Zithromycin or a Z-pack is a common antibiotic used by us dentists. You really shouldn't have a problem with it. I was wondering why your dentist just didn't prescribe amoxicillin or penicillin? Are you allergic to them? Anyways it just a 4-5 day course, not too long. I agree with previous posts, double up on the probiotics. If you have any specific questions please feel free to post them.
Some dentists will prescribe antibiotics before or prior to a root canal to decrease the amount of infection that is there. Don't forget that a root canal is done to treat an infected tooth. There is often pus and abscess formation associated with these kinds of situation. The antibiotics will lessen the pain because it decreases the pressure of the infection which is the cause of the pain in the first place. Second the effect of the local anesthetics is diminished when there is an active infection. Hence you will hear of stories of people who were given numerous "shots" of novacaine and it still hurt. Third, once the root canal is completed often dentists will prescribe antibiotics to get rid of any lingering infection.
Hope that helps.
Just to clarify my answer because I believe it to be true. In my opinion, both as a dentist and a person, I do believe that doctors over prescribe antibiotics which does more harm than good in the short and long run. No doubt about that from my point of view.
If patients are in a lot of pain often antibiotics will be prescribed for teeth that need root canals. This lessens the pain and deals with the problem quickly because the medication is treating a true infection. If a patient doesn't have that much pain or no pain often we will not prescribe antibiotics. It's as simple as that. Yes there are nuances but we don't need to go over that at this point.