Hi
Oboist here who's just had two root canals done and is still managing to play the oboe perfectly well, although I have to say mine were on molars not front teeth (which is, I assume, where yours has to be done?).
The process isn't entirely pleasant (what is at the dentist

?) but it's nothing to get worried about. I find that things are a bit tender for 24-48 hours afterwards but then it all quietens down and you'd not know anything had been done. Local anesthetic will almost certainly be used too which will wear off quickly.
I'm sure you'll be just fine for your exam therefore but do make sure your dentist knows you play the clarinet (I assume he/she already does) and about the exam. A root canal can sometimes need more than one appointment to sort and you'll need that to be planned around the exam. However, kerioboe's view is reasonable. If the exam's only too weeks away, can you postpone the dentist until after the exam is over?
Best of luck with dentist and exam!!
Oboist