Do "empirical" and "imperial" share a common etymology? [closed]

Solution 1:

No: imperial is from Latin imperium "power". Empirical is from Greek empeiros "experienced", from peira "attempt".

Solution 2:

Emperor is from Latin imperator, a commander, from imperare, to order. A related word is imperative as in the grammatical term imperative voice, i.e. the voice in which commands are given. "Empire", "emperor" came from this via Old French.
Empirical is from Greek experienced by way of Latin empiricus: no relation to imperator.