Show $e^x$ is irrational for rational $x \neq 0$

Question

I want to show that if $x$ is rational and nonzero then $e^x$ is irrational.

Clearly $e^{\frac{r}{s}} = \frac{p}{q} \Rightarrow q^s e^r = p^s$, but this doesn't seem helpful. The usual proof that $e$ is irrational doesn't look like it can be extended either.

Answer 1

You are on the right track.

Knowning that $e$ is transcendental, the algebraic equation $q^sz^r-p^s=0$ cannot have $e$ as a root.