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Fact: Hair needs to be cleansed, preferably on a regular basis. Never washing your hair is unsanitary.
With that out of the way, let's address why the idea of dirty hair promoting hair growth is still so prevalent. As you probably already know, black hair doesn't require the same frequent cleansing that naturally straight hair needs. You can shampoo once a week and have healthy locks. The days of believing that once-monthly washes are fine are over, however. One of the reasons some people believe dirty hair promotes growth is because too-frequent washing encourages dryness, brittleness and breakage. If you shampoo too often (for black hair, this could be daily with a harsh cleanser), yes, your hair will dry out and break. The alternative -- never washing -- isn't suitable. Natural oils, hair products, scalp buildup and air pollution all build up on your hair. Think about never washing it out, or only washing once a month or so. This isn't helping where growth is concerned. A clean scalp is a healthy scalp, so shampoo with a gentle product on a regular schedule. All hair grows and it grows an average of 1/4 to 1/2 inch per month, dirty or clean.
Bottom line: Dirty hair doesn't grow any faster than clean hair, so you may as well have a sweet-smelling scalp and fresh tresses. They'll look better, feel better and be healthier, too.