IMHO if it's for a senior engineer position, you should at least be able to explain how that would be implemented (bitwise operations) even if you wouldn't get it right in a whiteboard.
I know how bitwise operators works and have a CS degree... But even with googling I couldn't find a straight forward way to implement a general exponential operator using only bit operations.
Sure, but that's not a very satisfying answer because it doesn't rely on any special insight about how exponention works. It would be better to just ask someone to code up addition and/or multiplication using bitwise operations if that's what you want to test. That's sort of a reasonable question in the case of addition. Asking someone to implement multiplication using bitwise operations in a phone interview would be pretty crazy, though.
You're not talking about an engineering problem, first of all (assuming it's a software implementation being talked about, of course). Secondly, it's difficult for me to see how this particular problem is a good motivation for discussion about the use of bitwise operations in general. I've written code in low level game engines, real time computational systems, physics and for drivers. The first and last items are the only ones where bitwise operations of this sort were even necessary to consider.
Unless this problem is for a job working on calculations on hardware that only supplies bitwise ops, it's a math trivia problem disguised as a coding interview question.
I know how bitwise operators works and have a CS degree... But even with googling I couldn't find a straight forward way to implement a general exponential operator using only bit operations.