Jeg skærer brødet med en skarp kniv.

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

• et brød = a bread / a loaf of bread
• brødet = the bread

So brødet is one word meaning the bread. English uses a separate word (the), but Danish often uses an ending instead.

Because en skarp kniv is indefinite (a sharp knife), while brødet is definite (the bread).

In Danish:

• indefinite singular usually uses a separate article: en/et
• definite singular usually uses an ending on the noun

So:

• en kniv = a knife
• kniven = the knife
• et brød = a bread / a loaf of bread
• brødet = the bread

Because kniv is a common gender noun, and common gender nouns take en in the singular indefinite form.

Danish singular nouns are mainly divided into two genders:

• common gender: takes en
• neuter: takes et

So:

• en kniv
• et brød

Unfortunately, noun gender usually has to be learned with each noun.

Because it is describing an indefinite singular common gender noun: en kniv.

In that pattern, the adjective usually stays in its basic form:

• en skarp kniv = common gender, singular, indefinite

Compare:

• et skarpt ... = neuter singular indefinite
• den skarpe kniv = definite
• skarpe knive = plural

So skarp is correct because kniv is an en-word.

Skærer is the present tense of at skære (to cut).

• at skære = to cut
• jeg skærer = I cut / I am cutting

A lot of Danish present-tense verb forms end in -r, including this one.

It can mean both, depending on context.

Danish usually does not make a separate present progressive form like English I am cutting. The ordinary present tense often covers both meanings:

• Jeg skærer brødet = I cut the bread or I am cutting the bread

Context tells you which meaning is intended.

Because Danish does not normally need a separate verb like English am for this kind of present action.

English:

• I am cutting the bread

Danish:

• Jeg skærer brødet

So the idea of an action happening now is often expressed just with the ordinary present tense.

Here med means with, in the sense of using something as a tool.

So:

• med en skarp kniv = with a sharp knife

In other sentences, med can also mean with in the sense of accompaniment, but here it clearly means the instrument used to cut.

The sentence follows a very common Danish pattern:

• Jeg = subject
• skærer = verb
• brødet = object
• med en skarp kniv = prepositional phrase

So the structure is:

Subject + Verb + Object + other information

That gives:

Jeg skærer brødet med en skarp kniv.

Then Danish uses its usual verb-second pattern in a main clause.

So you can say:

Med en skarp kniv skærer jeg brødet.

Notice that the verb skærer still comes second in the clause, and jeg moves after it.

This is a very important feature of Danish word order.

No. Danish does not capitalize jeg all the time.

It is capitalized here only because it is the first word of the sentence:

• Jeg skærer brødet.
• Han siger, at jeg skærer brødet.

In the second example, jeg is lowercase because it is not sentence-initial.

They are separate Danish vowels.

• ø in brødet has no exact English equivalent. It is somewhat like a rounded vowel made with the lips more rounded than in normal English.
• æ in skærer is somewhat similar to the vowel in cat, but not exactly the same.

The main thing for a learner is to recognize that ø and æ are not just fancy versions of o and a. They are different letters with different sounds.

The dictionary form is at skære (to cut).

When you look up Danish verbs, you usually find them in the infinitive form with at:

• at skære = to cut

Then the present tense is:

• skærer

So if you want to learn or look up the verb, search for skære, not skærer.