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Yes, but it's important to take some precautions to make sure the lifespan and health of both your original hair and the extensions.
Here are some tips for washing and styling hair with tape-in extensions:
Washing:
Use a Gentle shampoo and conditioner, as sulfates can be harsh and may loosen the adhesive of the tape in Extensions
Gently massage the scalp and the roots of your hair, avoiding vigorous rubbing or pulling on the extensions.
Rinse this to remove all shampoo and conditioner residue. Avoid applying conditioner directly to the tape tabs, as this can cause them to slip.

Drying: slowly squeeze excess water from your hair with a towel, carefully the extensions.
Try to dry your hair in air whenever possible to minimize heat damage.
If using a blow dryer, use a low to medium heat setting and avoid direct heat on the tape tabs.

Brushing: Use a soft comb or a special extension brush to apart your hair, start from the top and working your way up to the roots.
Be gentle when brushing near the tape tabs to avoid pulling the extensions.

Styling:
You can style your hair with tape-in extensions using heating tools such as flat pressing machine and iron rod and hair dryers.

Use a heat protecting gel or serum before apply heat to your hair to minimize damage.
Avoid applying heat directly to the tape tabs to prevent them from loosening.

By following these tips and taking proper care of your hair and tape-in extensions, you can enjoy beautiful, long-lasting results without compromising the health of your natural hair.