Hi Peter,
I'm not sure we can answer your question but we can give you some information. We have received what is essentially this question before but then it was "I am a middle school teacher who is looking for a precise explanation of why zero raised to the zero power is undefined." The answer to "the question", regardless of which way you ask it depends on what you mean by m^{n} in situations where not both m and n are zero.
One meaning of m^{n} is in the response I received to your question from Claude.
The "advanced combinatorial way'' is that m^{n} counts the number of functions from a set with n elements to a set with m elements. The empty function is the only function from the empty set to itself, hence 0^{0} = 1.
This is somewhat abstract and esoteric but it is a useful way to interpret m^{n}, especially for mathematicians whose interest is combinatorics or counting. But what if m or n is not an integer? (I am going to insist that m can't be negative for otherwise I am faced with, for example (2)^{1/2} which is either undefined or I have to move to the world of complex numbers.)
One way to attempt to answer your question is to take a dynamitic approach. That is to evaluate m^{n} where m and n are close to zero, let m and n get closer and closer to zero and watch the value of m^{n}. The hope is that as m and n approach zero, m^{n} will approach some value and I can say "this is the value I should take for 0^{0} ". This is a calculus approach to the question and is the approach that Penny took when she responded to "I am a middle school teacher who is looking for a precise explanation of why zero raised to the zero power is undefined."
Another of my consultants, Sue, pointed me to more information on this "debate" on a web page at the University of Waterloo.
So we don't have an answer to your question or even a definite answer as to whether 0^{0} should be 1 or undefined. Contrary to what many people think there is not always one definite right answer to a mathematical question.
Harley
