"This week I am unhappy because I have eight exams."
Translation:Bu hafta mutsuzum çünkü sekiz tane sınavım var.
"Sınavlarım" means "more than one exam". But you've already said "eight", so that would be like saying: "I have eight exams, but make no mistake! I have more than one!" -- Yeah we kinda understood that.
We don't use the plural when we use a number.