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

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 $0\text{\hspace{0.17em}}$, or $1\text{\hspace{0.17em}}$, or $7\text{\hspace{0.17em}}$, or $\mathrm{99,999,999,999}\text{\hspace{0.17em}}$, and this equation will still be true!
For this reason, mathematicians say that ${0}^{0}$ is undefined.