Does the zeroth root exist? Definition of $Nth$ root:
$3rd$ order inverse group $1$ hyperoperation.
Division is how many times you can subtract a certain divisor from the dividend before it becomes negative.
Likewise Nth root is the result of repeated division by a certain divisor before it becomes $1$ or a decimal. The number of times you divide it before it becomes a decimal is the index.
Ex: $\sqrt [3]{8} = (8/2)/2$
Is the zeroth root even defined and if so what is $\sqrt [0]{x}$

The $n$-th root is the inverse of the operation $x \mapsto x^n$. (Let's take $x>0$ to avoid complications.)
For $n=0$, the operation $x \mapsto x^0$ is not invertible, since it is a constant function.
So, there is no $0$-th root operation.
