سأكتب الجواب في الدفتر بعد الدرس.

The base verb is أكتب = I write / I am writing.
The prefix سـ is attached to the imperfect verb to make it future:

أكتب → سأكتب = I will write

So the أ is not extra; it is part of the verb form itself, marking first person singular (I).

Yes. سوف أكتب also means I will write.

In Modern Standard Arabic, both سأكتب and سوف أكتب are correct. Very roughly:

• سأكتب = shorter, very common
• سوف أكتب = a bit fuller or more formal in tone

In many contexts, the difference is small.

Arabic often does not need a separate subject pronoun, because the verb already shows who is doing the action.

In أكتب, the أـ tells you the subject is I. So the subject is understood as a hidden أنا.

You could say:

أنا سأكتب الجواب في الدفتر بعد الدرس.

But أنا is optional here and usually added only for emphasis or contrast.

Because this is a normal verbal sentence in Arabic. In MSA, it is very common to begin with the verb:

سأكتب الجواب...

This is perfectly natural. Arabic can also use other word orders for emphasis, but verb-first is one of the most basic and common patterns.

الـ makes the noun definite, so:

• الجواب = the answer
• جواب = an answer

In this sentence, the speaker means a specific answer, so الجواب is used.

الجواب is the direct object of سأكتب, because it is the thing being written.

So in fully vowelled Arabic, it would be:

الجوابَ

with fatḥa at the end, because the direct object is normally accusative.

In ordinary unvowelled writing, that final case ending is usually not written, so you just see الجواب.

في means in. After a preposition like في, the following noun is normally genitive.

So in full vocalization:

في الدفترِ

with kasra on the end of الدفتر.

Again, in normal writing the final short vowel is usually omitted, so you simply see الدفتر.

بعد itself already means after. It commonly functions as a time expression (after the lesson, after class, etc.).

Here it forms an iḍāfa construction:

• بعدَ = after
• الدرسِ = the lesson

So the full form is:

بعدَ الدرسِ

Grammatically:

• بعدَ is in the accusative here as an adverbial expression of time
• الدرسِ is genitive because it is the second part of the iḍāfa

A fully vowelled version is:

سَأَكْتُبُ الجَوَابَ فِي الدَّفْتَرِ بَعْدَ الدَّرْسِ

This shows:

• سَأَكْتُبُ — verb in the indicative
• الجَوَابَ — direct object, accusative
• فِي الدَّفْتَرِ — after a preposition, genitive
• بَعْدَ الدَّرْسِ — time expression + iḍāfa

In everyday printed Arabic, most of these short vowels are not written.

This is because of sun letters and moon letters.

The د in الدفتر and الدرس is a sun letter, so the ل of الـ is not pronounced. Instead, the next consonant is doubled:

• الدفتر → pronounced ad-daftar
• الدرس → pronounced ad-dars

But ج in الجواب is a moon letter, so the ل is pronounced normally:

• الجواب → al-jawāb

So the spelling stays the same, but the pronunciation changes depending on the first letter after الـ.