Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

What is 0 to the power 0?

I assume you are familiar with powers.

The problem is similar to that with division by zero. No value can be assigned to 0 to the power 0 without running into contradictions. Thus 0 to the power 0 is undefined!

How could we define it? 0 to any positive power is 0, so 0 to the power 0 should be 0. But any positive number to the power 0 is 1, so 0 to the power 0 should be 1. We can't have it both ways.

Underlying this argument is the same idea as was used in the attempt to define 0 divided by 0. Consider a to the power b and ask what happens as a and b both approach 0. Depending on the precise way this happens the power may assume any value in the limit.

Fine print, your comments, more links, Peter Alfeld, PA1UM