Why connect the hood in the bathroom?
But there should be a hood, there are even wires, but for some reason there is no current on them, well!
Please help, either somehow in the current situation with these wires, or some other option with the hood connected to the bathroom. The only thing that comes to my mind is to put the cable on the tile, close it with a cable car and plug it into a socket, put the breaker on when you need it, but agree - it would be as unaddressable as possible to lead a cable car over the tile and then wind it into the ventilation shaft ... Therefore, I and here you are knowledgeable people, professionals, maybe you know how it was possible to mount all this or to make some kind of decision. Thanks in advance!
In any case, you will have to disassemble the ceiling or part of it. What kind of ceiling is made? You can also check if there is access and what happens in the junction boxes closest to the bathrooms. I cannot give an exact answer to the remaining questions “why there is no current in the wires,” because it is unrealistic. By the way, if the lamp is built into the ceiling - try to remove it, what if these wires pass through the lamp? If 3 wires, then it is quite possible ...
What is cable cab? Cable channel? I agree that this would not be aesthetically pleasing, but also to drive the cable into the ventilation ducts.