Take part in and participate in mean the same thing: to be actively involved in something.
- We’re here to take part in/ participate in the race. Where can we sign up?
- I don’t really want to take part in/ participate in this discussion.
Partake in also means to participate, however, it is less common and is generally used in more formal contexts:
- Fifty volunteers partook in the experiments.
- Have you ever partaken in any illegal activities?
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