# Zero to the Power of Zero

What is ${0}^{0}$ ? On one hand, any other number to the power of $0\text{\hspace{0.17em}}$ is $1\text{\hspace{0.17em}}$ (that's the Zero Exponent Property ). On the other hand, $0\text{\hspace{0.17em}}$ to the power of anything else is $0\text{\hspace{0.17em}}$ , 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 ${a}^{b}×{a}^{c}={a}^{\left(b+c\right)}$

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

${0}^{2}×{0}^{0}={0}^{\left(2+0\right)}={0}^{2}$

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

$0×{0}^{2}=0$

Notice that ${0}^{0}$ can be equal to $0\text{\hspace{0.17em}}$ , or $1\text{\hspace{0.17em}}$ , or $7\text{\hspace{0.17em}}$ , or $99,999,999,999\text{\hspace{0.17em}}$ , and this equation will still be true!

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