A student asked me this exercise of a past exam:
Find the smallest solution to the equation: $$\exp(\sqrt{2x})=16x^3-1.$$ Give the anwser up to $4$ decimal places.
Is there a way to find an exact solutions? I tried to substitute $y=\sqrt{2x}$, obtaining the equation $e^y=2y^6-1$ but that leads to nothing. Also tried a bunch of other substitutions that don't work.
I believe that this equation cannot be solved exactly using elementary methods, but I'm not sure.