I am a middle school teacher who is looking for a precise explanation of why zero raised to the zero power is undefined. I am hoping to get an explanation using something other than the fact that diividing by zero is undefined.
Thank you!!

Hi,

We have had this question before and answered it but I want to expand on the answer. I want to start with a⁰ where a is a positive number. To see what this "should be" my technique will be to calculate a^b for values of b that get close to zero. One way to do this is to calculate

Try it. Choose some positive number and use your calculator to take successive square roots. The sequence of numbers you get approaches 1. This is the type of reasoning that convinced mathematicians to use the convention that a⁰ = 1 if a is positive.
   Trying this same technique with a^b leads to confusion. If you calculate 0^b for values of b that get close to zero you always get 0^b = 0 which might lead you to suggest that 0⁰ should be 0. However if you calculate a⁰ with a positive and close to zero you always get a⁰ = 1 which might lead you to suggest that 0⁰ should be 1. If you start with a and b both not zero but getting close to zero you can produce an argument to say that 0⁰ should be anything you want. For example if

both of which get close to zero as n increases then you get

The larger n, the closer a^b gets to 1/2 so maybe 0⁰ should be 1/2.
   It is this ambiguity that had led mathematicians to use the convention that 0⁰ is undefined.

Cheers,
Penny