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Lesson 8: 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.
• For those still confused, hopefully this helps:

In terms of the number 2, think of going forward as multiplying 2 by 2 each step, and going backwards you are DIVIDING 2 by 2 each step.

With that in mind:

EXPONENT NUMBER LINE
⟵ ⟵ ⟵ ⟶ ⟶ ⟶
-3 -2 -1 0 1 2 3

Forwards (right direction):
2¹ = 2
2² = 2*2
2³ = 2*2*2

to get from 2³ to 2² we need to DIVIDE by 2
2*2*2/2 = 2*2

to get from 2² to 2¹ we need to DIVIDE by 2
2*2/2 = 2

So what is 2⁰?
Well, to get from 2¹ to 2⁰, we follow the same pattern or logic of DIVIDING by 2.

2⁰ = 2/2
2 cakes divided among 2 people, how many cakes each person gets? That's right, one!
2⁰ = 1

You can continue this logic into the negative and you would get this pattern:

2¯¹ = 1/2
2¯² = 1/2/2/2
2¯³ = 1/2/2/2/2

Hopefully that makes intuitive sense!
• Anyone else just scroll through comments laughing and commenting for fun? :)
• heck ya
• is it just me or do the teachers over explain it to make it confusing
• would the exponents be consider as the absolute values?
• 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?
• i think it is in every exponent equation but it wont make a difference in those but in these you do need the one.
• 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?
• Also wird 0 hoch zu jeder Potenz einfach Null sein