Should I study with the TV turned on?
Probably not it's a huge distraction potential and maybe information overload to do both.
I find when I am studying or reading on the web my comprehension and retention is significantly better without the TV on in here but probably not as important to have it off with casual web surfing or social media use .
There is a good reason we try to make library's quiet places to study and work or use the web or reference / study media .
At some point multitasking is just not doing anything well and just trying to do more at one time period instead of time /task managing well and scientific studies back that argument .
This depends entirely on your ability to study with the TV turned on. Most people would say that the visual distraction of the TV will probably inhibit your ability to study. If you need something else going on in the room, radio is probably the better option.