Раз тебе всё равно нужно написать преподавательнице, спроси заодно про зачёт.

Here раз does not mean one time.

At the start of a sentence, раз can be a colloquial conjunction meaning since, seeing that, or if, given that. It introduces a reason that the speaker treats as already established.

So:

• Раз тебе всё равно нужно написать...
  = Since you need to write anyway...

It sounds more conversational than поскольку or так как.

Mostly since.

Раз usually suggests that the fact is already accepted and the speaker is building on it:

• Раз ты уже идёшь в магазин, купи молоко.
  = Since you're already going to the store, buy milk.

English might sometimes translate it with if, but in Russian it usually feels less hypothetical than если.

So this sentence is not really saying if you happen to need to write. It is more like:

• Since you have to write to her anyway, ask about the credit too.

Because Russian often uses the pattern:

• кому-то нужно + infinitive

This means someone needs to do something.

So:

• тебе нужно написать
  literally = to you it is necessary to write
• natural English = you need to write

The person who has the obligation is put in the dative case:

• мне нужно уйти = I need to leave
• ему нужно позвонить = He needs to call
• нам нужно поговорить = We need to talk

That is why you get тебе, not ты.

Because нужно here is not the short form adjective meaning needed in agreement with a noun. It is an impersonal predicative word meaning it is necessary / one needs to.

Compare:

• Мне нужен учебник.
  = I need a textbook.
  Here нужен agrees with учебник.

But:

• Мне нужно написать преподавательнице.
  = I need to write to the teacher.
  Here нужно is part of the fixed pattern кому-то нужно + infinitive.

So in your sentence, нужно is the correct form.

Not here.

Всё равно can mean different things depending on context:

1. I don't care / it makes no difference to me

   • Мне всё равно.

2. Anyway / all the same / in any case

   • Тебе всё равно нужно написать.

In this sentence, it has the second meaning:

• тебе всё равно нужно написать
  = you need to write anyway

So it means the action is necessary regardless of other circumstances.

Because написать is perfective, and here the speaker means one complete act of writing: sending a message, email, or note.

• писать = to write, be writing, write in general
• написать = to write and finish / send

In this sentence, the idea is:

• you need to write to the teacher once
• not you need to be engaged in writing or you need to write regularly

So написать fits better.

Because написать takes the person written to in the dative case:

• написать кому? = to write to whom?

So:

• преподавательница = female teacher / instructor
• преподавательнице = to the female teacher / instructor

This is the dative singular form.

A similar pattern:

• Я написал другу. = I wrote to a friend.
• Она написала матери. = She wrote to her mother.

Not exactly.

Преподавательница usually means a female instructor/lecturer/teacher, often in higher education or courses, not necessarily a schoolteacher.

Учительница is more specifically a female schoolteacher.

So:

• преподавательница = instructor, lecturer, teacher in a broader/academic sense
• учительница = schoolteacher

Also, in some contexts Russian speakers may use преподаватель even for a woman, especially in more official or professional contexts. But преподавательница is a normal understandable word.

Спроси is the singular informal imperative of спросить (to ask).

It matches the informal тебе earlier in the sentence, so the speaker is talking to one person informally.

• спроси = ask
• спросите = ask (plural or formal)

Since the whole sentence is giving advice or a suggestion to one person, спроси is exactly what you would expect.

Because спросить is perfective, and the speaker means ask once, as a single action.

• спроси = ask (once, get it done)
• спрашивай = ask / keep asking / ask in general

In this sentence, the meaning is:

• When you write to her, ask about the credit too.

That is a one-time action, so perfective спроси is the natural choice.

Заодно means something like:

• while you're at it
• at the same time
• also, as part of the same action

It suggests combining one action with another because the opportunity is already there.

So:

• спроси заодно про зачёт
  = ask about the credit too while you're at it

This word is very common in everyday Russian and often sounds quite natural in practical suggestions.

Both are possible, but they differ a bit in style.

• про + accusative is more conversational
• о + prepositional is more neutral or a bit more formal

So:

• спроси про зачёт = casual, everyday
• спроси о зачёте = also correct, slightly more formal/bookish

Because the whole sentence is conversational (раз, всё равно, заодно), про зачёт fits the tone well.

Зачёт is a term from the Russian education system.

It often means a pass/fail credit rather than a graded exam. Depending on context, it can refer to:

• the requirement itself
• the result (pass)
• the assessment session connected with that subject

So in this sentence, спроси ... про зачёт means something like:

• ask about the course credit
• ask about the pass requirement
• ask about the pass/fail assessment

The exact English equivalent depends on the school system, because there is no perfect one-word match in English.

Because Russian often omits subject pronouns when they are clear from the verb form.

Here спроси already tells you that the speaker is addressing:

• you
• singular
• informal

So Russian does not need to say ты.

The same thing happens constantly in Russian:

• Иду домой. = I’m going home.
• Знаешь его? = Do you know him?
• Сделай это. = Do this.

Adding ты is possible, but usually only for emphasis or contrast.

You would change the informal forms to formal/plural ones:

• тебе → вам
• спроси → спросите

So:

• Раз вам всё равно нужно написать преподавательнице, спросите заодно про зачёт.

That means:

• Since you need to write to the teacher anyway, ask about the credit too.

Everything else can stay the same.

Because Раз тебе всё равно нужно написать преподавательнице is a subordinate clause introduced by раз.

Russian normally separates that kind of clause with a comma:

• Раз ..., ...

So the structure is:

• [Since you need to write to the teacher anyway], [ask about the credit too].

Russian commas are often more regular and mandatory than English commas in similar sentences.