Tandlægen vil tage et røntgenbillede.

Because Danish usually makes a noun definite by adding the article to the end of the noun.

• en tandlæge = a dentist
• tandlægen = the dentist

So -en here is the definite ending for a common-gender noun.

The dictionary form is tandlæge.

You would normally learn it with its article:

• en tandlæge = a dentist

From that, you get:

• tandlægen = the dentist
• tandlæger = dentists
• tandlægerne = the dentists

Vil is a modal verb. It can mean things like:

• will
• wants to
• is going to

Danish does not have a special future-tense ending like English sometimes seems to. Instead, future meaning is often expressed with words like vil or just the present tense, depending on context.

So vil tage can suggest intention, plan, or future action.

Because after a modal verb like vil, Danish uses the infinitive of the main verb.

So:

• vil tage = will take / wants to take

Not:

• vil tager ❌

This is similar to English:

• will take ✔
• will takes ❌

In Danish, the infinitive marker is at, which often corresponds to English to.

But after modal verbs such as:

• vil
• kan
• skal
• må

the at is normally left out.

So you say:

• Tandlægen vil tage et røntgenbillede.

not

• Tandlægen vil at tage ... ❌

Because billede is a neuter noun in Danish, so it takes et.

• et billede = a picture/image

In compound nouns, the last part usually determines the gender. Since the final part is billede, the whole compound is also neuter:

• et røntgenbillede = an X-ray image
• røntgenbilledet = the X-ray image

Yes. Danish very often writes compound nouns as one word.

So:

• røntgen + billede → røntgenbillede

This is very normal in Danish. English often uses separate words where Danish prefers a single compound.

Yes. That is also a correct sentence, but the nuance is a little different.

• Tandlægen tager et røntgenbillede = the dentist takes / is taking an X-ray
• Tandlægen vil tage et røntgenbillede = the dentist will / wants to / is going to take an X-ray

So vil tage sounds more like intention, decision, or future action.

You normally put ikke after the finite verb, which here is vil.

So:

• Tandlægen vil ikke tage et røntgenbillede.

That is a very common Danish word-order pattern:

• subject + finite verb + ikke + infinitive/main content

In a main-clause yes/no question, Danish usually puts the finite verb first.

So:

• Vil tandlægen tage et røntgenbillede?

This is the standard verb-first question pattern.

Compare:

• Statement: Tandlægen vil tage et røntgenbillede.
• Question: Vil tandlægen tage et røntgenbillede?

They are separate letters in Danish, not just spelling variations.

A rough guide:

• æ sounds somewhat like the vowel in cat, though the exact Danish sound is different
• ø has no exact English equivalent; it is similar to German ö or French eu

So in this sentence:

• tandlægen contains æ
• røntgenbillede contains ø

English speakers usually need some practice with both of these sounds.