proof verification - 0 to the power of any number

I have proof that $0^n$ = undefined.

Since,
$2^5 = 32$,
$2^4 = 16$,
$2^5/2 = 32/2 = 16 = 2^4$.

Similarly if $0^n = 0$.

Then,
$0^{n-1} = 0$
$0^0/0 = 0/0 = 0^{n-1}$.

But $0/0$ is undefined.
Therefore $0^n = 0$.

But calculators give the result of $0^n$ as $0$. Can you explain where I am going wrong.