HINT: I expect that you know that a number is rational if and only if its decimal expansion is eventually periodic. Suppose that the expansion eventually repeats with period $p$.
- Show that there are two consecutive powers of $2$ whose after the initial aperiodic segment whose lengths (when written in the usual way in base ten) are the same multiple of $p$.
I’ve left a further hint in the spoiler-protected block below; mouse-over to see it.
Show that on the one hand these two powers of $2$ must end in the same digit, and on the other hand that they cannot end in the same digit.