Learn Dutch: Ik zie de zanger optreden op het festival. - I see the singer perform at the festival.

Because zien (to see) is a verb of perception. When you use verbs like zien, horen (to hear), or voelen (to feel) to describe someone doing an action, the second action is a bare infinitive (the whole verb without te). This is exactly like English, where you say 'I see the singer perform' instead of 'to perform'.

I learned that optreden is a separable verb. Why isn't it separated here?

Separable verbs only split when they act as the main, conjugated verb in a standard sentence (like ik treed op). Because zie is the main verb here, optreden stays in its full, unseparated infinitive form.

Could I also say 'Ik zie de zanger op het festival optreden'?

Yes, absolutely! In Dutch, the infinitive normally goes at the very end of the sentence. However, prepositional phrases indicating time or place (like op het festival) are often allowed to come after the infinitive. Both word orders are correct and natural.

Why do we use op for 'at the festival' instead of bij or in?

In Dutch, you generally use op (on) for public events, parties, and festivals. You can think of it like being 'on the festival grounds'.