republikanisch
republican
Relating to or supporting a republic as a form of government, as opposed to a monarchy.
Die französische Revolution führte eine republikanische Regierung ein.
The French Revolution introduced a republican government.

republican
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republican
Relating to or supporting a republic as a form of government, as opposed to a monarchy.
Die französische Revolution führte eine republikanische Regierung ein.
The French Revolution introduced a republican government.

The adjective 'republikanisch' (republican) is derived directly from the noun 'die Republik' (the republic). It describes everything that has to do with this form of government.
Viele Länder haben eine republikanische Verfassung.
Many countries have a republican constitution.
To distinguish the meanings, pay attention to the context. If it is about forms of government or history, the general meaning is intended. If it is about current politics in the United States, it usually refers to the party.
Im Geschichtsunterricht sprachen wir über die republikanische Bewegung in Deutschland.
In history class, we talked about the republican movement in Germany.
In the general meaning, 'republikanisch' (republican) is often combined with nouns such as 'Staatsform' (form of government), 'Regierung' (government), 'Verfassung' (constitution), or 'Gedanke' (idea).
Die Weimarer Republik war Deutschlands erster Versuch einer republikanischen Demokratie.
The Weimar Republic was Germany's first attempt at a republican democracy.
When it is about the U.S. party, 'republikanisch' (Republican) often comes before words like 'Partei' (party), 'Kandidat' (candidate), 'Wähler' (voter), or 'Senator' (senator).
Er hat bei der letzten Wahl für den republikanischen Kandidaten gestimmt.
He voted for the Republican candidate in the last election.
democratic
'demokratisch' and 'republikanisch' can both be political words, but they describe different things. 'demokratisch' says that power comes from the people and that people can take part in decisions. 'republikanisch' refers to a republic, meaning a state without a monarchy, or to the Republican Party in the United States. The words can be confused because a republic can often also be democratic. Even so, the question is different: 'demokratisch' asks about participation in decision-making, while 'republikanisch' asks about the form of government or about a specific party.

Im Geschichtsunterricht sprechen wir über republikanische Ideen nach dem Ende der Monarchie.
In history class, we talk about republican ideas after the end of the monarchy.
Die Wahl soll frei und demokratisch ablaufen.
The election should be free and democratic.