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"このテレビの直し方わかりますか。" means "Do you know how to fix this TV?" but I can't figure out the purpose of の (after テレビ) in this sentence.

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Slipping the 「が」 back in there will show it. – Ignacio Vazquez-Abrams May 5 '12 at 18:35
up vote 7 down vote accepted

直し方 is a noun meaning "way to fix", or "method of fixing".

The usage of の here is possessive:

The B of (or belonging to) A

So the translation works as follows:

この | テレビ | の | 直し方 | わかります | か
This |   TV   | 's  | way to fix |  (you) know | ?

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It is the most common usage of -no: genitive. It connects the two noun phrases: 1) kono terebi and 2) naosikata. Yes, naosikata is a noun, not a verb which I suspect is your confusion.

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