Learn Hardcore Russian: Тебе следует отдохнуть после долгого дня.

Because следует is an impersonal modal verb that requires its “subject”—the person who should do something—in the dative. Literally it means “to you it is advisable…”

What does следует mean, and can I think of it as an auxiliary verb?

Следует is an impersonal verb meaning “it is advisable/necessary” or simply “should.” It isn’t an auxiliary in a technical sense, but it functions like a modal: you pair it with an infinitive to express advice or mild obligation.

Why is the verb отдохнуть in the infinitive form?

After impersonal modals like следует, the main action is always an infinitive. Here отдохнуть (“to rest” in perfective) tells you what you should do.

Why is the perfective aspect отдохнуть used rather than the imperfective отдыхать?

Perfective verbs highlight completion of an action. Since the speaker wants you to “take a (complete) rest” after a long day, отдохнуть is more appropriate. The imperfective отдыхать would stress the ongoing process rather than achieving a restful state.

Why does после take долгого дня in the genitive case?

The preposition после always governs the genitive. Therefore долгий день (“a long day” in nominative) becomes долгого дня in genitive.

How do the adjective and noun endings change in долгого дня?

День is a masculine noun. In genitive singular it becomes дня. The masculine adjective долгий takes the genitive ending -ого, giving долгого. Together: долгого дня.

Can I replace тебе следует with тебе нужно or тебе надо, and is there any nuance?

Yes. Тебе нужно отдохнуть or тебе надо отдохнуть are equivalent in meaning (“you need to rest”). Нужно/надо are more common in everyday speech, while следует sounds slightly more formal or bookish.

Is it acceptable to drop тебе and start with Следует отдохнуть?

Yes. Следует отдохнуть после долгого дня is perfectly fine, especially in written or general advice. Omitting тебе makes it impersonal; including it simply specifies you.

Does changing word order like Отдохнуть тебе следует affect the emphasis?

Yes. Russian word order is relatively free.

Тебе следует отдохнуть is neutral.
Отдохнуть тебе следует stresses the action “to rest.”
Тебе отдохнуть следует would emphasize you need rest (less common).
All convey the same basic advice, but fronting a word highlights it.