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Braids and perms both do a number on your strands, leaving them with nicks and abrasions. Combining the two treatments in a small window is a definite don't: It can leave you with strand breakage, and your perm can take unevenly. Give your mane a rest after wearing braids so you can look your best and save your hair when you perm it.
Your hair needs a few days to decompress after coming out of braids before you subject it to perm chemicals. When your hair is braided, it's under pressure. Allow the hair to breathe and relax before subjecting strands to a different kind of pressure. If your hair gets too stressed, it can break during the perm process or sustain nicks and abrasions that lead it to break months down the road. You're better off safe than sorry here.
As a general rule, wait at least five to seven days to perm your hair after taking down braids. This allows the strands to relax. You can wait longer if you like, but definitely give yourself -- and your hair -- a week off.
Use the week of down time to care for your hair. Help fill in rough spots that occur when you remove braids by using a deep reconstructing treatment for hair, which you can find at local beauty supply shops. This treatment will add protein and moisture to your hair, prepping it for the perm process. If you don't have time for a fancy hair treatment, use shampoo and deep conditioner to replenish hair nutrients.
If you want to put permed hair back in braids, wait four to six weeks. After this time, your scalp won't be as sensitive from the chemicals. Using a deep reconstructing treatment will minimize hair breakage and your perm will take more evenly. However, it won't totally stop breakage that can occur during the braiding process.