Karijes je veliki, tako da će stomatolog izvaditi zub.

Karijes means tooth decay, dental caries, or in many contexts simply a cavity.

A small nuance:

• karijes is often used like a general medical term for decay
• in everyday translation, English speakers often understand it as cavity

In this sentence, it means the tooth decay/cavity is severe enough that the tooth will need to be removed.

Je is the 3rd person singular present of biti = to be.

So:

• Karijes = the cavity/tooth decay
• je = is
• veliki = big/large

Literally: The cavity is big.

Serbian often needs je when a noun is linked to an adjective, just like English needs is.

Veliki agrees with karijes.

Since karijes is:

• masculine
• singular
• here in the nominative

the adjective also appears in a matching form: veliki.

So the grammar is:

• karijes → masculine singular
• veliki → masculine singular form of big

Yes. Both can be heard.

Traditionally, velik is the shorter predicate form, so Karijes je velik is very natural.
But in modern everyday Serbian, many speakers also use veliki in this kind of sentence.

So:

• Karijes je velik = very normal
• Karijes je veliki = also common in speech

For a learner, the important thing is that both are understandable, and the sentence you were given is acceptable Serbian.

Here tako da means so, therefore, or as a result.

It introduces a consequence:

• Karijes je veliki = The cavity is big
• tako da će stomatolog izvaditi zub = so the dentist will remove the tooth

In other contexts, tako da can sometimes be translated as so that, but in this sentence it clearly shows a result, not a purpose.

The future here is formed with:

• će = auxiliary for future tense
• izvaditi = infinitive of the main verb

So:

• će izvaditi = will remove / will extract

This is a standard Serbian future construction.

Compare:

• Stomatolog će izvaditi zub. = The dentist will extract the tooth.
• Tako da će stomatolog izvaditi zub. = So the dentist will extract the tooth.

Because će is a clitic. Clitics in Serbian usually go near the beginning of the clause, often in the second position.

After tako da, it is very natural to place će immediately after that connector:

• tako da će stomatolog izvaditi zub

This may feel unusual to an English speaker, because English keeps will directly next to the subject:

• the dentist will remove...

But Serbian clitic placement follows different rules.

Izvaditi is a very common verb for take out, remove, or extract.

With teeth, izvaditi zub is the normal way to say:

• remove a tooth
• pull a tooth
• extract a tooth

It is also a perfective verb, which fits well here because the action is seen as a single completed event: the dentist will remove the tooth.

Because zub is the direct object of izvaditi, so it is in the accusative case.

The important point is that zub is:

• masculine
• inanimate

For many masculine inanimate nouns in Serbian, the accusative singular looks exactly the same as the nominative singular.

So:

• nominative: zub
• accusative: zub

That is why the form does not change here.

Because Serbian does not have articles like English a/an/the.

So Serbian often says simply:

• karijes
• stomatolog
• zub

and the listener understands from context whether the meaning is:

• a cavity or the cavity
• a dentist or the dentist
• a tooth or the tooth

English requires articles, but Serbian does not.

Yes. Stomatolog is a standard and correct word for dentist.

A useful distinction is:

• stomatolog = more formal, medical, standard
• zubar = more everyday, colloquial

Both are common, but stomatolog sounds a bit more professional or clinical, which fits this sentence well.