Я возвращаю книгу в библиотеку.

Breakdown of Я возвращаю книгу в библиотеку.

книга
book
в
in / to (direction)
библиотека
library
возвращать
to return, to give back

Questions & Answers about Я возвращаю книгу в библиотеку.

Why is it в библиотеку and not в библиотеке?
Even though возвращать (to return) isn't a traditional verb of motion, it still implies movement toward a destination. Because you are taking the book to the library, you use в with the accusative case (библиотеку). You would only use the prepositional case (в библиотеке) to talk about a location, like if you were already inside the library reading.
Why did книга change to книгу?
Because the book is the direct object of the sentence—it is the thing being returned. Just like the transitive verbs of carrying you are learning in this lesson, возвращать requires an object. For feminine nouns ending in -а like книга, you must change the ending to -у to form the accusative case.
Could I use a carrying verb like нести to say this, since that's what the current lesson is about?
Yes! Since you are carrying the book on foot in one direction, you could literally say Я несу книгу обратно в библиотеку (I am carrying the book back to the library). However, using the verb возвращать is the most natural and common way to specifically talk about returning a borrowed item.
What is the infinitive form of возвращаю?
The infinitive is возвращать (to return an object). It is a regular First Conjugation verb ending in -ать, which makes it very predictable to conjugate. To get the я (I) form, you drop the -ть and add -ю: я возвращаю.

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