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Exponent properties with products

Learn how to simplify exponents when the numbers are multiplied with each other. We'll learn that (a*b)^c is the same as a^c*b^c, a^c*a^d is same as a^(c+d) and (a^c)^d is equal to a^(c*d). We will also solve examples based on these three properties. Created by Sal Khan and CK-12 Foundation.

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  • orange juice squid orange style avatar for user DanV
    Anything to zero power is one? Does that include zero to the zero power? - thanks
    (189 votes)
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  • male robot donald style avatar for user mansi rawat
    at where did the -1 go ?
    (27 votes)
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  • hopper cool style avatar for user Colin
    Why is n^0 = 1? Since n^1 = n(n times itself once) and n^2 = n * n(n times itself twice), shouldn't n^0 = 0(n times itself 0, or no times)? I understand Sal's point of view on this, but I think that my explanation is right. Why doesn't mine work?
    (24 votes)
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    • male robot hal style avatar for user Joshua Kennedy
      You don't subtract n each time, you divide by n. Using that pattern:

      n^2 = n * n *1, (divide by n) n^1 = n * 1, (divide by n) and n^0 = 1.

      I had the same problem too for a while. If it still doesn't make sense, just do what I did for a long time before I understood:

      Remember that mathematicians are slightly crazy. I only understand because I became near-mathematician-level crazy. This is not an insult. It means that you are one step closer to becoming as smart as Einstein (because you are harder to understand and only a few people could understand Einstein).
      (22 votes)
  • blobby green style avatar for user niffler390
    0 to the 0 power is undefined
    anything else to the 0 power is 1.
    (10 votes)
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  • blobby green style avatar for user msdarwin04
    I see in the comments that this has been covered. However, don't think the explanation is satisfactory as the answer does not fit within the rules set for ALL other exponents.
    So if x to the 3rd is x*x*x, x to the 2nd is x*x, and x to the first is just x. Then why would x to the zero not be 0? What is the justification to make zero mean something other than a null or nothing?
    Example, take any of the above I mentioned. I ask you to write the expression x to the 3rd. Then you would write down x three times, same with the second and first.
    So if you are writing down x however many times the exponent shows and if it shows zero times, then you write down nothing.
    I have found nothing that justifies the breaking of the rules used for all other exponents just for a null value.
    If math is logical, then what makes a zero exponent separate rule logical. This is what I could not find. Sorry for the long version for the question, but had to share my thinking.
    (4 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      First, don't apologize for the long question. It's well thought out, cohesive, and my pleasure to answer.
      We can determine the zero property of exponentiation by looking at ratios of exponents. If we had 2^5 / 2^4, we could write this as:
      (2 * 2 * 2 * 2 * 2) / (2 * 2 * 2 * 2)
      Using what we know about division, four 2's cancel out from the numerator and denominator, leaving us with just two. 2 is the same as 2^1, and the difference in the exponents that we started with was 1. Now, if we make the exponents the same, every 2 would cancel out, leaving us with 1, as x / x = 1.

      We could look at taking an x away as dividing by x. If we write x three times (x*x*x), and then want to write x squared, it would be x*x*x/x, which simplifies to x*x. If we want to go from x^1 to x^0, we divide by x, just like before. x/x = 1, regardless of what x is.

      Does this help with your question? It's not really a completely arbitrary rule made so that the rest of math would make sense, it works out that way. With the higher and higher abstractions of exponents and further, it becomes difficult to intuitively justify some of the rules surrounding them, especially with 0, as they aren't as often seen directly in nature.
      (13 votes)
  • leafers seedling style avatar for user WhoDoWeBooMSU
    Is anyone else here during quarantine school? Also, I know lots of people asked about 0^0 but I didn't quite get it. Can someone explain? -thanks
    (6 votes)
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  • leaf orange style avatar for user 8013247
    what ever happend to the (-1)^2 at ? Wouldn't it just be 1? and if so were did it go?
    (5 votes)
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  • duskpin tree style avatar for user Gabrus
    What if the exponent has a exponent? this question keeps making me bluescreen
    (4 votes)
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    • starky sapling style avatar for user astronomical
      I'm assuming that you mean (2^2)^2 but to do this you basically multiply the powers together and then raise the base to that number. For example, (2^2)^2, we would multiply 2 to 2, and then we would raise 2 to the 4th power. Another way to think about it is that (2^2)^2 is equal to (2^2)(2^2) which is equal to 2^4.

      I hope this helped
      Comment back if you have any questions:)
      (4 votes)
  • duskpin seedling style avatar for user 🖤SenpaiLiah🖤
    I knew about the number and adding part, but when Mr. Sal started talking about the variables with exponents, my mind went like "Into the unknown!".
    (5 votes)
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  • aqualine ultimate style avatar for user aum.mundhe
    In the last video, Sal said 0^0 is undefined. The Khan Academy calculator says it is 1. Which is correct? IM confused.
    (4 votes)
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Video transcript

In this video, I want to do a bunch of examples involving exponent properties. But, before I even do that, let's have a little bit of a review of what an exponent even is. So let's say I had 2 to the third power. You might be tempted to say, oh is that 6? And I would say no, it is not 6. This means 2 times itself, three times. So this is going to be equal to 2 times 2 times 2, which is equal to 2 times 2 is 4. 4 times 2 is equal to 8. If I were to ask you what 3 to the second power is, or 3 squared, this is equal to 3 times itself two times. This is equal to 3 times 3. Which is equal to 9. Let's do one more of these. I think you're getting the general sense, if you've never seen these before. Let's say I have 5 to the seventh power. That's equal to 5 times itself, seven times. 5 times 5 times 5 times 5 times 5 times 5 times 5. That's seven, right? One, two, three, four, five, six, seven. This is going to be a really, really, really, really, large number and I'm not going to calculate it right now. If you want to do it by hand, feel free to do so. Or use a calculator, but this is a really, really, really, large number. So one thing that you might appreciate very quickly is that exponents increase very rapidly. 5 to the 17th would be even a way, way more massive number. But anyway, that's a review of exponents. Let's get a little bit steeped in algebra, using exponents. So what would 3x-- let me do this in a different color-- what would 3x times 3x times 3x be? Well, one thing you need to remember about multiplication is, it doesn't matter what order you do the multiplication in. So this is going to be the same thing as 3 times 3 times 3 times x times x times x. And just based on what we reviewed just here, that part right there, 3 times 3, three times, that's 3 to the third power. And this right here, x times itself three times. that's x to the third power. So this whole thing can be rewritten as 3 to the third times x to the third. Or if you know what 3 to the third is, this is 9 times 3, which is 27. This is 27 x to the third power. Now you might have said, hey, wasn't 3x times 3x times 3x. Wasn't that 3x to the third power? Right? You're multiplying 3x times itself three times. And I would say, yes it is. So this, right here, you could interpret that as 3x to the third power. And just like that, we stumbled on one of our exponent properties. Notice this. When I have something times something, and the whole thing is to the third power, that equals each of those things to the third power times each other. So 3x to the third is the same thing is 3 to the third times x to the third, which is 27 to the third power. Let's do a couple more examples. What if I were to ask you what 6 to the third times 6 to the sixth power is? And this is going to be a really huge number, but I want to write it as a power of 6. Let me write the 6 to the sixth in a different color. 6 to the third times 6 to the sixth power, what is this going to be equal to? Well, 6 to the third, we know that's 6 times itself three times. So it's 6 times 6 times 6. And then that's going to be times-- the times here is in green, so I'll do it in green. Maybe I'll make both of them in orange. That is going to be times 6 to the sixth power. Well, what's 6 to the sixth power? That's 6 times itself six times. So, it's 6 times 6 times 6 times 6 times 6. Then you get one more, times 6. So what is this whole number going to be? Well, this whole thing-- we're multiplying 6 times itself-- how many times? One, two, three, four, five, six, seven, eight, nine times, right? Three times here and then another six times here. So we're multiplying 6 times itself nine times. 3 plus 6. So this is equal to 6 to the 3 plus 6 power or 6 to the ninth power. And just like that, we/ve stumbled on another exponent property. When we take exponents, in this case, 6 to the third, the number 6 is the base. We're taking the base to the exponent of 3. When you have the same base, and you're multiplying two exponents with the same base, you can add the exponents. Let me do several more examples of this. Let's do it in magenta. Let's say I had 2 squared times 2 to the fourth times 2 to the sixth. Well, I have the same base in all of these, so I can add the exponents. This is going to be equal to 2 to the 2 plus 4 plus 6, which is equal to 2 to the 12th power. And hopefully that makes sense, because this is going to be 2 times itself two times, 2 times itself four times, 2 times itself six times. When you multiply them all out, it's going to be 2 times itself, 12 times or 2 to the 12th power. Let's do it in a little bit more abstract way, using some variables, but it's the same exact idea. What is x to the squared or x squared times x to the fourth? Well, we could use the property we just learned. We have the exact same base, x. So it's going to be x to the 2 plus 4 power. It's going to be x to the sixth power. And if you don't believe me, what is x squared? x squared is equal to x times x. And if you were going to multiply that times x to the fourth, you're multiplying it by x times itself four times. x times x times x times x. So how many times are you now multiplying x by itself? Well, one, two, three, four, five, six times. x to the sixth power. Let's do another one of these. The more examples you see, I figure, the better. So let's do the other property, just to mix and match it. Let's say I have a to the third to the fourth power. So I'll tell you the property here, and I'll show you why it makes sense. When you add something to an exponent, and then you raise that to an exponent, you can multiply the exponents. So this is going to be a to the 3 times 4 power or a to the 12th power. And why does that make sense? Well this right here is a to the third times itself four times. So this is equal to a to the third times a to the third times a to the third times a to the third. Well, we have the same base, so we can add the exponents. So there's going to be a to the 3 times 4, right? This is equal to a to the 3 plus 3 plus 3 plus 3 power, which is the same thing is a the 3 times 4 power or a to the 12th power. So just to review the properties we've learned so far in this video, besides just a review of what an exponent is, if I have x to the a power times x to the b power, this is going to be equal to x to the a plus b power. We saw that right here. x squared times x to the fourth is equal to x to the sixth, 2 plus 4. We also saw that if I have x times y to the a power, this is the same thing is x to the a power times y to the a power. We saw that early on in this video. We saw that over here. 3x to the third is the same thing as 3 to the third times x to the third. That's what this is saying right here. 3x to the third is the same thing is 3 to the third times x to the third. And then the last property, which we just stumbled upon is, if you have x to the a and then you raise that to the bth power, that's equal to x to the a times b. And we saw that right there. a to the third and then raise that to the fourth power is the same thing is a to the 3 times 4 or a to the 12th power. So let's use these properties to do a handful of more complex problems. Let's say we have 2xy squared times negative x squared y squared times three x squared y squared. And we wanted to simplify this. This you can view as negative 1 times x squared times y squared. So if we take this whole thing to the squared power, this is like raising each of these to the second power. So this part right here could be simplified as negative 1 squared times x squared squared, times y squared. And then if we were to simplify that, negative 1 squared is just 1, x squared squared-- remember you can just multiply the exponents-- so that's going to be x to the fourth y squared. That's what this middle part simplifies to. And let's see if we can merge it with the other parts. The other parts, just to remember, were 2 xy squared, and then 3x squared y squared. Well now we're just going ahead and just straight up multiplying everything. And we learned in multiplication that it doesn't matter which order you multiply things in. So I can just rearrange. We're just going and multiplying 2 times x times y squared times x to the fourth times y squared times 3 times x squared times y squared. So I can rearrange this, and I will rearrange it so that it's in a way that's easy to simplify. So I can multiply 2 times 3, and then I can worry about the x terms. Let me do it in this color. Then I have times x times x to the fourth times x squared. And then I have to worry about the y terms, times y squared times another y squared times another y squared. And now what are these equal to? Well, 2 times 3. You knew how to do that. That's equal to 6. And what is x times x to the fourth times x squared. Well, one thing to remember is x is the same thing as x to the first power. Anything to the first power is just that number. So you know, 2 to the first power is just 2. 3 to the first power is just 3. So what is this going to be equal to? This is going to be equal to-- we have the same base, x. We can add the exponents, x to the 1 plus 4 plus 2 power, and I'll add it in the next step. And then on the y's, this is times y to the 2 plus 2 plus 2 power. And what does that give us? That gives us 6 x to the seventh power, y to the sixth power. And I'll just leave you with some thing that you might already know, but it's pretty interesting. And that's the question of what happens when you take something to the zeroth power? So if I say 7 to the zeroth power, What does that equal? And I'll tell you right now-- and this might seem very counterintuitive-- this is equal to 1, or 1 to the zeroth power is also equal to 1. Anything that the zeroth power, any non-zero number to the zero power is going to be equal to 1. And just to give you a little bit of intuition on why that is. Think about it this way. 3 to the first power-- let me write the powers-- 3 to the first, second, third. We'll just do it the with the number 3. So 3 to the first power is 3. I think that makes sense. 3 to the second power is 9. 3 to the third power is 27. And of course, we're trying to figure out what should 3 to the zeroth power be? Well, think about it. Every time you decrement the exponent. Every time you take the exponent down by 1, you are dividing by 3. To go from 27 to 9, you divide by 3. To go from 9 to 3, you divide by 3. So to go from this exponent to that exponent, maybe we should divide by 3 again. And that's why, anything to the zeroth power, in this case, 3 to the zeroth power is 1. See you in the next video.