"How can I show that I am free?"
Translation:Come faccio a dimostrare di essere libero?
I'm pretty sure you use dimostrate to demonstrate something, mostrare to show something (e.g. "look what I found!") and indicare to point something out (e.g. "show me the one you want.") English allows "show" for all three, but I don't believe Italian does. So, no, I don't think you can use mostrare here, but maybe a native speaker will chime in.
Also, would 'Come posso mostrare che sono libero?' also be correct? I don't understand why faccio is being used here (it seems difficult to render a literal translation here that makes sense in English: 'I do(make?) to demonstrate of being free').
I put: "come posso dimostrare di essere libero?" It was marked correct, but I feel like it's wrong for some reason - I suppose I'm asking: so it's correct to not use "fare," as in the ideal translation?
Is there a rule when to use "di" or "a" with the infinitive form? I cannot see a pattern of when they are needed and when they are not.
In italian you cannot use the same subject in a second claus
So you can't say "Penso che sono una brava persona"
instead it would have to be "Penso di essere una brava persona"
I think it's complicated - see http://italian.about.com/od/verbs/a/aa031908a.htm. It appears that the use of 'di' or 'a' is associated more with the preceding verb than the infinitive.
I also came across another comment from an Italian speaker that proposes a more general rule (see http://answers.yahoo.com/question/index?qid=20130906004535AA9sS0r), but I can't really see how to map it easily onto the list of verbs from the first link.
Finally, here's a link that suggests that it's easiest to memorize a shorter list of verbs that take 'a', while most of the others take 'di': http://tutorino.ca/grammatica/2007/10/26/uses-of-the-infinitive-in-italian.html (though 'fare' is not on the list :().