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^{0} = 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**.