All of the words above are adverbs and can be translated as "again," and they are often interchangeable. However, they do have slightly different connotations, and this means that one adverb may be used in certain situations or expressions exclusively.

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The adverb wieder is used to indicate repetition or a return to a previous state, though it may be uncertain exactly when this will occur. The connotation is "back again" or "repeatedly." You would not say nochmal da, nor would you switch out wieder in the two other expressions below. 

Wir sind wieder da auf Ganovenjagd

We are back again, chasing hoodlums

Caption 24, Die Pfefferkörner: Abschied tut weh

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Das sehen wir in dieser Stadt immer wieder.

We see this again and again in this city.

Caption 13, Architektur: der Stadt Karlsruhe

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Wir sind jetzt leider schon wieder am Ende der Sendung,

Unfortunately, we are now once again at the end of the program,

Caption 20, Abenteuer und Sport: Fallschirmspringen

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A casual shortening of noch einmal (literally "still one time"), the adverb nochmal suggests an immediate or additional repetition. The connotation is "another time," "a second time," or "one more time." As you might guess, danke nochmal is a standard expression, and neither wieder nor erneut would be used here. 

Danke nochmal, dass ihr mir geholfen habt, aber ich muss los.

Thanks again for helping me, but I have to go.

Caption 32, Die Pfefferkörner: Alles auf Anfang

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Ich wünschte, wir könnten uns heute nochmal begegnen.

I wish we could meet each other again today.

Caption 3, LOTTE: Zu jung

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Then there is erneut, which indicates "once again" and can have the connotation of a fresh attempt or starting over. You can remember this simply by the fact that the word neu is contained in the word, or by the related verb erneuern ("to renew," "to refresh"). Erneut is also slightly more formal than the other two adverbs, and can imply that something is inevitable or expected to repeat. 

Erneut rüsten sich die Murmeltiere für den Winter.

Once again, the marmots prepare for winter.

Caption 9, Wilde Schweiz: Sommer

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Die Prinzessin bat ihren Vater erneut um Rat.

The Princess asked her father for advice again.

Caption 65, Märchen - Sagenhaft Der Froschkönig

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Aber die Königin starb bald darauf und der König heiratete erneut.

But the Queen died soon afterwards and the king married again.

Caption 11, Märchen - Sagenhaft Schneewittchen

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Further Learning
You'll find many more examples to compare on Yabla German.

This week, we'll look at German verbs, adjectives, and adverbs that relate to taking care and being careful. 

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You may already know the word vorsichtig. As is normal for German adjectives, it is also an adverb, so it means both "careful" and "carefully."

Seid vorsichtig und macht bloß nichts kaputt.

Be careful and don't break anything.

Caption 47, Playmobil: Skispringen mit Familie Hauser

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Fahr vorsichtig, ja?

Drive carefully, OK?

Caption 35, Die Pfefferkörner: Alles auf Anfang

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Other words that can mean "careful" are sorgfältig, which implies a thoroughness or accuracy, and behutsam, which implies gentleness and caution.

Hast du die schon mal so sorgfältig bügeln sehen?

Have you ever seen them iron so carefully?

Caption 19, Playmobil: Skispringen mit Familie Hauser

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Wir alle sollten unsere Kontakte auch weiter behutsam einschränken.

We should all continue to cautiously restrict our contact.

Caption 57, Wir müssen über Corona reden: Mit Olaf Scholz, Luisa & Dr. Daniel Zickler

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Und ich nahm ihn ganz behutsam in die Hand

And I took it very gently in my hand

Caption 5, Nicole: Papillon

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Then there is also the word schonend. This relates to the verb schonen, which has a number of possible translations, including "to treat with care," "to protect," "to preserve," and "to rest" or "to go easy on."

Und es wird schonend mit dem Boden umgegangen, damit er viele Jahre Gemüse bringen kann.

And the soil is treated carefully so that it can produce vegetables for many years.

Captions 27-28, Berlin: Domäne Dahlem

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Um die Pferde zu schonen, werden die Vierbeiner übrigens selbst auch gedoubelt.

In order to treat the horses with care, by the way, these four-legged actors also have doubles.

Caption 21, Für Tierfreunde: Pferde beim Film

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Na ja, Frankfurt ist ja eine Stadt, die sehr früh als Großstadt politische Entscheidungen getroffen hat, Umwelt zu schonen. 

Well, yes, Frankfurt is a city that, very early on, as a major city, made political decisions to protect the environment.

Captions 50-51, Erstes Frankfurter „Schuljahr der Nachhaltigkeit“ Abschlussfeier

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Other verbs that relate to being careful or watching out are aufpassen and achtgeben, which can both relate to watching out for your own wellbeing or that of others. 

Da muss man aufpassen, wenn man die Straße überqueren will.

You have to be careful when you want to cross the street.

Caption 22, Shuah Auf der Straße in Berlin

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Kannst du auf die anderen achtgeben?

Can you watch over the others? 

Caption 12, Heidi Kann Heidi Peter retten?

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Further Learning
You can look up more examples with the words vorsichtig, sorgfältig, and behutsam on Yabla German in order to better understand the slight differences in their meanings.