Words with “rate” in them. There are kind of a lot. Like, three parts a lot. Which I’m fine with since it saves me from having to come up with new ideas.
Just plain rate first showed up in the early fifteenth century as a noun and a little later as a verb, both coming from the Old French rate, price or value. Before that, it was the Medieval Latin rata pars, fixed amount, and classical Latin rata, which has meanings like ratified, custom, firm, and fixed. Rata is the past participle of reri, to think, which is the origin word for reason and ratio. Reri can actually be traced even further back to the Proto Indo European re, to reason or count, and also the origin word for read. So basically, rate has a big extended family. Also, there’s another version of rate that showed up in the late fourteenth century as, seriously, to scold. That’s actually where the word berate comes from, and it’s descended from the Old French reter, blame, and classical Latin reputare, which shouldn’t surprise you as the origin word for reputation.
Next, we’re going to look at separate. It showed up in the late fourteenth century from the classical Latin separatus and separare, both of which are just different tenses of separate. That word is a mix of a little used prefix se-, which means apart and parare, prepare. Well, that was kind of boring.
Operate is fairly recent, having showed up in the early seventeenth century, although operation showed up over two centuries earlier. And of course neither word had anything to do with surgery or operating a machine. Back then, it was just an action. Operation comes from the Old French operacion and classical Latin operationem, which means operation, but only in the in action sense.
Finally today, we’re looking at corporate. It showed up in the early fifteenth century meaning united as one body, coming from the classical Latin corporatus and corporare, which both just mean corporation and corporate. They of course come from corporeus (physical) and corpus (body), the origin words for things like corpse and corporeal. Those words can be traced back to Proto Indo European, where he word is kwrep- (don’t ask me how to pronounce it), body or appearance.
TL;DR: -Rate words aren’t related.