Solve $x^{\frac{1}{x}}=0$

Question

When answering this question, I contemplated across the following problem:

Find $x$ such that $x^{\frac{1}{x}}=0.$

I thought RHS $ = 0$ is a special case, so I attempted solving $$\frac{1}{x} \ln x = \ln c$$ from some constant $c > 0,$ but I do not know how. I heard about Lambert W, but I am not familiar with it.

BTW, Maple solve(x^(1/x) = 0) gives $x=0$, and WolframAlpha gives (no solution exist)!

How can I solve this?

Answer 1

The only way you can ever have $a^b=0$ is to have $a=0$, but in the current case this leads to $0^{1/0}$, which is undefined. So, there's no solution.

Answer 2

The functin $f(x)=x^{1/x}=\exp(\frac{1}{x}\log x)$ never equals zero, however the limit as $x\to 0^+$ evaluates to zero, so if we consider the domain $[0,\infty)$ we could remove the discontinuity at $x=0$ if we desired.