I think you'll find more people would prefer "I've already taken the exam" or "I took the exam last week."
Americans are more likely to say "I already took the exam" than other English users, who would say "I"ve already taken it.' (In fact, I expect a BrE speaker may pop in to say "I already took it" to be unacceptable in BrE.)
However, you can't use the present perfetct (have t