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

$a^b\times a^c=a^{(b+c)}$

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

$0^2\times 0^0=0^{(2+0)}=0^2$

We know that $0^2=0$ . So this says

$0\times 0^2=0$

Notice that $0^0$ can be equal to $0$ , or $1$ , or $7$ , or $\mathrm{99,999,999,999}$ , and this equation will still be true!

For this reason, mathematicians say that $0^0$ is undefined .