Rational solutions to $e^x - \frac{1}{2} = \sqrt{ x^3 + 1/4}$ Show that the only rational solution to the title curve is $x=0$. 
My attempt: Squaring both sides we have $e^{2x} - e^x + \frac{1}{4} = x^3 + \frac{1}{4}$, which yields $e^{2x} - e^x = x^3$. I suspect that the left hand side is irrational for any rational $x\neq 0$, and if so, we reach a contradiction and conclude that the only possibility is $x=0$.

We can write equation as $$e^x(e^x-1)=x^3$$ therefore $e^x-1=\frac{x^3}{e^x}$  now we know that $e$ is irrational(2.781...) so any value of $x$ else than $0$ makes the the equation irrational so $0$ is the only rational solution.
