Sender: "Salina Young"
Subject: About the power of "0"

Hello! I have two questions to ask about the power of "0".

First, what is the value of "0' to the power of "0"? And why?

Second, what is the value of "0" to the power of "2" or '3" etc? And why?

Thank you for your help.

Hi Selina,

02 is easier, 02 = 0x0 = 0. Similarly 0n = 0 for any positive integer (or even postive fraction).

0-1 = 1/0 is undefined (as is any non-zero number divided by 0). Similarly 0-n is undefined for any positive integer n.

That leaves the 'middle case' 00 somewhere between 0 and undefined.

If you got to 00 through some kind of sequence of values - you look at the limit of that sequence. For example 0x = 0 for any positive x, so as x gets very close to zero it seems to indicate that 00 should be zero. Also x0 = 1 for any x that is not zero so as x gets close to zero it seems reasonable to say 00 is one. There are many answers depending on how you got there. The convention in math is that this is not well defined, so in general, no answer is given. In short 00 is undefined. In any particular problem if you want to know more you have to tell me more about where this came from!

Walter and Harley


Go to Math Central