So we take a look at some of the exponent rules now, and when we saw something like the product and quotient rule, we only just saw positive exponents. But that's not the only type of exponent you'll see. In some problems whether it's by using the quotient rule or sometimes the problem actually will just have it already, you might actually run into 0 or negative exponents. I'm going to show you how to deal with those in this video because you'll need to know what those things evaluate to. It's actually pretty straightforward. So let's just go ahead and take a look at the next two rules in our table, which have to do the 0 and negative exponents. Alright? So let's take a look at our example. Let's say we have something like 24/24. What the quotient rule tells us, because we can use that, we have the same thing on the top and bottom, is that this actually just turns out to be 24-4, which is just 20. So what does that actually mean? What does 20 mean? 24 means 2 multiplied by itself 4 times. How do I take 2 and multiply by itself 0 times? So, well, it turns out that we can actually basically just evaluate and just sort of expand out what these expression means. And that's what I'm going to do over here. What is 24? Well, 2 times 2 times 2 times 2, if you work it out, actually ends up just being 16. So in other words, we have 16 over 16. And what happens when you have the same thing on the top and bottom of a fraction? What does this always end up being? So you're just dividing by the same number. So in other words, you just get 1. So in other words, when I looked at this from the quotient rule, I got 20, but when I expanded everything and then just divided, I am just getting 1. It turns out these two things mean the exact same thing. So whenever you have something 20, really anything to the 0 power, it always basically just means 1. Alright? And by the way, you're always going to have this because the top and bottom exponents will be the same, so they'll always just cancel out to 0. So the rule is that anything to the 0 power, anything to the 0 exponent always equals 1. The one exception, however, is where you have 0. You can't have 0 to the 0 power because then you get 0 over 0, and this is just one of those weird math things that you can't do. Alright? So anything except for 0 raised to the 0 power is always equal to 1. That's what the 0 exponent rule means. Alright? So now let's take a look at our next next example over here, and now we have a different situation. Now we have 22/25. So but it's the same idea. We're going to do the exact same thing. So using the quotient rule, this ends up being 22-5, which is 2-3. So, again, 2]4 means 2 times itself 4 times. How do I take 2 and multiply it by itself negative 3 times? What does that even mean? Well, again, let's just expand it out and rewrite this. So 22 is just actually equal to 2 times 2, and I'm I want you to write this write it this way for now because you're going to see what happens. And then 25 is just 2 times itself 5 times. So if you remember from fractions, what happens is when you have the same thing on the top and the bottom, you can cancel out the terms. So I can cancel out 2 of the pairs of twos. And when you cancel everything out, there's still, like, an invisible one that's hidden here on top. So what happens when I expanded and divided everything? This just turns into 1 divided by 2 times 2 times 2. In other words, this really just becomes 1 divided by 23. Alright? So, again, when I did this using the quotient rule, I just got 2-3. But when I expanded and divided, I get 1 over 23. These mean the exact same thing. So look at the difference here. Here, I don't have a fraction, but I have a negative exponent. Here, now the 2 is on the bottom of a fraction, and the exponent became positive. That's what the rule says. Basically, what the negative exponent does is it basically just flips it to the bottom of a fraction. So 2-3 becomes 1 over 23. So when you have a negative on the top, you flip it to the bottom, you rewrite it with positive exponents. And by the way, you actually may see this the other way around. You may see a number with a negative exponent on the bottom, and you do the exact same thing, except you just flip it to the top. So if you have one over one over 2-3, you actually just flip this to the top, and this becomes 23. Alright? So it's basically just the reciprocal. When you have a negative exponent on the top, you flip it to the bottom. And we have it, when you have it on the bottom, you flip it to the top, and you always rewrite it with positive exponents. Alright? So that's what the negative exponent means. So in other words, what this actually just becomes over here is just becomes 1 over 8. So that would be your final answer. Alright? So that's really it for these next couple of rules. Let's go ahead and take a look at some examples over here. So we're going to simplify these expressions using the two rules that we just learned. Let's take a look at the first one. We have a parenthesis x y to the negative three power. So what happens here? Well, basically, what happens is I'm going to take this entire term just like I had 2-3, and I flipped it to the bottom of an expression. That's exactly what it says to do here. This just becomes one over, I flipped to the bottom. This is going to be x y to the third power, and I can't really do anything else with this. So it turns out that this is just my final answer. So this is just my final answer. I can't use any of the other rules. I can't use the product rule, the quotient rule, anything like that. Okay. So what about this one? Doesn't this just look exactly like what I just had in part a? Well, yes. Except for one key difference, which is that in this case, we had the parentheses, and in this one, we didn't. And so what happens in this case is this is actually really like x times y to the negative third power. Okay? And so what happens here? Well, basically, what this becomes is it becomes x times and then remember, y to the negative third power means we have to flip it to the bottom of a fraction. So this becomes 1 over y to the third power. Okay? So in other words, when this thing was in the parentheses, we had to kinda treat it as one object, and so we move this whole entire thing to the bottom of the expression. But when you don't have a parenthesis, the negative three only just pertains to the term that's immediately in front of it. So what has actually ends up happening here and your what your final answer is, is it ends up ends up being x over y to the 3 power. Alright? So make sure that you understand the difference between these two when you have parentheses versus no parentheses and negative exponents because they're very, very different things. Alright. And last but not least, we have our last example, 90/9-4. So remember, this is just 90. What does that mean? Well, I remember anything to the 0 power except for 0 is just equal to 1. So that that's what this becomes. In other words, we have 1 over 9-4. And how do we simplify this? Well, we don't want negative exponents here on the bottom. So what we can do is we can basically just flip it to the top and rewrite with a positive exponent. So in other words, this actually is just 94. Alright? So that's it for this one, folks. Let me know if you have any questions, and let's move on.
Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
0. Review of Algebra
Exponents
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