Learn Turkish: Siz kütüphanede çalışıyorsanız sakız çiğnememelisiniz. - If you are studying in the library, you should not chew gum.

This uses the current lesson's grammar. It takes the verb root çalış- (study/work), adds the present continuous suffix -ıyor (is doing), then the conditional -sa (if), and finally the "you plural/formal" ending -nız. Together, çalış-ıyor-sa-nız means "if you are studying".

I thought çalışmak meant "to work". Why is it translated as "studying" here?

Çalışmak means both "to work" at a job and "to study" (like doing homework or preparing for a test). Because this action is happening in a library (kütüphanede), "studying" makes the most sense in context.

Why are there two me syllables in a row in çiğnememelisiniz?

The root is çiğne- (chew). The first -me is the negative suffix ("not"). The second -me is the beginning of the -meli suffix ("should" or "must"). When you add the personal ending -siniz, çiğne-me-meli-siniz becomes "you should not chew".

Why do we need the pronoun siz at the beginning if the verbs already tell us who is doing the action?

You don't strictly need it! Turkish verbs already show the subject with their endings (like -nız and -siniz). Adding the pronoun siz here is completely optional and just adds a little extra emphasis to "you".