Zero to the Power of Zero

What is 0 0 ? 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
a b × a c = a (b+c)

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

0 2 × 0 0 = 0 (2+0) = 0 2

We know that 0 2 =0 . So this says

0× 0 2 =0

Notice that 0 0 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 0 0 is undefined.