grammaticality - Is this a complete sentence?


There was an ad on telly I saw, saying



Relax, knowing your home is safe



Is this a complete sentence that is grammatically correct? Could this go in an essay? What is the technical word for "knowing"? I know it's a verb. Would you call a present continuous verb and that's it?


What about this sentence



Become part of this website, exploring new worlds and games



Is this an okay sentence?




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