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The 0 & 1st power

Discover a pattern that explains why any non-zero number to the zero power equals one. Created by Sal Khan.

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  • leafers ultimate style avatar for user benjamin
    how do you type the multiplication symbol on a keyboard?
    (108 votes)
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  • starky ultimate style avatar for user Rainbow of the Lionhearts
    At Sal says it will make sense. I never really got past the 2 to the 0 power. I don't understand how he took two, put a zero above it, and then turned it into a one?! How does that work?
    (6 votes)
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    • hopper cool style avatar for user 🤔 ᴄᴏᴅᴇᴅ ɢᴇɴɪᴜȿ 😎
      What he showed helped me understand why it does that, so now let me try to explain it for you:
      He changed the way he did the exponents to multiplying 1 times how many numbers (the number that the exponent is) to that one.

      So when it is 2⁰ = 1 because there aren't any 2s to multiply by.
      So something as big as 1,000,000⁰ = 1

      Now let's do it regular:
      When you have 4⁶ = 1 × 4 × 4 × 4 × 4 × 4 × 4 = 4096
      Well, that is a bit too big of a number so let's do 3² = 1 × 3 × 3 = 9

      Ask me to clarify anything.
      (16 votes)
  • piceratops ultimate style avatar for user RITIK   BOMPILWAR
    Why any number (except 0 ) raise to power 0 is always 1 ?
    (2 votes)
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    • leaf green style avatar for user Nathan
      It has to do with the properties of exponents.
      Here's a good example:
      Lets say you have a number a^x/a^y and x = y.
      Using a property of exponents you can rewrite the equation as a^(x-y) and since x=y that becomes a^0.
      Also since x = y, a^x = a^y and so it becomes 1/1 or just 1.
      (3 votes)
  • ohnoes default style avatar for user Dakota Bailey
    () I really don't understand how any non-zero number to the power of 0 is 1.
    (6 votes)
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  • female robot grace style avatar for user nymphachan
    Where did you get the 1?
    (3 votes)
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    • blobby green style avatar for user Larry Kuck
      Hello,
      We are permitted to multiply by 1 because it does not change the outcome, whether it's a simple multiplication problem of
      3 x 3 = 1 x 3 x 3 = 9
      or explaining why
      n^0 = 1
      where n is any number.

      Just as multiplying by 1 does not change the outcome, not multiplying by 1 should produce the same outcome that was reached by multiplying by 1.

      I see many people are asking the same or similar questions, and I provided a similar, but more detailed response to the first question under this video,
      (6 votes)
  • starky sapling style avatar for user http://facebookid.khanacademy.org/100006957275444
    Where can we get a video of a problem I encountered: 0.2^4

    I get something like 0.0016 when I believe it's 1.6 - The decimals are funky. May not have found them yet.
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      You need to use the rules for multiplying decimals to get the decimal point in the correct place.
      0.2^4 = 0.2 * 0.2 * 0.2 * 0.2
      Each 0.2 has one decimal place. Add them up 1+1+1+1 = 4 decimal places in your answer.
      So, answer = 0.0016
      Hope this helps.
      (2 votes)
  • blobby green style avatar for user Deepak Somarajan
    if 2^0 is 1, would 0^0 be 1, 0, or undefined?
    (5 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Interesting question!

      Consider the following two rules.

      1) Any nonzero number to the 0 power is 1.

      2) Zero to any positive power is 0.

      If we attempt to extend both of these rules to define 0^0, we get two different answers. Because of this situation, it is best to call 0^0 indeterminate (though 0^0 is often interpreted as 1).

      Have a blessed, wonderful day!
      (3 votes)
  • blobby green style avatar for user Arbaaz Ibrahim
    Why do we have do bring in one to work out hour exponents?
    anything to the zero power should be 0?
    (4 votes)
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  • cacteye yellow style avatar for user Karlar
    What does the little circle above the 2 mean?
    (5 votes)
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  • purple pi teal style avatar for user minkikwak
    Hi, I’m really confused like Sal is saying if exponents are repeated multiplication what is 2^0? In that case, it would have to be 0 right? And a lot of people (most of them) are saying that anything to the power of zero is one? Can anybody tell me which one is correct?
    (2 votes)
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    • aqualine ultimate style avatar for user 管墨涵
      Anything to the 0th power is ONE.
      Why is that?
      Let us start with a simple pattern.
      2^4 = 16
      2^3 = 8
      2^2 = 4
      2^1 = 1.
      As you can notice, as the exponent goes down 1, the result has been divided by 2. So 2^3 is half of 2^4, 2^2 is half of 2^3, and 2^1 is half of 2^2.
      If we continue this pattern, 2^0 should be half of 2^1.
      Since 2^1 = 2, 2^0 = 2/2 = 1.
      I hope it helps!
      (PS If you don't believe me, you can check the Wikipedia or any maths or science website for that. And you can get that anything except 0 to the 0th power is 1!)
      (6 votes)

Video transcript

What I want to do in this video is think about exponents in a slightly different way that will be useful for different contexts and also go through a lot more examples. So in the last video, we saw that taking something to an exponent means multiplying that number that many times. So if I had the number negative 2 and I want to raise it to the third power, this literally means taking three negative 2's, so negative 2, negative 2, and negative 2, and then multiplying them. So what's this going to be? Well, let's see. Negative 2 times negative 2 is positive 4, and then positive 4 times negative 2 is negative 8. So this would be equal to negative 8. Now, another way of thinking about exponents, instead of saying you're just taking three negative 2's and multiplying them, and this is a completely reasonable way of viewing it, you could also view it as this is a number of times you're going to multiply this number times 1. So you could completely view this as being equal to-- so you're going to start with a 1, and you're going to multiply 1 times negative 2 three times. So this is times negative 2 times negative 2 times negative 2. So clearly these are the same number. Here we just took this, and we're just multiplying it by 1, so you're still going to get negative 8. And this might be a slightly more useful idea to get an intuition for exponents, especially when you start taking things to the 1 or 0 power. So let's think about that a little bit. What is positive 2 to the-- based on this definition-- to the 0 power going to be equal to? Well, we just said. This says how many times are going to multiply 1 times this number? So this literally says, I'm going to take a 1, and I'm going to multiply by 2 zero times. Well, if I want to multiply it by 2 zero times, that means I'm just left with the 1. So 2 to the zero power is going to be equal to 1. And, actually, any non-zero number to the 0 power is 1 by that same rationale. And I'll make another video that will also give a little bit more intuition on there. That might seem very counterintuitive, but it's based on one way of thinking about it is thinking of an exponent as this. And this will also make sense if we start thinking of what 2 to the first power is. So let's go to this definition we just gave of the exponent. We always start with a 1, and we multiply it by the 2 one time. So 2 is going to be 1-- we're only going to multiply it by the 2. I'll use this for multiplication. I'll use the dot. We're only going to multiply it by 2 one time. So 1 times 2, well, that's clearly just going to be equal to 2. And any number to the first power is just going to be equal to that number. And then we can go from there, and you will, of course, see the pattern. If we say what 2 squared is, well, based on this definition, we start with a 1, and we multiply it by 2 two times. So times 2 times 2 is going to be equal to 4. And we've seen this before. You go to 2 to the third, you start with the 1, and then multiply it by 2 three times. So times 2 times 2 times 2. This is going to give us positive 8. And you probably see a pattern here. Every time we multiply by 2-- or every time, I should say, we raise 2 to one more power, we are multiplying by 2. Notice this, to go from 2 to the 0 to 2 to the 1, we multiplied by 2. I'll use a little x for the multiplication symbol now, a little cross. And then to go from 2 to the first power to 2 to the second power, we multiply by 2 and multiply by 2 again. And that makes complete sense because this is literally telling us how many times are we going to take this number and-- how many times are we going take 1 and multiply it by this number? And so when you go from 2 to the second power to 2 to the third, you're multiplying by 2 one more time. And this is another intuition of why something to the 0 power is equal to 1. If you were to go backwards, if, say, we didn't know what 2 to the 0 power is and we were just trying to figure out what would make sense, well, when we go from 2 to the third power to 2 to the second, we'd be dividing by 2. We're going from 9 to 4. Then we'd divide by 2 again to go from 2 to the second to 2 to the first. And then it seems like we should just divide by 2 again from going from 2 to the first to 2 to the 0. And that would give us 1.