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.
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 ab 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.
both of which get close to zero as n increases then you get
The larger n, the closer ab gets to 1/2 so maybe 00 should be 1/2.