EX: Do you wind your watch, or does it have a battery?
"battery-powered" (hyphen optional) sounds odd if used to describe a watch, but "battery-operated" sounds fine to my North American ears. "powered" implies power; watched don't really have any power. "run on a battery" sounds OK but it's iffy. It's the "run" part. Again, "power" is implied. All in all, "have a battery" works well.
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