Learn Finnish: Eikö opiskelija voi läpäistä tenttiä näkemättä vaivaa? - Can't a student pass the exam without making an effort?

It comes from the verb nähdä (to see). To make the "without doing" form (the third infinitive abessive), you take the strong stem of the verb—which is näke-—and add the ending -mättä (or -matta, depending on vowel harmony). Literally, näkemättä means "without seeing".

If näkemättä means "without seeing", why does the translation say "without making"?

This is because of a very common Finnish idiom. The phrase nähdä vaivaa literally translates to "to see trouble" or "to see effort", but in English it means "to make an effort". Therefore, näkemättä vaivaa means "without making an effort".

Why is the object tenttiä (exam) in the partitive case here?

The sentence starts with Eikö, which makes the entire main clause negative. In Finnish, if a sentence is negative, its object must be in the partitive case. If this were a positive statement ("The student can pass the exam"), it would take a total object: Opiskelija voi läpäistä tentin.

How does the word Eikö work to form the question?

It is made of the negative verb ei (used here for the third-person singular subject, opiskelija) and the question ending -kö. Adding a question ending to a negative verb is how you ask "Doesn't...?" or "Can't...?" in Finnish.