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Lesson 5: Exponents with negative bases

# 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.
• is it just me or do the teachers over explain it to make it confusing
• Is 0^0 one or zero? Technically it could be 1, because 0^1 = 1*0, so 0^0 must be 1 with no zeros to multiply it to. It could also be 0, because you do not have to add the 1 * at the beginning. Would that mean 0^0 is nothing?
• Yeah, `0^0` is indeterminate (since there is more than one possible value for it). Same with `0/0` (here, it could be any real number)
• Big thanks to Khan Academy for teaching me that (a^b)^c = a^(b*c).
Now if someone could explain why my calculator keeps trying to fight me on this...
• If you're using a scientific calculator don't forget the parentheses. Enter the numbers in the form `(a^b)^c`, not `a^b^c`. `a^b^c` is read top exponent first, as `a^(b^c)`!
• I hear what he's saying, I just don't understand it. You always have to include a 1 before you start multiplying a number with itself. Is it required with every exponent equation or, is this just a different method he's showing us?