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.
Salina

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 0^x = 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!

Cheers,
Walter and Harley