Zero to the Power of Zero

What is 00? On one hand, any other number to the power of 0 is 1 (that's the Zero Exponent Property). On the other hand, 0 to the power of anything else is 0, 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
ab × ac = a(b + c)

Let's let a = 0, b = 2, and c = 0. Substituting, we have:

02 × 00 = 0(2 + 0) = 02

We know that 02 = 0. So this says

0 × 00 = 0

Notice that 00 can be equal to 0, or 1, or 7, or 99,999,999,999, and this equation will still be true!

For this reason, mathematicians say that 00 is undefined.