Tandlægen skal undersøge tanden.

Because Danish usually puts the definite article at the end of the noun instead of using a separate word like English the.

• en tandlæge = a dentist

• tandlægen = the dentist

• en tand = a tooth
• tanden = the tooth

Both tandlæge and tand are common gender nouns, so their definite singular ending is -en.

Compare:

• common gender: en bil → bilen
• neuter: et hus → huset

So -en here is basically the Danish way of saying the.

For the same reason: Danish often uses a suffix article instead of a separate article.

So English:

• the dentist
• the tooth

becomes Danish:

• tandlægen
• tanden

A separate definite word can appear in some other structures, especially with adjectives:

• den store tand = the big tooth

But in your sentence there is no adjective, so the normal form is just the noun with the ending:

• tanden

Yes, it is one word. Danish forms compound nouns very freely, much more like German than English.

• tand = tooth
• læge = doctor

Together:

• tandlæge = dentist

This is very normal in Danish. The last part is usually the main noun, and the first part describes it.

Examples:

• tandbørste = toothbrush
• hospitalslæge = hospital doctor
• skoletaske = school bag

So tandlæge is not unusual at all; it is exactly how Danish commonly builds nouns.

Skal is the present tense of skulle. In a sentence like this, it usually expresses one of these ideas:

• must / has to
• is supposed to
• sometimes will / is going to, depending on context

So Tandlægen skal undersøge tanden can mean something like:

• The dentist must examine the tooth
• The dentist is supposed to examine the tooth
• The dentist is going to examine the tooth

The exact nuance depends on context. In everyday Danish, skal is very common and often covers both obligation and planned future action.

Because after a modal verb in Danish, the next verb normally appears in the bare infinitive without at.

Common modal verbs include:

• kan = can
• vil = will / want to
• må = may / must
• skal = shall / must / is going to
• bør = should

So you get:

• Jeg kan svømme = I can swim
• Hun vil spise = She wants to eat / will eat
• Tandlægen skal undersøge tanden

Not:

• skal at undersøge

The word at is used before infinitives in many other cases, but generally not after modal verbs.

Undersøge is the infinitive form of the verb, similar to English to examine.

Some useful forms are:

• infinitive: undersøge
• present tense: undersøger
• past tense: undersøgte
• past participle: undersøgt

In your sentence, it stays in the infinitive because it follows the modal verb skal:

• skal undersøge

That is the normal pattern in Danish.

The basic order here is:

Subject + finite verb + infinitive + object

So:

• Tandlægen = subject
• skal = finite verb
• undersøge = infinitive
• tanden = object

This looks fairly similar to English:

• The dentist must examine the tooth

But Danish is a V2 language, which means the finite verb usually comes in the second position in main clauses.

So if you move something else to the front, skal still stays second:

• I dag skal tandlægen undersøge tanden = Today the dentist will examine the tooth

That second-position rule is very important in Danish.

In your sentence, tanden means the tooth, singular.

The noun tand has these common forms:

• en tand = a tooth
• tanden = the tooth
• tænder = teeth
• tænderne = the teeth

Notice that the plural changes the vowel:

• tand → tænder

This is similar to irregular English patterns like tooth → teeth.

So if you wanted the dentist must examine the teeth, you would say:

• Tandlægen skal undersøge tænderne

These are two vowels that English speakers often find tricky.

• æ in tandlægen is roughly like the vowel in English cat, but not exactly.
• ø in undersøge has no exact English equivalent. If you know German ö or French eu in deux, it is similar.

A rough learner-friendly approximation might be:

• tandlægen ≈ tan-lay-en
• undersøge ≈ oon-er-suh-eh or un-er-suh-yeh

But those are only approximations. Real Danish pronunciation is usually softer and more reduced than English speakers expect.

The most helpful thing is to learn the words by listening, because Danish spelling often does not match pronunciation very closely from an English perspective.

Grammatically, skal is in the present tense, but the whole sentence often refers to something that is still going to happen.

So depending on context, it may mean:

• a present obligation: The dentist has to examine the tooth
• a planned future action: The dentist is going to examine the tooth

This is very normal in Danish. A present-tense modal verb can point to a future event when the context makes that clear.

Yes, but it would change the meaning.

Your sentence uses the definite forms:

• tandlægen = the dentist
• tanden = the tooth

If you used indefinite forms instead, you would get:

• En tandlæge skal undersøge en tand = A dentist must examine a tooth

That sounds much less specific. The original sentence refers to a particular dentist and a particular tooth, or at least treats them as known in the situation.