Wir können die Aufgabe lösen, ohne den Taschenrechner zu benutzen, wenn wir ruhig nachdenken.

This is a standard modal verb + infinitive pattern.

• können is the conjugated modal verb: wir können
• lösen is the main action verb in the infinitive, placed at the end of the main clause

So:

• Wir können die Aufgabe lösen. = We can solve the task/problem.

This is very similar to English can solve.

Because die Aufgabe is the direct object of lösen.

The verb lösen takes an accusative object: you solve something.

• nominative: die Aufgabe
• accusative: die Aufgabe

Since Aufgabe is feminine singular, the article stays die in both nominative and accusative, so the form does not change here.

This structure means without doing something.

In the sentence:

• ohne den Taschenrechner zu benutzen = without using the calculator

It is a very common German pattern:

• ohne zu warten = without waiting
• ohne nachzudenken = without thinking
• ohne ihn zu fragen = without asking him

So ohne ... zu introduces an infinitive clause that describes something not being done.

Because the subject is understood to be the same as in the main clause.

Here, wir are the ones who können, and also the ones who would benutzen. German usually does not repeat the subject in an ohne ... zu clause when it is the same subject.

So:

• Wir können die Aufgabe lösen, ohne den Taschenrechner zu benutzen.

naturally means:

• We can solve the problem without using the calculator.

If the subject were different, German would often use a different structure, for example ohne dass ...

Because Taschenrechner is masculine, and here it is in the accusative.

That happens because benutzen takes a direct object in the accusative:

• den Taschenrechner benutzen = to use the calculator

So:

• nominative: der Taschenrechner
• accusative: den Taschenrechner

This is a good form to notice, because unlike die Aufgabe, the article does change here.

Because these are two different constructions.

1. After a modal verb like können, German uses a bare infinitive:

   • Wir können die Aufgabe lösen.
   • not Wir können die Aufgabe zu lösen

2. In an ohne ... zu infinitive clause, German does use zu:

   • ohne den Taschenrechner zu benutzen

So:

• können + infinitive → no zu
• ohne ... zu + infinitive → zu is required

There are commas for two reasons.

First, the ohne ... zu infinitive clause is set off with a comma:

• ..., ohne den Taschenrechner zu benutzen, ...

Second, the wenn clause is a subordinate clause, and subordinate clauses are separated by commas in German:

• ..., wenn wir ruhig nachdenken.

German uses commas more consistently than English in structures like these, so comma placement is something learners need to pay attention to.

Because wenn introduces a subordinate clause, and in German subordinate clauses send the conjugated verb to the end.

So:

• main clause: Wir denken ruhig nach.
• subordinate clause: wenn wir ruhig nachdenken

That final position is normal after wenn, weil, dass, obwohl, and many other subordinating conjunctions.

Wenn can mean either if or when, depending on context.

In this sentence, it is most naturally understood as if or when/whenever in a general sense:

• if we think calmly
• when we think calmly
• whenever we think calmly

English usually chooses one, but German wenn often covers both kinds of idea. If the sentence is presenting a condition, English often prefers if.

Here ruhig means something like calmly, carefully, or without panicking.

So ruhig nachdenken is not mainly about being physically quiet. It means thinking in a calm, composed way.

This is a very common use of ruhig in German. It often carries the idea of take it easy or just stay calm.

Yes. German can move the subordinate clause to the front:

• Wenn wir ruhig nachdenken, können wir die Aufgabe lösen, ohne den Taschenrechner zu benutzen.

That is completely natural.

Notice what happens in the main clause after the fronted wenn clause:

• ..., können wir ...

The verb können comes before the subject wir because the first position in the sentence is already occupied by the whole wenn clause.

Because nachdenken is a separable verb.

Its basic infinitive is:

• nachdenken

But in a normal main clause, the prefix separates:

• Wir denken nach.

In a subordinate clause or infinitive form, it stays together:

• wenn wir nachdenken
• nachzudenken

So the sentence is showing the unsplit form because nachdenken is at the end of a subordinate clause.

Yes. That would also be grammatical, but it changes the nuance slightly.

• ohne den Taschenrechner zu benutzen = without using the calculator
• ohne einen Taschenrechner zu benutzen = without using a/any calculator

The original sentence sounds like it is referring to the calculator as a known or contextually relevant object. Using einen makes it more indefinite.