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Powers of zero

Any non-zero number to the zero power equals one. Zero to any positive exponent equals zero. So, what happens when you have zero to the zero power? Created by Sal Khan.

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  • hopper happy style avatar for user Micah Kunkle
    there is no such thing as +0 or -0 right? my friends think that and i wasn't for sure.
    (195 votes)
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  • duskpin sapling style avatar for user Arundhati
    Why is 0 raised to the power of a negative number undefined
    (8 votes)
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    • piceratops sapling style avatar for user neilakhan
      When a number is raised to the power of a negative number, it is put under one and the exponent turns positive. For example, 2^-2 would be written as 1/2^2 or 1/4.
      Now if zero is raised to a negative power, it would be like: 0^-1 what simplifies to 1/0^1 what simplifies to 1/0. When a number is divided by zero, it results in undifined.
      (19 votes)
  • female robot grace style avatar for user Vanessa Eva Sky
    Hey, everyone, I understand everything Sal is explaining but I still feel I need a deeper understanding of why a^0=1. You see my dilemma is not in understanding how for example when 2^4=16 is also like saying 2^4=1x2x2x2x2. It's just if I applied that same logic to say 2^0 then I would get 2^0=1x(nothing) and from what I've gathered any number multiplied by zero is always zero. I'm confused as to how this becomes intuitive or logical. I can just accept it, but there doesn't seem to be any logical explanation here and I know math is a formal/logical system and it's meant to be understood so would someone please explain to me what I am missing to logically understand this :)?
    (8 votes)
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    • stelly blue style avatar for user Kim Seidel
      2^0 is not "1 x nothing".
      2^0 = 1 x "no 2's". This leaves just the 1.

      Of, work it backwards...
      2^3 = 8
      2^2 = 4
      How do you change 2^3 or 8 into 4? You divide by 2: 2^3 / 2 = 8/2 = 4
      2^2 = 4
      2^1 = 2
      Again, 2^2 / 2 = 4/2 = 2
      2^1 = 2
      So, 2^0 = ?. Use the same logic. 2^1 / 2 = 2/2 = 1, NOT zero.

      Hope this helps.
      (14 votes)
  • blobby green style avatar for user Mohammad Kalaf
    For those still confused, hopefully this helps:

    In terms of the number 2, think of going forward as multiplying 2 by 2 each step, and going backwards you are DIVIDING 2 by 2 each step.

    With that in mind:

    EXPONENT NUMBER LINE
    ⟵ ⟵ ⟵ ⟶ ⟶ ⟶
    -3 -2 -1 0 1 2 3

    Forwards (right direction):
    2¹ = 2
    2² = 2*2
    2³ = 2*2*2

    to get from 2³ to 2² we need to DIVIDE by 2
    2*2*2/2 = 2*2

    to get from 2² to 2¹ we need to DIVIDE by 2
    2*2/2 = 2

    So what is 2⁰?
    Well, to get from 2¹ to 2⁰, we follow the same pattern or logic of DIVIDING by 2.

    2⁰ = 2/2
    2 cakes divided among 2 people, how many cakes each person gets? That's right, one!
    2⁰ = 1

    You can continue this logic into the negative and you would get this pattern:

    2¯¹ = 1/2
    2¯² = 1/2/2/2
    2¯³ = 1/2/2/2/2

    Hopefully that makes intuitive sense!
    (10 votes)
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  • duskpin ultimate style avatar for user Lollipop
    Anyone else just scroll through comments laughing and commenting for fun? :)
    (8 votes)
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  • blobby green style avatar for user Maui Lumb-broadnax
    is it just me or do the teachers over explain it to make it confusing
    (8 votes)
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  • aqualine seed style avatar for user SHANNISE WALKER
    would the exponents be consider as the absolute values?
    (6 votes)
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  • piceratops sapling style avatar for user brittcar001
    Also wird 0 hoch zu jeder Potenz einfach Null sein
    (5 votes)
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  • aqualine ultimate style avatar for user cayden
    why does a power of zero always equal to 1?
    (5 votes)
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  • female robot grace style avatar for user 41986
    I thought 0 times 0 is undefined.
    (5 votes)
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Video transcript

- [Instructor] In this video, we're going to talk about powers of zero. And just as a little bit of a reminder, let's start with a nonzero number, just to remind ourselves what exponentiation is all about. So if I were to take two to the first power, one way to think about this is we always start with a one, and then we multiply this base that many times times that one. So here, we're only gonna have one two, so it's gonna be one times two, which is, of course, equal to two. If I were to say what is two to the second power? Well, that's going to be equal to one times, and now I'm gonna have two twos, so times two times two, which is equal to four. And you could keep going like that. Now, the reason why I have this one here, and we've done this before, is to justify, and there's many other good reasons why two to the zero power should be equal to one. But you could see, if we use the same exact idea here, you start with a one, and then you multiply it by two zero times. Well, that's just going to end up with a one. So, so far I've told you this video's about powers of zero, but I've been doing powers of two. So let's focus on zero now. So what do you think zero to the first power is going to be? Pause this video, and try to figure that out. Well, you do the exact same idea. You start with a one, and then multiply it by zero one time. So times zero, and this is going to be equal to zero. What do you think zero to the second power is going to be equal to? Pause this video and think about that. Well, it's going to be one times zero twice, so times zero times zero. And I think you see where this is going. This is also going to be equal to zero. What do you think zero to some arbitrary positive integer is going to be? Well, it's going to be equal to one times zero that positive integer number of times. So once again, it's going to be equal to zero. And in general, you can extend that. Zero to any positive value exponent is going to give you zero. So that's pretty straightforward. But there is an interesting edge case here. What do you think zero to the zeroth power should be? Pause this video and think about that. Well, this is actually contested. Different people will tell you different things. If you use the intuition behind exponentiation that we've been using in this video, you would say, all right, I would start with a one and then multiply it by zero zero times. Or in other words, I just wouldn't multiply it by zero, in which case I'm just left with the one, the zero to the zeroth power should be equal to one. Other folks would say, "Hey, no, I'm with the zero and that's the zeroth power, maybe it should be a zero." And that's why a lot of folks leave it undefined. Most of the time, you're going to see zero to the zeroth power, either being undefined or that it is equal to one.