Although both of the words above are most often translated as "together," and there are instances in which either one can be used, there are also some nuances to be aware of. 

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The word gemeinsam will often be used when there is a mutuality, an act of cooperating or working together, or an idea of having a common goal. Another translation could be "collectively" or "conjointly." With gemeinsam, there is an indication that a group exists or has been formed.

Heute wollen wir mal wieder gemeinsam kochen.

Today we want to cook together again.

Caption 2, Bundesländer und ihre Rezepte - Hessen

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Der Lehrer bespricht die Aufgaben gemeinsam mit den Schülern.

The teacher discusses the assignments with the students.

Caption 50, Deutsch mit Eylin - Pronomen

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Heute machen wir gemeinsam einen Einbürgerungstest.

Today we'll do a citizenship test together.

Caption 2, Bundesrepublik Deutschland - Einbürgerungstest

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The word zusammen refers to the state of being together with another person, but both people might still act independently or have different goals. 

Wie verbringt ihr Silvester?

How do you spend New Year's Eve?

-Mit meiner Familie und wir essen schön zusammen.

-With my family and we have a nice supper together.

Caption 59, Silvester Vorsätze für das neue Jahr - Karlsruhe

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Aber ihr seid doch erst seit ein paar Wochen zusammen. 

But you've only been together a few weeks. 

Caption 18, Nicos Weg - A2 Folge 15: Kinder

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Wir haben zum Beispiel Karten zusammen gespielt.

We played cards together, for example.

Caption 32, Angelique Kerber - Ihre tennisfreie Zeit

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This distinction is easy to remember if you can remember that there is also the phrase etwas gemeinsam haben, which means "to have something in common." 

Wir haben ja vieles gemeinsam.

We have a lot in common.

Caption 12, Weihnachtsmann gesucht - Der Engel

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Further Learning
When you watch videos on Yabla German, note when the word gemeinsam is used rather than zusammen. You can also write some sentences about what you and your friends have in common.