Main content

## Algebra basics

### Unit 1: Lesson 3

Exponents- Intro to exponents
- Exponent example 1
- Exponent example 2
- Squaring numbers
- Intro to exponents
- The 0 & 1st power
- Powers of zero
- Meaning of exponents
- 1 and -1 to different powers
- Comparing exponent expressions
- Exponents of decimals
- Powers of whole numbers
- Evaluating exponent expressions with variables
- Variable expressions with exponents
- Exponents review

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# Powers of zero

CCSS.Math:

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.

## Want to join the conversation?

- there is no such thing as +0 or -0 right? my friends think that and i wasn't for sure.(177 votes)
- Hi! There Isn't such a thing as positive zero or -0. Zero is an undefined number, meaning that it is not - or +. I hope this helps!(221 votes)

- Why is 0 raised to the power of a negative number undefined(8 votes)
- 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.(13 votes)

- 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 :)?(5 votes)
- 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.(8 votes)

- Good video. However, you also mentioned that
*0^x=0**(where x is any non-0 real no.)* and i think that is wrong because NEGATIVES are also real numbers. in my opinion,`0^(-x)= ∞`

`because that becomes`

`1/(0^x)= 1/0 = ∞`

`, right? Also, whenever we reduce powers, we divide the power by`

*(base^1)*. Thus, to obtain *`0^0`

`*, we have to do`

, or`(0^1) / (0^1)`

`, which is *_∞_ (UNDEFINED)*. Is that the logic that certain mathematicians used do say, "`0/0`

" ?`0^0 = ∞`

(5 votes)- I agree with you that 0^x=0,where x is any non zero real number seems a bit odd because when we raise zero to negative powers the result is undefined.

Secondly,you stated that 1/0=**infinity**,the logic that makes you conclude the result is infinity,I am assuming,is that if 1/x=y then the smaller the x,the greater the y,this can be mathematically stated using limits(I'm avoiding the mathematical statement for simplicity) "the limit of 1/x as x approaches 0 equals infinity",but this statement(whether in English or in math) can be contradicted if you approach 0 from the negative side(1/-1=-1,1/-0.5=-2,1/-0.25=-4),because then when you take values closer to zero you start getting closer to**NEGATIVE INFINITY**,since you get 2 answers for 1/0(+infinity and -infinity) mathematicians have left 1/0,undefined.(2 votes)

- I would think that zero to the negative 1 would be undefined as it would be the same as 1/0 isn't it?(4 votes)
- Correct, and all powers of zero, because you're basically multiplying 0*0*0 or whatever, end with the answer of zero. (All possible ones anyways without going into super complicated math.)(3 votes)

- I have a simple question.... so if anything is to the power of 0 then it would just be 0? I would appreciate it if someone had answered quickly so please(3 votes)
- 0⁰ is
*indeterminate*or*undefined*

Any other number to the power of zero is 1. For example:

1⁰ = 1

5⁰ = 1

20⁰ = 1

100⁰ = 1

(-8)⁰ = 1

(-49)⁰ = 1

(-212)⁰ = 1

Hope this helps!(4 votes)

- What if -1 to the power of 0?

The TI-84 Plus CE is saying it is -1, but I am not sure.(4 votes)- Any number to the power of 0 would equal 1. Any negative number to the power of 0 would equal -1, unless it is within parentheses, like (-1)^0 which would instead equal 1. Hope that helps!(2 votes)

- Would -1/2 to the zeroth power be 1 or -1?(3 votes)
- -1/2^0 would be 1. A random number "n" to the zeroth power is always 1.(3 votes)

- then why is 10 to the power of zero equal 1?(3 votes)
- Think of it this way:

If you have already learned dividing exponents, you know that it uses subtraction. You also know that something divided by that same something is just one. Let's say we have x^5/x^5. This would be equal to x^5-5 = x^0. But since those are both the same, they cancel out and make 1.(3 votes)

- what would be 0 to -9 power be?(3 votes)
- Interesting question!

0^(-9) = 1/(0^9) = 1/0, which is undefined.

In general, 0 to any negative power is undefined.(3 votes)

## 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.