Can you use a power drill with the exchangeable screwdriver bits on a manual screwdriver?
Screwdriver bits are generally considered consumable. Regardless of the brand, they eventually break/wear out. Hardware stores sell them in packs of 5-10 for exactly that reason. I wouldn't put too much stock in what kind of tool the bit originally came with.

The only exception is if you're using an impact driver you might want to be a little more picky. Impact drivers (or impact wrenches) put a huge amount of torque on the bit, and a cheap one could fail catastrophically. There are special "impact driver bit sets" you can buy that advertise high torque strength.