Exponents - Video Tutorials & Practice Problems

Introduction to Exponent Rules

Welcome back, everyone. We saw how to simplify expressions by combining like terms. So for example, in this expression, we could combine three X squared and two X squared into five X squared and make the expression shorter. But combining like terms isn't always going to work. And this complicated expression over here, we can't combine anything because there's no pluses and minus signs. So it turns out that when we, when this happens and we can't combine like terms, we're gonna need some new rules to simplify uh expressions that have exponents in them. What I'm gonna do in this video is I'm gonna show you by using all these rules. We're gonna talk about that. This expression actually just simplifies down to something like X times Y. It's pretty cool. Let me just show you how it works. Feel free to use the page right before this. That has a master table of all these rules. So you don't have to fill this out uh multiple times and you'll have all your notes in one place. Let's just go ahead and get started here. So let's say I had something like one to the fourth power, one to any power. And I wanted to know what that evaluates to. Well, that just, just uh, one to the fourth power just ends up being one times, one times, one times one. And it doesn't matter how many multi, how many times you multiply one by the end result is always just one. And that's the rule, one to any power, always just equals one. All right. So that's a pretty straightforward one. It's called the base one rule. The names are the least important thing about the rule. It's just really important that you learn how they work. Let's go ahead and move on to the second one here. A negative to even power. So let's say I had negative three squared, that just means negative three and negative 33 times three just equals nine. What happens to the negative signs? Well, as long as you have a pair of negative signs, the negative sign always just gets cancer outs doesn't matter if the exponent is two or four, as long as it's any even number. So for example, negative three to the fourth power just looks like this and we'll see that three multiplied by itself four times is 81. And what happens is the negative gets canceled with this one and this negative gets canceled with this one. So anytime you have a negative number to an even power, you basically just drop the negative sign or it just gets canceled out. That's the rule. Um Now let's see what happens when you have negatives raised to odd powers, something like negative two to the third power. Well, let's write this out. This is negative two times negative two times negative two. So two times two times two is just eight. What happens to the negative sign? Well, this gets canceled out with this one. What about this one? This third negative sign doesn't have another one to cancel it out the negative. So it actually just gets kept there. So this is negative eight. So this rule is the opposite. Whenever you have a negative to an odd power, you actually end up keeping the negative sign on the outside. So you keep the negative sign here, right? So pretty straightforward, let's take a look at another couple of rules here. Now we're gonna get into like multiplication and division. Um Let's see what happens when you have something like four squared times four. The first power we'll just write this out four squared is four times four. Then we multiply by another factor of four. Remember the dot And the X just mean the same thing. It's all multiplication. So it's basically like I just have three fours mold multiplied together. But the easiest way to represent that is actually just four to the third power. That's the simplest way I can do that. And so if you look at what happened here with these exponents, there's two and the one we basically just added them. And that's actually what the rule ends up being. Anytime you're multiplying numbers of the same base, you actually just add their exponents together. So when you multiply, you add one way, you can kind of remember this is that the multiplication symbol and the addition symbol, they kind of just look the like the same, but one is tilted. So it's an easy silly way to remember this. But they actually uh turns out to be a really, really important uh rule in a shortcut because sometimes you're gonna have expressions where you don't want to write out all the terms like why do the 30th and why to the 70th? And you could actually really simply figure this out. This actually just ends up being why to the 100 because it's just 30 plus 70. All right. So pretty straightforward. Now, that's called the product rule by the way. And now let's take a look at the last one where you're now dividing terms that have the same base. So it's not four times four, it's four divided by four. And we'll see here that this is just four times four times four, divided by one factor of four. And remember from uh from fractions, we can always cancel out one of these things and we're just left with like a one that's out here. It's kind of like an invisible one. And the easiest way to represent this is, it's just four times four, but that's just four squared. All right. So here we actually ended up adding the exponents. Uh but here to get the two, we actually ended subtracting the three and the, and the one. And so that's the rule. Whenever you are dividing terms of the same base, you subtract their exponents. All right. So when you divide, you subtract and one way to remember this is that you're doing division, which kind of looks like a little minus sign. So division is subtraction. Now, one tiny difference here uh is that when you added the exponents, the order doesn't matter because two plus one is the same thing as one plus two. But in subtraction, it does matter, you always have to sub uh subtract the top exponent from the bottom. Sorry. Uh So always do top minus bottom. All right. So that's really important. Don't mess that up. All right, everyone. So that's it. For the first couple of rules. We'll take a look at more later on. Let's get some practice with these rules over here. We're going to simplify these expressions by using the exponent rules. Let's take a look at the first one. We have negative five to the ninth power divided by negative five to the sixth power. So in other words, we have the same base that's being divided with different exponents. That just means we're gonna use the quotient rule and we're gonna subtract the exponents. All right. So in other words, we're gonna take this and this is gonna be negative five to the nine minus six power. This becomes negative five to the third power. But remember this actually is now a negative number raised to an odd power. So we can use the negative to odd power rule and we keep the negative sign on the outside. And then if we wanted to evaluate this as a single number, this would just be negative 100 and 25. All right. So let's move on now to part B and part B. Now we're gonna start mixing up numbers and variables. We have two X to the fourth power and seven X squared all divided by X to the fifth power. So we have multiplication and division. Let's just deal with the multiplication first on top. So what happens is all this stuff is multiplied? So in other words, the two and the seven multiplied to 14 and then you have X fourth X squared. So in other words, you're multiplying numbers are terms at the same base. So that means we can actually add their exponents. So the two and the seven become 14 and the X to the fourth and X squared becomes X to the four plus two. And then we just have X to the fifth on the bottom. So in other words, what happens is this just becomes 14, X to the six divided by uh X to the fifth power. And now, what happens is we have the same base on the top and the bottom. So now we can use the quotient rule for this and we subtract the exponents. In other words, this is just 614 X to the six minus five and this just becomes 14 X uh to the, you know, X to the one power. In other words, just 14 X. All right. So that's it for the second one. Let's go to the, take a look at the third one here, we have just multiplication of a bunch of these terms here. Um We can do the exact same thing that we did with the numerator. In this term, everything is multiplied, there's no division. So the six and the four basically just become 24. And now if you'll notice here, so that's what the six and the four become, you have an X cubed and then an X squared. So in other words, when you multiply those two things together, you get X to the three plus two power using the product rule. And then we also have Y squared Y the fifth power. So when you drop those two down and use the product rule, you get Y to the two plus five power. So this whole thing is just the product rule, you just sort of match up all of the terms that are similar. And then you use the product rule for each one of them in other words, this just becomes 424 and this is X to the fifth power, so X to the fifth power and then we have Y to the seventh power and that's your final answer. All right folks. So that's it for this one. Thanks for watching and watch. I'll see you in the next video.

Simplify the expression using exponent rules.

$\left(-5a^2\right)\left(3a^8\right)$

• A. $-8a^{16}$
• B. $-10a^{10}$
• C. $-15a^{16}$
• D. $-15a^{10}$

Simplify the expression using exponent rules.

$-\frac{12b^{11}}{4b^7}$

• A. $-3b^{18}$
• B. $-3b^{-18}$
• C. $-3b^4$
• D. $-3b^{-4}$

Zero and Negative Rules

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 zero or negative exponents. I'm gonna 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 zero and negative exponents. All right. So let's take a look at our example. Let's say we have something like two to the fourth power over two to the fourth power with 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 two to the four minus four power, which is just two to the zero power. So what does that actually mean? What does two to the zero power, mean two to the fourth power means two, multiplied by itself four times. How do I take two and multiply it by itself zero times. So, well, it turns out that we can actually basically just evaluate and just sort of expand out what these expression means. That's what I'm gonna do over here. What is two to the fourth power? Uh Well, two times two times, two times two, if you work it out actually ends up just being 16. So in other words, we have 16/16. 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 divided by the same number. So in other words, you just get one. So in other words, when I looked at this from the quotient rule, I got two to the zero power. But when I expanded everything and then just divided, I am just getting one. It turns out these two things mean the exact same thing. So whenever you have something, two to the zero power, really anything to the zero power, it always basically just means one. All right. And by the way, you're always going to have this because the top and bottom X ones will be the same. So they'll always just cancel out to zero. So the rule is that anything to the zero power, anything to the zero exponent always equals one, the one exception, however is where you have zero, you can't have zero to the zero power because then you get zero over zero. And this is just one of those weird math things that you can't do. Right? So anything except for zero race to the zero power is always equal to one, that's what the zero exponent rule means. All right. So now let's take a look at our next example over here and now we have a different situation and we have two squared over two to the fifth power. So, but it's the same idea, we're gonna do the exact same thing. So using the quotient rule that ends up being two, the, the two minus five power, which is to the negative three power. So again, two to the fourth power means two times itself four times. How do I take two and multiply it by itself negative three times. What does that even mean? Well, again, let's just expand it out and rewrite this. So two squared is just actually equal to two times two. And I'm, I'm gonna want you to write this, write it this way for now because you're going to see what happens. And then two to the fifth power is just two times itself five 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 two of the pairs of twos. And when you cancel everything out, there's still like an invisible one that's hidden here on the top. So what happens when I expanded and divided everything? This just turns into one, divided by two times, two times two. In other words, this really just becomes one divided by two to the third power. All right. So again, when I did this using the quotient rule, I just got two to the negative three power. But when I expanded and divided, I get 1/2 to the third power. These mean the exact same thing. So look at the difference here here, I don't have a fraction but I have a negative exponents here. Now the two 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 two to the negative three power becomes 1/2 to the third power. So when you have a negative on the top, you flip it to the bottom, you rewrite it with a 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 1/1 over two to the negative three power you actually just flip it to the top and this becomes two to the positive third power. All right. 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 a uh when you have it on the bottom, you flip it to the top and you always rewrite it with a positive exponents. All right. So that's what the negative exponent means. So in other words, with this actually just becomes over here, it just becomes 1/8. So that would be your final answer. All right. 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 gonna simplify these expression using the two rules that we just learned. Let's take a look at the first one. We have a parentheses xy to the negative three power. So what happens here? Well, basically what happens is I'm gonna take this entire term just like I had two to the negative three power and I flipped it to the bottom of an expression. That's exactly what it says to do here. This just becomes one over uh I flip to the bottom. This is going to be XY 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, a quotient rule or anything like that? OK. So what about this one? Doesn't this just look exactly like I just, what I had just had in part a, well, yes, except for one key difference, which is that in this case, we had the parenthesis 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. OK? 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 one over Y to the third power. OK? So in other words, when this thing was in the parentheses, we had to kind of 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 happened actually ends up happening here? And your, what your final answer is is it ends up, it ends up being X over Y to the net to the three power. All right. So make sure that you understand the difference between these two when you have parentheses versus no parentheses and negative exponents because they are very, very different things. All right. And last but not least we have our last example, nine to the zero over nine to the negative four. So remember this is just nine to the zero power. What does that mean? Well, remember anything to the zero power except for zero is just equal to one. So that, that's what this becomes. In other words, we have 1/9 to the negative four power. 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 exponents. In other words, this actually is just nine to the fourth power. All right. So that's it. With this one, folks, let me know if you have any questions and let's move on.

Power Rules

Hey, everyone. So we're gonna look at a few more rules called the power rules. So we're gonna continue on with our exponents tables. I'm just going to show you how they work because there's 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 for the to the third power 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 three times four and that's raised to the second power or you could even see fractions like 12/4 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 gonna use these power rules. Let's take a look at the first one here. Here, I have four cube to the second power. 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, four thirds times, four thirds. So how does this work out? Well, we've actually seen how this works out from the products rule. You basically just add their exponents three plus three and you get four to the sixth power. But I'm gonna 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 gonna do is you're gonna take these two exponents and you're just gonna multiply them. So in other words, you're gonna take three and a two and now the power is gonna be three times two and what do you get, you still just get four of the six power. So in other words, this is just two 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. All right, pretty straightforward. So let's take a look at the second one here. Uh Here I have three times four rays to the second power. I'm just gonna show you how this works. Basically, the idea is that this, this exponent here distributes to everything that's inside of the parentheses. It's actually kind of how, like we use the distributive property with something like two parentheses, three plus four 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? There's three times four parentheses to the second power. This is the same thing as three squared plus four, sorry times four squared. So this is just nine times 16. And if you work this out, this ends up being 100 and 44 here's another way to think about it. This is really just a, the CS with an exponent. So we can use pem A S order of operations. This just becomes 12 squared. What is 12 squared? It's also 100 and 44. All right. So this is called the power of a product rule. Um So power of product means that you just distribute exponents to each term to each term inside the parentheses. All right. So if you see something like this, distribute it to everything that's inside. And for last, but not least you're gonna see sometimes fractions with exponents. And the idea is the same, you distribute the exponent to everything that's inside of the parentheses. So this 12 divided by four raised, the second power just becomes 12 squared, divided by four squared. So this just becomes 100 and 44 divided by 16. And what do you get, you get nine? Another way you could have done this is, you could have just done 12 divided by four, which is just three and what's three squared and also is nine. 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 distributed exponents to the top and the bottom. So the distributed exponents numerator and denominator. So this is basically how all these rules work. Let's go and take a look at some examples and see how um see a few more situations here. So here we have m to the negative two power to the negative five power. So I have a power over here that's on top of another power. So I use the power rule. Power rule says I just multiply their exponents. So this just becomes M to the negative two times negative five power and this just becomes M to the 10th power. And that is my answer. All right. 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 Xy to the third power to the fourth power. All right. So what do you think this becomes? Well, some of you might be thinking that this should really just become Xy to the three times four power. But this is actually incorrect. All right, this is uh 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 four to the third power over here. This is two terms I have an X and A Y cube. Those are two separate things. So that actually this is more of a power of a product, you distribute the four to the Y cubed and also the X. So what happens here is this actually just becomes X to the fourth power, Y to the three times fourth power. So remember what happens is this basically just now becomes a product rule. So this becomes X to the fourth Y to the 12th, right? And that is what your answer is. And for our last but not least we have five divided by X raised to the negative three power. All right. So what happens? I distribute the negative three power to everything inside the parentheses and this just becomes five to the negative three power divided by X to the negative three power. 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'm going to write this five to the uh three power, but then I write it with a positive exponents. 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 Xon on the bottom, the opposite, I just flip it to the top. So in other words, this becomes XX cubed over 5 to 5 to the third power. All right. 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, right? So first we had five over X, now we have X over five and then we just have the distributive, we distributed the three into each one of those terms. That's it for this, for folks. Thanks for watching.

Rewrite the expression using exponent rules.

$\left(4x^2\right)^3$

• A. $64x^6$
• B. $64x^5$
• C. $4x^6$
• D. $64x^3$

Rewrite the expression using exponent rules.

$\left(\frac{3x^4}{y^{-2}}\right)^3$

• A. $3x^{12}y^6$
• B. $\frac{27x^{12}}{y^6}$
• C. $27x^{12}y^2$
• D. $27x^{12}y^6$

Simplifying Exponential Expressions

Hey, everyone. So by now, you should have your completed table of all of the exponent rules. That whole page should be fully filled out. And in some problems, what you're gonna see is you're gonna see a generic exponential expression and they won't tell you which rules to use. What I'm gonna show you in this video is that in these types of problems, you're usually gonna have to use multiple exponent rules, combinations of them to fully simplify expressions. So what I want to do is rather than show you a step by step process, it's actually really more of a checklist. You're going to sort of navigate through this checklist in really no particular order just to make sure that you've checked all of these things. And then your expression is fully simplified. Let's just jump into this first problem. So I can show you how this work. So we had this first problem three X to the negative fifth to the in that whole expression squared. And then we have negative two X to the fourth and that whole expression is huge. One of the first, you want to first things you want to check for is that you have no powers raised to other powers. So for example, I see a power on top of a power. In fact, I have a power on top of this whole expression over here. So I can use the power rules, I can basically multiply exponents and distribute them into everything that's inside the parentheses. So let's go ahead and do that. First. This three goes into the X to the negative fifth and the three. And so, or sorry, this two, so this becomes three squared and then this becomes X to the negative fifth power um squared. And then over here, what happens is I have the three that distributes to each one of these things and this becomes negative two to the third power and then I have X to the fourth to the third power. All right, now I'm still not done because I still have powers on top of powers. So I have to sort of simplify this again, one step further. This just becomes X three squared. And what happens to the X to the negative fifth uh to the second power, you have to multiply the exponents. So this just becomes X to the negative 10. And now you can just drop the parentheses. Now what happens over here here, I have negative to the third power. Then I have a three outside of a four multiply those exponents. And then this just becomes X to the T 12. So that's what these two expressions became. All right. So now we have no more powers on top of power. So we're done with that step. One of the other things you want to check for is that you have no parentheses when that's all left over. Now, I only see one parenthesis over here and that's actually a number. And so we're actually gonna use another rule right here. Um or another thing in this checklist real quick, make sure that all your numbers with exponents get evaluated. So in other words, the three square just becomes nine, I have nine X to the 10th X to the negative 10th times and this becomes negative eight. Um Over here, I'm actually just gonna drop that negative eight X to the 12. All right. So here we have all numbers with exponents have been evaluated. All right, let's keep going. One of the other things you want to check for is that you have none of the same bases that are multiplied or divided here. What I have, I have an X to the negative 10th power later on, I have another term. It's X to the 12th power. That's not as simplified as it could be because I could really just merge those into one term by using the product and quotient rules. I can either add or subtract the exponent based on whether they're products or quotients. So here's what I'm gonna do here. I'm gonna do nine and then I'm just gonna flip the order of some of these things, all these things are multiplied. So I can flip the order. This is nine times negative eight and then we have times X to the negative 10 and then times X to the 12th. But because I have the same base, I can just add their exponents. So I'll just do that right here. This is gonna be negative 10 plus 12 power. All right. Now what also happens here is I notice that I have some numbers that are now multiplied nine times negative eight. So one of the other things that you can do is just make sure that all your operations have been performed between numbers nine times negative eight, been simplified to negative 72. And then what happens here as a result of this product rule is I just get X squared. All right. So we've done, we've done no parentheses. We've done no same bases. The last couple things you want to check for here is that you have no zero exponents or negative exponents because then you can just evaluate it to one or you can use the negative exponent rule to flip stuff on the bottom or the top depending on where it is and then rewrite it with a positive exponent. All right. So we've checked for no zero and no negative exponents. So that means we're done here. And this is as simple as this expression could possibly be, I can't simplify this any further. All right, that's all there is to it. Let's take a look at the second here. Um So here, what I've got here is I've got X to the square X squared Y to the seventh and then I've got in the nominator, X to the fifth and then Y to the fourth. So I also have that whole entire expression here that's raised to the negative one power. So if you go through this checklist, one of the things you might think that you want to do first is sort of distribute the powers on inside of everything that's over here. But actually, that's going to lead to more work because if you distribute the negative, then you're gonna have a bunch of negative exponents everywhere. So what I wanted to sort of give you is a little bit of a pro tip. Um Usually there's no correct order in using these rules, but usually it's simplest to kind of work from the innermost expression to the outermost expression. So if you have something in the parentheses, sort of simplify that down first and then work your way outwards, that's usually the best way to go about things. So here's what I mean by this, right? So um I'm just gonna ignore the powers and other powers for right now, what I want to do is I have parentheses, but I also have the same bases that are multiplied and divided. So I've got X to the two and X to the fifth. So what is, what happens here is I'm just gonna sort of use the quotient rule. Um X to the two power over X to the fifth power. This really just becomes um I have X to the third power over here. All right. And then what happens is Y to the seventh power over Y to the fourth power, you can cancel out four powers of Y and subtract four powers of Y over here. And what you're left with is you're left with Y to the third power up top. So in other words, all the X is canceled out fully from the denominator and all the Y is canceled out fully from sorry, the numerator and all the Y is canceled out fully from the denominator and you're left with some powers of X here on the bottom and the top. So here was a three and this was a three. So now I have this expression raised to the negative one power. All right. So usually if you work your way from inside out, you'll see that the expression on the inside will simplify. And then now what we can do is we can distribute this negative one into the top and the bottom because now there's fewer terms. All right. So again, no correct order usually just kind of work your way inside out. But if you did it, you know, uh if you wanted to distribute this first. You would have gotten the same answer. All right. So what I've got here is I've got now Y to the negative three power and then X to the negative three power. All right. So I've got no more powers on top of powers and I have no parentheses. I have also have no same base multiplied or divided. I have no zero exponents, but I do have some negative exponents. So I'm going to have to deal with those if, remember if I have a negative on the top and it flips to the bottom. So this becomes Y cubed. And if I have a negative exponent on the bottom, I flip it to the top and it becomes a positive exponent. So in other words, this whole thing just becomes the reciprocal. I have X cubed over Y cubed. Now, I also got no numbers that are evaluated. Um I've got no numbers in this problem and I've performed all my operations. So this thing is fully simplified. All right. So that's all there is to it. Hopefully this made sense. Thanks for watching.