Learn Hardcore Russian: Студенты готовятся к завтрашнему тесту вместе в библиотеке. - The students are preparing for tomorrow’s test together in the library.

они готовят тест = they are preparing the test (they are organizing or writing it).
они готовятся к тесту = they are preparing for the test (they are studying).

In the sentence, готовятся is 3rd person plural, present tense of готовиться:
они готовятся – they are preparing / getting ready.

Why do we use к завтрашнему тесту and not a direct object like завтрашний тест in the accusative?

The verb готовиться always takes the preposition к with the dative case:

готовиться к чему? – to prepare for what?

Because of this pattern, we must say:

готовиться к тесту,
готовиться к экзамену,
готовиться к поездке, etc.

So к завтрашнему тесту is “for tomorrow’s test” and both завтрашнему and тесту are in the dative singular after к.

Using an accusative like завтрашний тест directly after готовиться would be ungrammatical.

What is the base form of завтрашнему, and why does it look like that?

The base adjective is завтрашний – tomorrow’s. It declines to agree with the noun in gender, number, and case.

тест is masculine, singular.
After к, we need the dative case.

Masculine/neuter dative singular of завтрашний is завтрашнему.
Masculine dative singular of тест is тесту.

So we get:

к завтрашнему тесту – for tomorrow’s test
- завтрашнему – masculine dative singular
- тесту – masculine dative singular

They match in gender, number, and case, as Russian adjectives must.

What is the difference between завтра and завтрашний / завтрашнему here? Could I just say к тесту завтра?

завтра is an adverb: tomorrow.
завтрашний is an adjective: tomorrow’s.

In this sentence:

к завтрашнему тесту = for tomorrow’s test (the test that will be tomorrow).

If you said готовятся к тесту завтра, it would mean:

They are preparing for the test tomorrow (the time when they prepare is tomorrow),
not
They are preparing now for the test that is tomorrow.

To keep the original meaning using завтра, you would need something like:

готовятся к тесту, который будет завтра – are preparing for the test that will be tomorrow.

So завтрашнему тесту is a compact way to say “the test that is tomorrow.”

Why is it в библиотеке and not в библиотеку?

The preposition в can take either:

Prepositional case (где? – where?) → location
Accusative case (куда? – where to?) → direction / motion into

Here we are talking about where they are preparing (location), not where they are going, so we use:

в библиотеке – in the library (prepositional case)

If you said:

в библиотеку – that would mean into the library / to the library and would normally go with a motion verb like идут в библиотеку – they are going to the library.

Where can I put вместе in this sentence? Is the word order fixed?

Word order in Russian is relatively flexible, and вместе can move around, with slight changes in emphasis but not in basic meaning.

All of these are possible:

Студенты готовятся к завтрашнему тесту вместе в библиотеке.
– Neutral: they prepare for tomorrow’s test together in the library.
Студенты вместе готовятся к завтрашнему тесту в библиотеке.
– Slight emphasis that they are together, not individually.
Студенты готовятся вместе к завтрашнему тесту в библиотеке.
– Emphasis on together as part of the preparation.
В библиотеке студенты вместе готовятся к завтрашнему тесту.
– Starts by emphasizing the place (in the library).

The original word order is natural and common, but not the only correct one.

What aspect and tense is готовятся? Could we use something like подготовятся instead?

Готовятся is:

Present tense,
Imperfective aspect,
From готовиться – to be preparing / to be in the process of getting ready.

It describes an ongoing process happening now.

Подготовиться or приготовиться are perfective:

они подготовятся = they will prepare (and finish preparing),
In the present form, perfective effectively refers to a single future completed action.

In this sentence we want to say they are currently preparing, not they will have prepared, so imperfective готовятся is the natural choice.

How would the sentence change if there was only one (female) student?

We change:

студенты (plural) → студентка (singular, feminine)
готовятся (3rd person plural) → готовится (3rd person singular)
We normally drop вместе, because one person cannot be together alone.

Result:

Студентка готовится к завтрашнему тесту в библиотеке.
– The (female) student is preparing for tomorrow’s test in the library.

For a male student:

Студент готовится к завтрашнему тесту в библиотеке.

Is тест different from экзамен in Russian? Could we say к завтрашнему экзамену?

Yes, you can say к завтрашнему экзамену, and it is fully correct.

Nuance:

тест – usually a test with many short questions, often multiple choice, more informal or smaller.
экзамен – a more serious, formal exam, often end-of-course or official.

The grammar is identical:

к завтрашнему тесту – to/for tomorrow’s test
к завтрашнему экзамену – to/for tomorrow’s exam

Both тесту and экзамену are masculine dative singular.

Does вместе в библиотеке mean they are together in the library, or could it just mean they are together, and also it happens in the library?

By default, Russian speakers will understand that:

They are preparing together, and
This preparation is happening in the library.

So: They are in the library together, preparing for the test.

Technically, вместе and в библиотеке are both adverbial modifiers of готовятся, so they independently describe the action:

how? вместе – together
where? в библиотеке – in the library

If you wanted to emphasize that being together is specifically connected with the library (as opposed to, say, together online), you could highlight place more strongly in context or with a different word order, but the standard interpretation is already together in the library.