Please check if I have done any mistakes in solving the equation $$x^{1/x}=0$$
Taking the natural logarithm on both sides, we get $$\frac{1}{x}\ln{x} = -\infty \tag1$$
On further simplification, the equation becomes $$-\ln{x} \cdot e^{-\ln(x)} = \infty \tag2$$ and taking the Lambert $W$ function on both sides we get $$-\ln{x} = \infty \quad\text{or}\quad\ln{x} = -\infty \quad\text{or}\quad x = 0 \tag3$$ which means that $$0^{1/0}=0 \tag4$$
However, we can't conclude that $0^0 = 0$.
I also checked WolframAlpha which says $\sqrt[0]{0} = 0$. If I have made any mistakes, please correct me.