If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

1 and -1 to different powers

Use odd and even rules to determine the sign of an exponential expression. Created by Sal Khan.

Want to join the conversation?

  • aqualine ultimate style avatar for user SH
    If you're not doing a number to the zeroth power, is the starting 1 really necessary?
    (11 votes)
    Default Khan Academy avatar avatar for user
    • old spice man blue style avatar for user Cave Johnson
      Yes, it's actually necessary!

      We know that a multiplication equation multiplied by 1 is just the same. It's the same for exponents.

      Let's try 3 for example:

      3^1 = 1 * 3
      3^2 = 1 * 3 * 3
      3^3 = 1 * 3 * 3 * 3
      3^4 = 1 * 3 * 3 * 3 * 3
      3^5 = 1 * 3 * 3 * 3 * 3 * 3

      See the pattern? So what will 3 to the zeroth power be:

      3^0 = 1

      Hope this helps!
      (15 votes)
  • piceratops ultimate style avatar for user Geovany Diaz
    At how does 1^0 equal 1?
    (9 votes)
    Default Khan Academy avatar avatar for user
  • starky sapling style avatar for user Max Rachman
    Wait if he did 2^4 it also means 2x2x2x2 and so if that is it why is there to put a 1 it is just going to be the same answer
    (4 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Leonardo Padro
      There are two different ways to "think" of the calculation of the exponent.

      The first is to multiply the number by itself as many times as the exponent says to do so. Example:

      5^3 is calculated as: 5x5x5=125

      The other way to picture the calculation of an exponent is to start from the number one and then multiply as the exponent says to. Example:

      5^3 is calculated as: 1x5x5x5=125

      This doesn't impact the process much. Only if you happen to be going to the power of zero, which is why everthing to the power of zero is 1. Although both proceses are correct which one is more correct is up for debate. As such untill we get a definitive answer best to learn both ways.
      (5 votes)
  • aqualine ultimate style avatar for user olmedopinedaa7201
    After a few days do you gt use to doig it
    (4 votes)
    Default Khan Academy avatar avatar for user
  • female robot grace style avatar for user Mariam Busari
    I still don't get why 2^0 = 1
    Is there any reason why you have to start with a 1 when multiplying exponents? Why not start with 2 or 10?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • marcimus pink style avatar for user Tyler
      I think the simplest way to understand it is this. Start by taking some powers of 2:

      2^2 = 4
      2^3 = 8
      2^4 = 16
      ...

      Notice, based on this, that it is pretty simple to go forwards from one power to the next. If you want to go from "2^2" to "2^3", just multiply by 2. For instance:

      2^2 * 2 = 2^3

      But what if we wanted to go backwards? For instance, what if we wanted to go from 2^4 backwards to 2^3? Simple: we divide.

      2^3 / 2 = 2^2

      So now, what if we wanted to find 2^0? Well, simple! Start with 2^1, and go backwards. How do we go backwards again? We divide!

      2^1 / 2 = 2^0

      And "2^1 / 2" is just 1. So that's why "1 = 2^0".


      And as for the other part of your question, the reason we start from 1 is because it makes the math work. For instance, if you said that "2^0 = 10", then "2^1" would be 10 * 2, which would be 20. And "2^3" would be "10 * 2^2", which would be 40. And if we do that, then what's the point of exponents? Exponents are supposed to be used to multiply the number by itself. But if we throw a 10 in there, then exponents are pretty useless. However, it is okay for us to multiply by 1, because multiplying by 1 does nothing; it doesn't change anything. So that's why we always start with 1.

      Anyways, I hope this helped a bit.
      (6 votes)
  • marcimus pink style avatar for user Ashna
    how would we solve 10^-4?
    does that mean -10x-10x-10x-10 or 10x-10x-10x-10 or neither?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • female robot ada style avatar for user 𝕴𝖟𝖟𝖞
    What if you multiply or square something by one and its a negative one would the outcome be negative
    (1 vote)
    Default Khan Academy avatar avatar for user
    • sneak peak green style avatar for user Samuel Baer
      If you raise a negative number to an even power you get a positive number.
      And if you raise a negative number to an odd power, you get a negative number.

      (-1)^1 = -1
      (-1)^2 = 1
      (-1)^3 = -1
      (-1)^4 = 1
      (-1)^5 = -1
      (-1)^6 = 1

      Hope that helps!
      (3 votes)
  • blobby green style avatar for user siddharth manoj
    Would one to the power of negative one be negative one?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • starky ultimate style avatar for user Warlover
    Is (-1)^infinity an indeterminate form? Or was it just 1^infinity that was an indeterminate form... because ln(-1) when we try and take the limit of it is undefined. So (-1)^k as k approaches infinity diverges?
    A series from K=0 to infinty of (-1)^k diverges because (-1)^infinty is infinity because it is not an indeterminate form? I am pretty lost how this diverges...I'm trying to learn properties of power series.
    (2 votes)
    Default Khan Academy avatar avatar for user
  • starky tree style avatar for user aaronreeves
    at isn't anything times 0=0
    (1 vote)
    Default Khan Academy avatar avatar for user

Video transcript

Let's think about exponents with ones and zeroes. So let's take the number 1, and let's raise it to the eighth power. So we've already seen that there's two ways of thinking about this. You could literally view this as taking eight 1's, and then multiplying them together. So let's do that. So you have one, two, three, four, five, six, seven, eight 1's, and then you're going to multiply them together. And if you were to do that, you would get well, 1 times 1 is 1, times 1-- it doesn't matter how many times you multiply 1 by 1. You are going to just get 1. You are just going to get 1. And you could imagine. I did it eight times. I multiplied eight 1's. But even if this was 80, or if this was 800, or if this was 8 million, if I just multiplied 1-- if I had 8 million 1's, and I multiplied them all together, it would still be equal to 1. So 1 to any power is just going to be equal to 1. And you might say, hey, what about 1 to the 0 power? Well, we've already said anything to 0 power, except for 0-- that's where we're going to-- it's actually up for debate. But anything to the 0 power is going to be equal to 1. And just as a little bit of intuition here, you could literally view this as our other definition of exponentiation, which is you start with a 1, and this number says how many times you're going to multiply that 1 times this number. So 1 times 1 zero times is just going to be 1. And that was a little bit clearer when we did it like this, where we said 2 to the, let's say, fourth power is equal to-- this was the other definition of exponentiation we had, which is you start with a 1, and then you multiply it by 2 four times, so times 2, times 2, times 2, times 2, which is equal to-- let's see, this is equal to 16. So here if you start with a 1 and then you multiply it by 1 zero times, you're still going to have that 1 right over there. And that's why anything that's not 0 to the 1 power is going to be equal to 1. Now let's try some other interesting scenarios. Let's start try some negative numbers. So let's take negative 1. And let's first raise it to the 0 power. So once again, this is just going, based on this definition, this is starting with a 1 and then multiplying it by this number 0 times. Well, that means we're just not going to multiply it by this number. So you're just going to get a 1. Let's try negative 1. Let's try negative 1 to the first power. Well, anything to the first power, you could view this-- and I like going with this definition as opposed to this one right over here. If we were to make them consistent, if you were to make this definition consistent with this, you would say hey, let's start with a 1, and then multiply it by 1 eight times. And you're still going to get a 1 right over here. But let's do this with negative 1. So we're going to start with a 1, and then we're going to multiply it by negative 1 one time-- times negative 1. And this is, of course, going to be equal to negative 1. Now let's take negative 1, and let's take it to the second power. We often say that we are squaring it when we take something to the second power. So negative 1 to the second power-- well, we could start with a 1. We could start with a 1, and then multiply it by negative 1 two times-- multiply it by negative 1 twice. And what's this going to be equal to? And once again, by our old definition, you could also just say, hey, ignoring this one, because that's not going to change the value, we took two negative 1's and we're multiplying them. Well, negative 1 times negative 1 is 1. And I think you see a pattern forming. Let's take negative 1 to the third power. What's this going to be equal to? Well, by this definition, you start with a 1, and then you multiply it by negative 1 three times, so negative 1 times negative 1 times negative 1. Or you could just think of it as you're taking three negative 1's and you're multiplying it, because this 1 doesn't change the value. And this is going to be equal to negative 1 times negative 1 is positive 1, times negative 1 is negative 1. So you see the pattern. Negative 1 to the 0 power is 1. Negative 1 to the first power is negative 1. Then you multiply it by negative 1, you're going to get positive 1. Then you multiply it by negative 1 again to get negative 1. And the pattern you might be seeing is if you take negative 1 to an odd power you're going to get negative 1. And if you take it to an even power, you're going to get 1 because a negative times a negative is going to be the positive. And you're going to have an even number of negatives, so that you're always going to have negative times negatives. So this right over here, this is even. Even is going to be positive 1. And then you could see that if you went to negative 1 to the fourth power. Negative 1 the fourth power? Well, you could start with a 1 and then multiply it by negative 1 four times, so a negative 1 times negative 1, times negative 1, times negative 1, which is just going to be equal to positive 1. So if someone were to ask you-- we already established that if someone were to take 1 to the, I don't know, 1 millionth power, this is just going to be equal to 1. If someone told you let's take negative 1 and raise it to the 1 millionth power, well, 1 million is an even number, so this is still going to be equal to positive 1. But if you took negative 1 to the 999,999th power, this is an odd number. So this is going to be equal to negative 1.