Exponents - Online Tutor, Practice Problems & Exam Prep

Understanding exponent rules is crucial for simplifying expressions. Key concepts include the base one rule, which states that any number raised to the zero power equals 1, and the negative exponent rule, which indicates that a negative exponent signifies a reciprocal. The product rule allows for the addition of exponents when multiplying like bases, while the quotient rule involves subtracting exponents when dividing. Mastering these rules aids in handling complex exponential expressions, ensuring clarity in calculations and enhancing problem-solving skills in algebra.

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 x2 and x2 into x2 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 pluses and minus signs. So, it turns out that when we can't combine like terms, we're going to need some new rules to simplify expressions that have exponents in them. And 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⁢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 14, 1 to any power, and I wanted to know what that evaluates to. Well, that just ends up being 1⁢1⁢1⁢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. One 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 −32. That just means negative three and negative three. 3⁢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, −34 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 −23. Well, let's write this out. This is −2⁢-2⁢2. So 2⁢2⁢2 is just 8. But 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 −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.

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 42 times 41. We'll just write this out. 42 is 4⁢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 4 all multiplied together. But the easiest way to represent that is actually just 43. That's the simplest way I can do that. And so if you look at what happened here with these exponents, there's 2 and the 1, you 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 like the same, 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

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 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.

Power Rules

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.

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 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. And 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 expressions 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-5 to the whole expression squared, and then we have -2x4, 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. 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. Let's go ahead and do that first. This 3 goes into the x to the negative 5th and the 3, and so or sorry, this 2, so this becomes 3 squared and then this becomes x to the negative 5th 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 4th 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 x3 squared, and what happens to the x to the negative 5th, to the second power? You have to multiply the exponents. So this just becomes x-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 x12. 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 10th, x to the negative 10th 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. What are 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 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 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 going to be negative 10 +12. 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 x2. 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 just evaluate it to 1 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.