Exponent rules - Online Tutor, Practice Problems & Exam Prep

Introduction to Exponent Rules

Welcome back, everyone. We saw how to simplify expressions by combining like terms. For example, in this expression, we could combine \(3x^2\) and \(2x^2\) into \(5x^2\) and make the expression shorter. But combining like terms isn't always going to work. In this complicated expression over here, we can't combine anything because there are no plus and minus signs.

So it turns out that when this happens and we can't combine like terms, we're going to need some new rules to simplify expressions that have exponents in them. What I'm going to do in this video is show you by using all these rules we're going to 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 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 \(1^4\), \(1\) to any power, and I wanted to know what that evaluates to. Well, \(1^4\) just ends up being \(1 \times 1 \times 1 \times 1\), and it doesn't matter how many times you multiply \(1\) by. The end result is always just \(1\), and that's the rule. \(1\) to any power always just equals \(1\).

Alright. 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 an even power. So let's say I had negative three squared. That just means negative three and negative three. \(3 \times 3\) just equals \(9\). What happens to the negative signs?

Well, as long as you have a pair of negative signs, the negative sign always just gets canceled out. It doesn't matter if the exponent is \(2\) or \(4\) as long as it's any even number. So for example, \((-3)^4\) just looks like this, and we'll see that \(3\) multiplied by itself 4 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.

Now let's see what happens when you have negatives raised to odd powers, something like \((-2)^3\). Well, let's write this out. This is \(-2 \times -2 \times -2\). So \(2 \times 2 \times 2\) is just \(8\). But what happens to the negative sign?

Well, this gets canceled out with this one. But 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 \(-8\). 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. Alright. So pretty straightforward. Let's take a look at another couple of rules here.

Now we're going to get into, like, multiplication and division. Let's see what happens when you have something like \(4^2 \times 4^1\). We'll just write this out. \(4^2\) is \(4 \times 4\), then we multiply by another factor of \(4\). Remember the dot and the \(x\) just mean the same thing.

It's all multiplication. So it's basically like I just have \(3\) fours multiplied together. But the easiest way to represent that is actually just \(4^3\). That's the simplest way I can do that. And so if you look at what happened here with these exponents, this \(2\) and the \(1\), 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 alike, but one is tilted. So it's an easy, silly way to remember this. But that actually turns out to be a really, really important rule and a shortcut because sometimes you're going to have expressions where you don't want to write out all the terms like \(y^{30}\) and \(y^{70}\), and you can actually really simply figure this out.

This actually just ends up being \(y^{100}\) because it's just \(30 + 70\). Alright? 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 \(4 \times 4\), it's \(4\) divided by \(4\). And we'll see here that this is just \(4 \times 4 \times 4\) divided by one factor of \(4\). And remember 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 just \(4 \times 4\), but that's just \(4^2\).

Alright. So here we actually ended up adding the exponents, but here to get the \(2\), we actually ended subtracting the \(3\) and then the \(1\). And so that's the rule. Whenever you are dividing terms with the same base, you subtract their exponents. Alright.

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 is that when you added the exponents, the order doesn't matter because \(2 + 1\) is the same thing as \(1 + 2\), but in subtraction, it does matter. You always have to subtract the top exponent from the bottom. So always do top minus bottom.

Alright? So that's really important. Don't mess that up. Alright, 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 \((-5)^9\) divided by \((-5)^6\).

So in other words, we have the same base that's being divided with different exponents. That just means we're going to use the quotient rule and we're going to subtract the exponents. Alright? So in other words, we're going to take this and this is going to be \((-5)^{9-6}\). This becomes \((-5)^3\).

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 \(-125\). Alright? So let's move on now to part b. In part b, now we're going to start mixing up numbers and variables.

We have \(2x^4\) and \(7x^2\) all divided by \(x^5\). So we have multiplication and division. Let's just deal with the multiplication first on the top. So what happens is all this stuff is multiplied. So in other words, the \(2\) and the \(7\) multiply to \(14\), and then you have \(x^4\), \(x^2\).

So in other words, you're multiplying numbers or terms of the same base. So that means we can actually add their exponents. So the \(2\) and the \(7\) become \(14\), and the \(x^4\) and \(x^2\) becomes \(x^{4 + 2}\), and then we just have \(x^5\) on the bottom. So in other words, what happens is this just becomes \(14x^6\) divided by \(x^5\). 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 \(14x^{6-5}\), and this just becomes \(14x\), to the, you know, \(x\) to the one power. In other words, just \(14x\). Alright. So that's it for the second one.

Let's go take a look at the third one. Here we have just multiplication of a bunch of these terms here. We could do the exact same thing that we did with the numerator in this term. Everything is multiplied. There's no division.

So the \(6\) and the \(4\) basically just become \(24\). And now if you'll notice he...

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 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 with the 0 and negative exponents. Alright? So let's take a look at our example. Let's say we have something like 2 to the 4th power over 2 to the 4th power. 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 2 to the 4 minus 4 power, which is just 2 to the 0 power.

So what does that actually mean? What does 2 to the 0 power mean? 2 to the 4th power means 2 multiplied by itself four times. How do I take 2 and multiply it by itself 0 times? So, well, it turns out that we can actually basically just evaluate and just sort of expand out what these expressions mean, and that's what I'm going to do over here.

What is 2 to the 4th power? 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 2 to the 0 power. 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 2 to the 0 power, 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 example over here, and now we have a different situation. Now we have 2 squared over 2 to the fifth power. 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 2 to the 2 minus 5 power, which is 2 to the negative 3 power.

So, again, 2 to the 4th power means 2 times itself four times. How do I take 2 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 2 squared is just actually equal to 2 times 2, and I don't want you to write this way for now because you're going to see what happens.

And then 2 to the fifth power is just 2 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 1 divided by 2 times 2 times 2. In other words, this really just becomes 1 divided by 2 to the 3rd power. Alright? So, again, when I did this using the quotient rule, I just got 2 to the negative 3 power. But when I expanded and divided, I get 1 over 2 to the 3rd 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 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 to the negative 3 power becomes 1 over 2 to the 3rd power. So when you have a negative on the top, you flip it to the bottom, you rewrite it with a positive exponent. 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 over one over 2 to the negative three power, you actually just flip this to the top, and this becomes 2 to the positive third power. 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 a positive exponent. 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 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, I flipped to the bottom. This is going to be x y to the 3rd 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 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 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 parentheses, the negative three only just pertains to the term that's immediately in front of it. So what have 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, 9 to the 0 over 9 to the negative 4. So remember, this is just 9 to the 0 power. 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 to the negative 4 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 exponent. So in other words, this actually is just 9 to the 4th power.

Alright? So that's it for this one, folks. Let me know if you have any questions, and let's move on.

Power Rules

Hey, everyone. So we're going to look at a few more rules called the power rules. So 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 4 to the third power and that's raised to another exponent. So it's almost like I have an exponent on top of another exponent. You could also see situations like 3 times 4 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 going to use these power rules. Let's take a look at the first one here. Here, I have \(4^3\) raised 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, \(4^3 \times 4^3\).

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 \(4^6\). 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 use the product rule.

So here is where the power rule comes in handy. What you're going to do is you're going to take these two 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 \(4^6\).

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. Alright? Pretty straightforward. So, let's take a look at the second one here.

Here I have \(3 \times 4\) raised to the second power. I'm just going to show you how this works. Basically, the idea is that this exponent here distributes to everything that's inside the parentheses. 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 \(3 \times 4\) inside the parentheses to the second power. This is the same thing as \(3^2 \times 4^2\).

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 parentheses with an exponent, so we can use PEMDAS order of operations. This just becomes \(12^2\). What is \(12^2\)?

It's also 144. Alright? So this is called the power of a product rule. The power of a 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 parentheses. So this \(12/4\) raised to the second power just becomes \(12^2/4^2\). So this just becomes 144/16.

And what do you get? You get 9. Another way you could have done this is you could have just done \(12/4\), which is just 3, and what's \(3^2\)? 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 and denominator. 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 \(m^{-2}\) raised to the -5 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 \(m^{(-2 \times -5)}\), and this just becomes \(m^{10}\).

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 xy to the third power raised to the 4th power. Alright? So what do you think this becomes? Well, some of you might be thinking that this should really just become xy to the \(3 \times 4\) power, 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 4 to the third power 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 x. So what happens here is this actually just becomes \(x^4\), \(y^{(3 \times 4)}\).

So remember what happens is this basically just now becomes a product rule. So this becomes \(x^4, y^{12}\). Alright? And that is what your answer is. And for our last but not least, we have \(5/x\) raised to the negative three power.

Alright? So what happens? I distribute the negative 3 power to everything inside the parentheses, and this just becomes \(5^{-3}/x^{-3}\). 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 if I have something on the top, I flip it to the bottom. So in other words, I want to write this \(5^3\), but then I write it with 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 exponent on the bottom?

The opposite. I just flip it to the top. So in other words, this just becomes \(x^3/5^3\). 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 reciprocal of the fraction.

Right? So first, we had \(5/x\), now we have \(x/5\), and then we just distributed the 3 into each one of those terms. That's it for this, 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 the exponent rules. That whole page should be fully filled out. And in some problems, what you're going to see is you're going to see a generic exponential expression, and they won't tell you which rules to use. What I'm going to show you in this video is that in these types of problems, you're usually going to 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 no particular order, just to make sure that you've checked all of these things, and then your expressions are fully simplified. Let's just jump into this first problem so I can show you how this works. So we have this first problem, 3x to the negative fifth to the second power, and then we have negative 2x to the fourth, and that whole expression is cubed. One of the first things you want to check for is that you have no powers raised to other powers.

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 3 goes into the x to the negative fifth and the 3, and so or sorry, this 2, So this becomes 3 squared, and then this becomes x to the negative fifth power, squared. And then over here, what happens is I have the 3 that distributes to each one of these things, and this becomes negative 2 to the third power, and then I have x to the fourth to the third power.

Alright. 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 3 squared, and what happens to the x to the negative fifth, 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 parenthesis.

Now what happens over here, here I have negative 2 to the third power, then I have a 3 outside of a 4, multiply those exponents, and then this just becomes x to the 12th. So that's what these two expressions became. Alright? So now we have no more powers on top of powers, 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 going to use another rule right here, or another thing in this checklist real quick. Make sure that all your numbers with exponents get evaluated. So in other words, the 3 squared just becomes 9. I have 9x to the negative tenth, x to the negative tenth times, and this becomes negative 8, over here.

I'm actually just going to drop that. Negative 8 x to the 12th. Alright. So here we have all numbers with exponents have been evaluated. Alright.

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 tenth power. Later on, I have another term that'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 going to do here. I'm going to do 9, and then I'm just going to flip the order of some of these things. All these things are multiplied, so I can flip the order. This is 9 times negative 8, and then we have times x to the negative tenth 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 going to be negative 10 plus 12 power. Alright. Now what also happens here is I noticed that I have some numbers that are now multiplied 9 times negative 8. So one of the other things that you can do is just make sure that all your operations have been performed between numbers.

9 times negative 8 can simplify to negative 72, and then what happens here as a result of this product rule is I just get x squared. Alright. So we've done we've done no parentheses, we've done no same bases. The last couple of things you want to check for here is that you have no 0 exponents or negative exponents because then you can evaluate it to 1, or you can use the negative exponent rule to flip stuff on the bottom or top depending on where it is, and then rewrite it with a positive exponent. Alright?

So we've checked for no 0 exponents 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. Alright? That's all there is to it.

Let's take a look at the second one here. 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 denominator 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 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 going to have a bunch of negative exponents everywhere.

So what I wanted to sort of give you is a little bit of a pro tip. 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 a parenthesis, 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, I'm just going to ignore the powers and other powers for right now. What I want to do is I have a parenthesis, but I also have the same bases that are multiplied and divided. So So I've got x to the 2nd and x to the fifth. So what happens here is I'm just going to sort of use the quotient rule, x to the second power over x to the fifth power.

This really just becomes, I have x to the third power over here. Alright. 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's canceled out fully from the denominator, and all the y's canceled out fully from sorry, the numerator, and all the y's 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 3, and this_was a 3. So now I have this expression raised to the negative one power. Alright? 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 are fewer terms.

Alright? So again, no correct order, usually just kind of work your way inside out. But if you did it, you know, if you wanted to distribute this first, you would have gotten the same answer. Alright. So what I've got here is I've got now y to the negative third power, and then I've got x to the negative third power.

Alright? So I've got no more powers on top of powers, and I have no parentheses. I also have no same bases multiplied or divided. I have no 0 exponents, but I do have some negative exponents, so I'm going to have to deal with those. If I remember, if I have a negative on the top, then it flips to the bottom, so this becomes y^3 on the bottom, 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^3 and y^3 on the top. Now I also got no numbers that are evaluated. I've got no numbers in this problem, and I've performed all my operations. So this thing is fully simplified.

Alright. So that's all there is to it. Hopefully, this made sense. Thanks for watching.