the 1/3, what is x? Well, 2 times 2, times 2 is equal to 8. And there's no really easy way, especially once you go to the fourth root, or the fifth root, and you have decimals of calculating these. You probably need a calculator most of the time to do these. But things like 8 to the 1/3, or 16 to the 1/4, or 27 to the 1/3 , they're not too hard to calculate. So this right here, let me be clear, is 2. Now, let's make it a little bit more confusing. What is 27 to the negative 1/3 power? Well, don't get too worried. We're just going to take it step by step. When you take the negative power, this is completely equivalent to 1 over 27 to the 1/3 power. These two are equivalent. You get rid of the negative and take 1 over the whole thing. And then what is 27 to the 1/3 power? Well, what number times itself 3 times is equal to 27? Well, that's equal to 3. So this is going to be equal to 1 over 3. Not too bad. Now I'm going to take it even to another level, make it even more confusing, even more daunting. Now, let me do something interesting. What is 8 to the 2/3 power? Now that seems a little bit scary. And all you have to remember is this is the same thing, using our exponent rules really, as 8 squared to the 1/3 power. How do I know that? Well, if I multiply these two exponents, this is 2/3. So 8 to the 2/3 is the same thing is 8 squared, and then the third root of that. But you could view it the other way. This should also be equal to 8 to the 1/3 power squared. Because either way, when I multiply these exponents, I get 8 to the 2/3. Let's verify for ourselves that we really do get the same value. So 8 squared is 64. And we're going to take that to the 1/3 power. Down here, we have 8 to the 1/3. We already figured out what that is. That's 2, because 2 to the third power is 8. So this is 2 squared. Now, what is 64 to the 1/3? What times itself 3 times is equal to 64? Well, 4 times 4, times 4 is equal to 64, or 4 to the third is equal to 64, which means that 4 is equal to 64 to the 1/3. So this is equal to 4. And, lucky for us, 2 squared is also equal to 4. So it doesn't matter which way you do it. You could take the square and then the third root, or you could take the third root and then square it. You're going to get the exact same answer. Now, everything I've been doing has been with actual numbers. Let me do a couple of problems that just bring everything we've done together using variables. So let's say we wanted to do a few expressions and we want to make sure there are no negative exponents in the answer. So let's add x to the negative 3 over x to the negative 7. There's a bunch of ways we could view this. We could view this as equal to x to the negative 3, times 1 over x to the negative 7. And what is 1 over x to the negative 7? This is the same thing as x to the seventh power, right? If you have 1 over something, you can get rid of the 1 over, and put a negative in front of the exponent. But if you're putting a negative in front of a negative 7, you're going to get x to a seventh. So this thing can simplify to x to the negative 3, times x to the seventh power. And then we can add the exponents, and that is x to the fourth power. Now, another way, a completely legitimate way we could have done this, is we could have just subtracted the exponents. We could have said, well, gee, this is the same base. This is going to be x to the negative 3, minus negative seventh power. Well, negative 3 minus negative 7, that's a negative 3 plus 7 which is equal to x to the fourth power. And then one final way-- I mean, actually, there's more than one final way we could have done this. We could have said x to the negative 3 over x to the negative 7-- sorry, not negative x-- over x to the negative 7. Well, x to the negative 3 is the same thing as 1 over x to the third-- that's that term right there-- times 1 over x to the negative 7, so this would have been equal to 1 over x to the third times x to the negative 7. You could add the exponents, so that's equal to 1 over 3 minus 7 is x to the negative 4. And then this-- if we just get rid of the inverse, we take the inverse of it, we can put a negative in front of this negative, making it a positive-- this is going to be equal to x to the 4. So no matter how we did it, as long as we're consistent with the rules, we got x to the fourth. Let's do one more slightly hairy one. And then I think we'll be done for now. Let's say we have 3x squared times y to the 3/2 power. And we're going to divide it by x times y to the 1/2 power. Well, once again, this is the same thing as 3 times the x terms right here, so 3 times x squared over x, times y to the 3/2 over y to the 1/2. Well, this is going to be equal to 3 times-- what's x squared over x? Or x squared over x to the first power? That's going to be equal to x to the 2 minus 1. And then this is going to be times y to the 3/2 minus 1/2. So what does the whole thing become? It becomes 3 times x. 2 minus 1 is just 1-- I can just write x there-- times 3/2 minus 1/2 is 2/2. So that's y to the 2/2. 2/2, or 2 2ths-- that's just the same thing is y. so this is equal to 3xy. Anyway, I encourage you to do many, many more examples of that. But, you'll see that just using the rules that we've been exposed to in the last few videos, you can pretty much simplify any exponent expression.