This may be unpopular, but I would only suggest asking for an exam extension under
four five conditions:
- A medical emergency
- A death in your immediate family
- A true scheduling conflict (usually occurs only during finals) with another exam
- The instructor changes the exam date that they posted on the syllabus and it causes a conflict with a professional obligation that you scheduled based on the syllabus
- Your work was accepted to a conference or seminar after the beginning of the semester that conflicts with the exam date (edit based on suggestion from ff524)
For point 3, I would still suggest only going to the instructor that you have a better personal relationship with. If it is back to back exams, then you are expected to manage your time and energy effectively at this point in your academic career. Some professor's are very clear that they are there to help and accommodate their students, but if yours has not given those overtones, then I would hesitate.
For point 4, the syllabus is a contractual arrangement with you as a student. It outlines the obligations of the course (your contractual obligation) and the commitments of the professor (their contractual obligations), one of which is keeping to the schedule barring unforeseen circumstances.
Understand that when you made the choice to do graduate work, you were making a decision about your career, one that should be extremely important to you. As a result you have a professional obligation to meet the requirements and conduct yourself in a professional manner.
If you were working in the corporate sector and had a major deliverable on a project you were working on, or you are a key member in the team that makes the sales or investor pitch that could make your start-up succeed or fail, would you go to your boss and ask for the time off or turn to the customer or venture capitalist and ask to change the meeting for this situation?
Why then would you treat an exam, which is part of the career obligations you chose to undertake, any differently? Also remember you are asking the professor to do extra work for your benefit. They may need to write a different exam for you.
All that being said, if you are in a graduate program (which you are technically supposed to be if you are asking on this site), then you should know people in your department fairly well and you should be able to get a feel for how amenable the professor will be to your request. Good personal lives make for successful careers, but just remember what that request is saying. These decisions are not easy.
EDIT: Regarding Comments
I think many people are trying to rationalize with institutional "rules" about exams or how nice of a person the professor might be. One of the best professors I had rescheduled our final for a number of us who had a difficult exam before her's and gave us the exam the day before she was leaving the country for an extended holiday. But it was her belief that she wanted to give us the best opportunity to demonstrate what we had learned and effort we put in and not what the results of a scheduling decision by the registrar were. But she in no way had to be that flexible or generous with her own free time.
The point I am trying to make with my post is about personal responsibility and conducting one's self professionally. And to do that, you need to chose obligations and needs over wants. Whether or not the school's policy allowed for it, it is unprofessional to expect the professor to have to do more work so that you can go on vacation. Most professor's have a host of other professional responsibilities, and you are either cutting into their professional time to do research or you are cutting into their personal time to do what ever it is they would do with the time it takes to figure out how they will accommodate your request, including likely rewriting the exam, administering the exam, or finding someone else to proctor the exam, all for something that is your obligation, to show up for a scheduled exam and be prepared.
There are certain things that trump that, and that list is above as they are about the obligations and needs for the student's well being and only someone completely selfish and lacking of compassion would begrudge the request, and that includes outside work obligations in a job that supports the student that were made prior to the syllabus being issued.
But for something that, for all intents and purposes, is a holiday, as the student would have no official responsibilities at the conference as they are not the one invited to present their work is not a professionally acceptable reason. Vacation time is scheduled in between semesters, and as a student, one should accept that it is their professional responsibility to meet those scheduling requirements, especially on the graduate level.