Zero to the Power of Zero
What is ? On one hand, any other number to the power of is (that's the Zero Exponent Property ). On the other hand, to the power of anything else is , because no matter how many times you multiply nothing by nothing, you still have nothing.
Let's use one of the other properties of exponents to solve the dilemma:
Product of Powers Property
Let's let , , and . Substituting, we have:
We know that . So this says
Notice that can be equal to , or , or , or , and this equation will still be true!
For this reason, mathematicians say that is undefined .