at my school, a person can only take an exam on a different date if they have a conflict with another exam given on the same day. And the person must take it the next day. Personal conflicts, at least at my school, do not give a student any right to take an exam at a different time than it is given. However, I do know that what is in the school guidelines, and what can actually be done is different. I would suggest to talk to the dean of academic affairs, explain your situation, see what can be done. I wouldn't go to your prof. that u have the exam with, because obviously, they will be biased toward you taking their exam.