Learn Hardcore Russian: В библиотеке можно взять книгу.

Because with the preposition в indicating location in something, Russian requires the prepositional case. It answers the question где, so библиотека changes to библиотеке.

What does можно mean and how is it used?

Можно is an impersonal modal verb meaning possible or allowed. It is always followed by an infinitive such as взять to express permission or possibility. There is no explicit subject; the action applies to people in general.

Why is there no subject like I or you in this sentence?

Because the construction with можно is impersonal: the performer of the action is indefinite or general (for example, any person). Russian often omits a subject when it is clear from context.

Why is взять used instead of брать?

Взять is the perfective form meaning a single or completed act of taking or borrowing. In contrast, the imperfective брать describes habitual or repeated actions (for example, в библиотеке можно брать книги).

Why is книгу in the accusative case?

Книгу functions as the direct object of the verb взять, so it takes the accusative case. For feminine nouns ending in -а, the accusative singular ending is -у.

Can the word order be changed?

Yes. Russian word order is fairly flexible for emphasis. For example: • Можно в библиотеке взять книгу (focus on the possibility)
• Книгу в библиотеке можно взять (focus on the book)
• В библиотеке книгу можно взять (slightly different emphasis)
The most neutral order is В библиотеке можно взять книгу.