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

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

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 .