Hey, everyone. So we're going to look at a few more rules called the power rules. We're going to continue on with our exponents tables. I'm just going to show you how they work because there are a few situations that we haven't seen yet. Let's just go ahead and get started. I'll show you how this works. So let's say I have an expression like 43 and that's raised to an exponent. So it's almost like I have an exponent on top of an exponent. You could also see situations like 3×4, and that's raised to the second power, or you could even see fractions like 124 raised to the second power. What's common about all of these is that you see either powers that are on top of powers or you see products or quotients that are raised to other powers. And so the whole thing here is you're going to use power rules. Let's take a look at the first one here. Here, I have 432. Remember, what that means is if I have a term that's raised to a power, it's basically like multiplying it by itself twice. So in other words, 43 times 43. So how does this work out? Well, we've actually seen how this works out from the product rule. You basically just add their exponents, 3 + 3, and you get 46. But I'm going to show you that in some cases, actually, the numbers here will be really, really big. You don't want to have to write this all out and do the product rule. So here's where the power rule comes in handy. What you're going to do is you're going to take these 2 exponents, and you're just going to multiply them. So in other words, you're going to take 3 and the 2, and now the power is going to be 3 times 2. And what do you get? You still just get 46. So in other words, this is just 2 different ways of representing the same thing, but this is what the power rule says. Anytime you have a power on top of a power, you just multiply their exponents. Alright? Pretty straightforward. So let's take a look at the second one here. Here I have 3×4 raised to the second power. I'm just going to show you how this works. Basically, the idea is that the exponent here distributes to everything that's inside of the parenthesis. It's actually kind of how, like, we use the distributive property with something like 2(3+4). We distribute it to everything that's inside. This is basically what that is. It's like the distributive property for exponents. So what happens here? This 3×4 to the second power is the same thing as 32 times 42. So this is just 9 times 16, and if you work this out, this ends up being 144. Here's another way to think about it. This is really just a parenthesis with an exponent, so we can use PEMDAS order of operations. This just becomes 122. What is 122? It's also 144. Alright. So this is called the power of a product rule. So the power of product means that you just distribute exponents to each term inside the parentheses. Alright? So if you see something like this, distribute it to everything that's inside. And for the last but not least, you're going to see sometimes fractions with exponents. And the idea is the same. You distribute the exponent to everything that's inside of the parenthesis. So this 12242 just becomes 144 divided by 16. And what do you get? You get 9. Another way you could have done this is you could have just done 124, which is just 3. And what's 32? It also is 9. So all these things are the same. So this is what the power of a quotient rule says. Just basically distribute exponents to the numerator. So the distributing exponents to the top and at the bottom. So this is basically how all these rules work. Let's go and take a look at some examples and see a few more situations here. So here we have m-2 to the negative 5 power. So I have a power over here that's on top of another power, so I use the power rule. The power rule says I just multiply their exponents. So this just becomes m10. And that is my answer. Alright. So pretty straightforward. This works for numbers, but it also just works for variables too. They work the same exact way. Let's take a look at the second one over here. So I have xy34. Alright. So what do you think this becomes? Well, some of you might be thinking that this should really just become xy3×4, but this is actually incorrect. Alright? This is wrong. Don't do this because what happens here is you actually have a product that's on the inside of this parenthesis. This isn't just one term like the 43 over here. This is two terms. I have an x and a y cubed. Those are 2 separate things. So that actually this is more of a power of a product. You distribute the 4 to the y cubed and also the x4. So what happens here is this actually just becomes x4, y3×4. So remember what happens is this basically just now becomes a product rule. So this becomes x4, y12. Alright? And that is what your answer is. And for our last but not least, we have 5x raised to the negative three power. Alright? So what happens, I distribute the negative three power to everything inside the parenthesis, and this just becomes 5-3x-3. So is this fully simplified, and could I leave this like this? Well, we actually saw from previous rules that you don't like to leave negative exponents inside of your expressions. So how do we take care of this? Well, if you remember the negative exponent rule says that I basically, if I have something on the top, I flip it to the bottom. So that, in other words, I want to write this 53, but then I write it with a positive exponent. So this just goes from the top to the bottom, and I rewrite with a positive. Now what happens if I have something with a negative exponent on the bottom? The opposite. I just flip it to the top. So in other words, this just becomes x353. Alright. So basically, what happens here is when you have a negative power like this with a fraction, you're just going to take the actual just reciprocal of the fraction. Alright. So first, we had 5 over x, now we have x over 5, and then we just have the distributive we distributed the 3 into each one of those terms. That's it for this fun, folks. Thanks for watching.
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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