We say "Powered by Microsoft"; can one also say "Microsoft powered" instead of this?
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It depends. Is this a legal question or a language question? If your contract with Microsoft stipulates that you use certain language, then that's what you must use. Otherwise you can say what you want. If you are going to use the adjectival version, though, I would hyphenate it: Microsoft-powered instead of Microsoft powered.
Again, though, there may be legal constraints. Check with the appropriate people in your company who handle that sort of thing.
Don't think they're equivalent.
You say "battery-powered" (note the hyphen), but you never (at least I didn't) hear "powered by battery".
"X-powered" means "X" being the actual source of (electrical in the case of batteries) power.
While "powered by X" would make me suspect, even if I'd never heard of X, that X is a company or a technology that's behind X or that makes X possible. I wouldn't (as long as I had no reason to from context) expect this to refer to an actual power source.