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Nothing more to my question, really. I just wonder if the words share an etymological root. Thanks.

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    With all due respect, I fail to understand why this question should be put on hold. Doesn't the website have a whole category for etymology? Your decision seems rather whimsical.
    – asef
    May 14, 2016 at 22:47

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

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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.

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