This is a really common problem that I see at my hospital. It is called an avulsed nail.
Usually, the exposed "quick" is covered in a new nail in about 2 weeks and a new nail is almost normal in length in 4 weeks. You will have to start trimming it by about 8 weeks!
So, I ask people to keep their dogs on sidewalks and out of sand, snow and mud for a couple of weeks.
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In terms of improving the strength of weak, brittle nails, you might want to try giving Essential Fatty Acids, orally. These improve hair coat and nail strength, but they take about 8 to 12 weeks before you can see any improvement. EFA-Z and DermCaps are a couple of brands.
I hope that this helps you to help your dog!
If this has been helpful, please accept my answer and leave feedback. If you need more information, just click on reply and I will be back to provide it!
The above is given for information only. Although I am a licensed veterinarian, I cannot legally prescribe medicines or diagnose your pet's condition without performing a physical exam. If you have concerns about your pet I would strongly advise contacting your regular veterinarian.
Best wishes to you and your dog!