Så fort oppvasken er tatt og klærne er på tørkestativet, kan vi endelig sette oss ned og slappe av.

Yes, literally så fort can mean so fast / so quickly, but in sentences like this it often means as soon as.

So:

• Så fort oppvasken er tatt ... = As soon as the dishes are done ...

This is a very common time expression in Norwegian. A close synonym is så snart.

Compare:

• Han løper så fort. = He runs so fast.
• Så fort han kommer hjem, ringer han. = As soon as he gets home, he calls.

The context tells you which meaning is intended.

This is because of the Norwegian V2 rule: in a main clause, the finite verb normally comes in second position.

Here, the whole first clause is placed first:

• Så fort oppvasken er tatt og klærne er på tørkestativet, ...

That entire part counts as position 1. So in the main clause, the finite verb must come next:

• kan vi endelig sette oss ned og slappe av

Not:

• ... vi kan endelig ...

This is very typical in Norwegian when a time clause comes first.

Norwegian often uses the definite form where English might use either the or a more general expression.

In this sentence:

• oppvasken = the dishes / the washing-up
• klærne = the clothes
• tørkestativet = the drying rack

These are understood as specific things in the situation: the dishes that need washing, the clothes that need drying, the drying rack in the home.

This is very natural in Norwegian household-context language. For example:

• Middagen er klar. = Dinner is ready.
• Barna sover. = The children are sleeping.
• Bilen står utenfor. = The car is outside.

It means something like:

• the dishes are done
• the washing-up is done
• more literally, the dishes have been taken care of

This is an idiomatic Norwegian way to talk about completed chores.

The verb ta often appears in expressions for doing tasks:

• ta oppvasken = do the dishes
• ta klesvasken = do the laundry
• ta rengjøringen = do the cleaning

So oppvasken er tatt focuses on the finished result, not just the action.

Because ta oppvasken is simply the normal Norwegian expression for do the dishes.

Even though ta usually means take, Norwegian uses it in many everyday expressions where English uses do:

• ta oppvasken = do the dishes
• ta en pause = take a break
• ta et bilde = take a picture
• ta bussen = take the bus

So you should think of ta oppvasken as a set expression, not translate it word for word.

Because Norwegian often just states the resulting state if that is what matters.

• klærne er på tørkestativet = the clothes are on the drying rack

This tells us the important thing: the clothes have been put there and are now drying.

You could say something more explicit, such as:

• klærne er hengt opp på tørkestativet

But that sounds a bit heavier and puts more focus on the action of hanging them up. In the original sentence, the simpler version sounds more natural.

Because Norwegian forms compounds as one word much more often than English does.

So:

• tørke = dry
• stativ = rack / stand
• tørkestativ = drying rack
• tørkestativet = the drying rack

This is extremely normal in Norwegian. English often writes these as two words, but Norwegian usually joins them:

• togstasjon = train station
• sommerferie = summer holiday
• kjøkkenbord = kitchen table

So tørkestativet may look long to an English speaker, but it is built in a very regular way.

Sette oss ned means sit down.

More literally:

• sette = set / place
• oss = ourselves / us
• ned = down

The full reflexive verb is sette seg ned, which means sit down.

It changes form depending on the subject:

• jeg setter meg ned = I sit down
• du setter deg ned = you sit down
• vi setter oss ned = we sit down

So oss is there because the subject is vi.

Because modal verbs in Norwegian are followed by the bare infinitive, without å.

Common modal verbs include:

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

So you get:

• kan sette oss ned
• skal gå
• vil spise
• må dra

Not:

• kan å sette
• skal å gå

In the sentence, both infinitives belong to kan:

• kan vi endelig sette oss ned og slappe av

Slappe av is a fixed verb expression meaning relax.

The word av is a particle, and many Norwegian verbs work this way. You should learn slappe av as a unit.

Examples:

• Jeg vil slappe av. = I want to relax.
• Vi slapper av hjemme. = We relax at home.
• Nå kan vi slappe av. = Now we can relax.

The meaning is not really the sum of slappe + av word for word. It is just the normal verb for relax.

They are related, but not identical.

• sette oss ned = sit down
• slappe av = relax

Together they create a natural sequence:

1. first, we sit down
2. then, we relax

In English we often say the same kind of thing:

• sit down and relax

So the sentence sounds very natural and conversational. It emphasizes the feeling of finally being finished with chores and being able to rest.