Eu desejo-te boa sorte no exame.

That’s right: Portuguese often leaves out the subject pronoun because the verb ending already shows who is doing the action.

So Desejo-te boa sorte no exame is perfectly natural and means the same thing.

Adding Eu can:

• add emphasis
• make the sentence slightly more explicit
• create contrast, as in Eu desejo-te boa sorte, mas ele não

So here, Eu is optional, not required.

Here, desejar means to wish.

Although desejar can sometimes mean to desire, in this sentence it clearly means to wish, because it is followed by boa sorte.

So:

• desejar boa sorte = to wish someone good luck

This is a very normal use of desejar in Portuguese.

Te means to you or you, referring to one person in an informal way.

In this sentence, it is the person receiving the wish:

• Desejo-te boa sorte = I wish you good luck

It corresponds to you when speaking to someone you would address as tu in European Portuguese.

In European Portuguese, unstressed object pronouns like me, te, se, lhe, nos, vos are often placed after the verb in ordinary affirmative sentences. This is called enclisis.

So:

• desejo-te = I wish you

The hyphen is the normal spelling when the pronoun is attached after the verb.

This is one of the things that often feels different from English and also from Brazilian Portuguese.

In European Portuguese, desejo-te is the normal choice in a simple affirmative sentence.

Te desejo is much more typical of Brazilian Portuguese. In Portugal, it can sound non-standard, foreign, or strongly marked in this context.

However, in European Portuguese, the pronoun does move before the verb in certain situations, for example after words like:

• não
• question words
• some conjunctions

For example:

• Não te desejo isso = I don’t wish that on you

So in your sentence, desejo-te is the expected European Portuguese form.

Because sorte is a feminine noun.

In Portuguese, adjectives usually agree with the noun in gender and number:

• boa = feminine singular
• bom = masculine singular

So:

• boa sorte = correct
• bom sorte = incorrect

Also, boa sorte is a very common fixed expression meaning good luck.

Sometimes you may hear very short expressions, but boa sorte is the standard and most natural way to say good luck.

So:

• Desejo-te boa sorte = natural
• Desejo-te sorte = understandable, but less idiomatic in most contexts

For a learner, boa sorte is the safest and best choice.

No is a contraction of:

• em
  • o = no

And exame is a masculine singular noun:

• o exame

So:

• em o exame becomes no exame

This kind of contraction is very common in Portuguese:

• em + a = na
• em + os = nos
• em + as = nas

So no exame literally means something like in the exam, though in English we usually say on the exam or simply keep the meaning natural as good luck on the exam.

It is informal singular, because of te.

You would use it with someone you address as tu, such as:

• a friend
• a sibling
• a classmate
• someone of a similar age in a casual situation

If you wanted to be more formal, you would normally say:

• Desejo-lhe boa sorte no exame.

Here, lhe means to you in a more formal or polite way.

Yes, it is correct and natural. But in everyday speech, people often choose a shorter version:

• Boa sorte no exame!

That is probably the most common and straightforward way to say it.

Your full sentence:

• Eu desejo-te boa sorte no exame.

sounds a bit fuller and more explicit, but it is still perfectly good Portuguese. It may sound slightly more deliberate or warm than the shorter version.

A rough breakdown is:

• Eu = I
• desejo = wish
• -te = to you / you
• boa sorte = good luck
• no exame = in/on the exam

So the structure is roughly:

I wish-you good luck on the exam

That word order is normal in Portuguese, especially with the pronoun attached to the verb in European Portuguese.