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## College Algebra

### Course: College Algebra>Unit 7

Lesson 1: Exponent properties (integer exponents)

# 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.

## 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.
• 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!
• Why is 0 raised to the power of a negative number undefined
• 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.
• 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 :)?
• 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.
• Anyone else just scroll through comments laughing and commenting for fun? :)
• heck ya
• 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
``(0^1) / (0^1)``
, or
``0/0``
`, which is *_∞_ (UNDEFINED)*. Is that the logic that certain mathematicians used do say, "
``0^0  = ∞``
" ?
• 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.
• I would think that zero to the negative 1 would be undefined as it would be the same as 1/0 isn't it?
• 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.)
• 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
• 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!
• What if -1 to the power of 0?

The TI-84 Plus CE is saying it is -1, but I am not sure.
• 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!
• Would -1/2 to the zeroth power be 1 or -1?