Water changes left prevent algae growth?

I have previously posted about the hair algae in my tank. Whenever I try to trim and remove affected parts of plants, there's always a small bit of hair algae left that eventually recovers and spreads back. My ammonia, nitrites, and nitrates are all 0. I have heard that water changes are supposed to significantly help against algae, but I'm not sure if it'll help me if they are all at 0 already.

Thoughts?