0 to the power of any number

Question

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.

Answer 1

The rule $x^{n-1}=x^n/x$ requires that you divide by $x$, which you can't when $x=0$. The rule comes from $$ x^n=x^{n-1} x, $$ and then dividing by $x$. But if $x$ is zero you cannot divide, and so the rule does not apply.

Answer 2

To calculate $0^{n-1}$, you divided by $0$, which is undefined. Notice that $0^n=0\cdots0=0$ (with $n$ factors); there is no division involved here.