Depois do exame, o alívio é enorme.

In Portuguese, the preposition de + the masculine singular article o always contracts to do.

• de + o = do
• de + a = da
• de + os = dos
• de + as = das

So:

• ❌ depois de o exame
• ✅ depois do exame

You could say depois de o exame if you were speaking very slowly for emphasis or in poetry, but in normal speech and writing it must be depois do exame.

Depois de is just after + a verb or a general idea; depois do is after the + a specific noun.

• Depois de estudar, fui dormir.
  After studying, I went to sleep. (verb in the infinitive: estudar)

• Depois do exame, fui dormir.
  After the exam, I went to sleep. (specific noun: o exame)

So:

• depois de + infinitive: depois de fazer o exame (after taking the exam)
• depois de + article + noun → contraction:
  • depois do exame (after the exam)
  • depois da prova (after the test)
  • depois dos exames (after the exams)

Yes. Both are correct:

• Depois do exame, o alívio é enorme.
• O alívio é enorme depois do exame.

In Portuguese, adverbial time expressions (hoje, amanhã, depois do exame, à noite) can go at the beginning or the end of the sentence.

Putting Depois do exame at the beginning slightly emphasizes the time frame:

• Depois do exame, o alívio é enorme.
  → The important point is what happens after the exam.

• O alívio é enorme depois do exame.
  → The important point is how big the relief is, and then you add when.

Because Depois do exame is a fronted adverbial phrase of time. In European Portuguese, when you put this kind of phrase at the beginning of the sentence, it is normally followed by a comma:

• Depois do exame, o alívio é enorme.
• À noite, estudo português.
• No verão, faz muito calor.

If you move the time expression to the end, you don’t use a comma:

• O alívio é enorme depois do exame.
• Estudo português à noite.

In Portuguese, singular countable and many abstract nouns usually take a definite article (o, a, os, as) even when English would omit it.

• O alívio é enorme.
  Literally: The relief is enormous.

Without the article (Alívio é enorme) it sounds wrong or very odd; it’s not how native speakers would say it.

Compare:

• A felicidade é importante. – Happiness is important.
• O medo é natural. – Fear is natural.
• O amor é complicado. – Love is complicated.

So: keep the article o in o alívio here.

Grammatically, we choose ser here because we’re describing a characteristic/quality of o alívio, not its location or a changing condition of a person.

• O alívio é enorme.
  → The relief (as a feeling/event) is enormous. (quality)

If you talked about a person feeling relieved, you’d use estar:

• Depois do exame, estou muito aliviado.
  After the exam, I am very relieved.

So:

• ser
  • adjective → describes the nature or degree of a thing:
    O alívio é grande / é enorme / é imediato.
• estar
  • adjective → describes someone’s current state:
    Estou aliviado.

Both exame and alívio are masculine nouns:

• o exame → do exame
• o alívio → o alívio é enorme

Gender in Portuguese is mostly lexical—you have to learn it with the word, but there are some useful patterns:

• Many nouns ending in -o are masculine: o livro, o carro, o alívio.
• Many nouns ending in -a are feminine: a casa, a prova.

Exame ends in -e, which is less predictable; you must learn its gender: o exame.

Always memorize new nouns with their article: o exame, a prova, o alívio, a alegria.

In this context, no. You want to refer to a specific exam, so you need the article:

• ✅ Depois do exame, o alívio é enorme.
• ❌ Depois de exame, o alívio é enorme. (ungrammatical here)

De exame can exist in other constructions, usually as a kind of adjective:

• resultados de exame – exam results
• marcação de exame – booking of an exam

But when you mean “after the exam”, you must use depois do exame.

Both express a similar idea, but from different angles:

• Depois do exame, o alívio é enorme.
  Focus on the feeling of relief itself as a big thing.

• Depois do exame, fico muito aliviado.
  Focus on you / someone becoming relieved.

Common patterns:

• o alívio é… → grande, enorme, imediato, incrível
• ficar aliviado → fico / ficas / fica / ficamos / ficam aliviados

Example:

• Depois do exame, fico muito aliviado. – I (male) feel very relieved.
• Depois do exame, fico muito aliviada. – I (female) feel very relieved.

Yes, enorme is an adjective meaning huge / enormous. You can use muito grande instead, with a slightly weaker tone:

• O alívio é enorme. – The relief is enormous. (stronger, more emphatic)
• O alívio é muito grande. – The relief is very big. (natural, slightly softer)

Both are perfectly correct. Word order is:

• O alívio é enorme.
• O alívio é muito grande.

You can also put the adjective before the noun for stylistic emphasis:

• Depois do exame, é um enorme alívio.
  (literally: it is an enormous relief)

Approximate breakdown (European Portuguese):

• Depois – /də-POISH/

  • unstressed de- like English duh; -pois like poish.

• do – /du/

  • close to English du in dude but shorter.

• exame – /e-ZA-m(ɨ)/

  • e- like eh
  • -xa- is pronounced like za (voiced s: z)
  • final -me is a very reduced /mɨ/, often almost just /m/.

• o – /u/

  • like English oo in food, but shorter.

• alívio – /a-LI-vyu/

  • a- like uh/ah
  • lí- stressed syllable: LEE
  • -vio like vyu (similar to view but shorter).

• é – /ɛ/

  • open é, like eh.

• enorme – /i-NÓR-m(ɨ)/

  • e- like ee but shorter;
  • -nor- stressed: like English NOR;
  • final -me again very reduced.

Spoken quickly, you will hear a lot of linking and reduction, for example:

• Depois do exame → /də-POISH-du-e-ZA-mɨ/

Both can mean exam/test, but usage varies:

• exame

  • often used for official, final, or medical exams:
    • exame final – final exam
    • exame nacional – national exam
    • exame de sangue – blood test

• prova

  • common for school/university tests and competitions:
    • prova escrita – written test
    • prova oral – oral exam
    • prova de atletismo – athletics event

In your sentence, if it’s a big official exam, exame is very natural:

• Depois do exame, o alívio é enorme.

If it were just a classroom test, Depois da prova, o alívio é enorme could also be used.

You make both the noun and the article plural:

• Depois dos exames, o alívio é enorme.
  After the exams, the relief is enormous.

Changes:

• do exame → dos exames
  • de + os = dos
  • plural of exame is exames.

The rest of the sentence stays the same:

• o alívio (relief in general, still singular and abstract)
• é enorme (still agrees with o alívio)

It’s neutral. You can use this sentence:

• in everyday conversation
• in writing (emails, messages)
• in more formal contexts, like a report or essay

It doesn’t use slang or very formal vocabulary; it’s standard, natural European Portuguese.