Learn Arabic (Modern Standard): جاء إلى مصر بهدف الدراسة. - He came to Egypt with the aim of studying.

The word مصر is typically treated as a diptote (a noun that doesn't take a tanween and takes a fatha instead of a kasra in the genitive case). So, in formal spoken MSA, you would say إلى مصرَ. Because it is a three-letter feminine name with a vowelless middle letter, Arabic grammar actually allows it to optionally take a kasra, but the fatha is much more commonly used.

What is the exact function of the prefix بـ in the word بهدف?

Here, the preposition بـ translates closest to 'with' or 'by.' Attached to the noun هدف (goal or aim), the phrase بهدف is a very common fixed expression meaning 'with the goal of' or 'aiming to.'

How do the words بهدف الدراسة relate to each other grammatically?

They form an Idafa (a possessive or 'of' construction). هدف is the first part and is in the genitive case because of the preposition بـ. الدراسة is the second part and is also genitive, marked by a kasra. Literally, it translates to 'with the goal of the study.'

Why is الدراسة translated as the English gerund 'studying' rather than the noun 'the study'?

In Arabic, the verbal noun or masdar الدراسة encompasses both the abstract concept of 'the study' and the active process of 'studying.' While English strongly prefers the '-ing' verb form after a preposition like 'of' ('with the aim of studying'), Arabic naturally uses the definite noun form in this construction.