In Roots, how are African values and cultural beliefs instrumental in Kunta Kinte's survival in America?  

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Kunta Kinte repeatedly refers to the fact that he knows who he is, and that he is sure of his identity. This is something that he comes to realise is a gift that few of the slaves he meets in America and works alongside actually possess. Note what he thinks about Josephus, the gardener who dies:

He wondered what the gardener's true name had been--the name of his African forefathers--and to what tribe they had belonged. He wondered if the gardener himself had known. More likely he had died as he had lived--withotu ever learning who he really was.

For Kunta Kinte, nothing is worse than not knowing who you are and being unsure of your identity. The very firm knowledge of who he is and where he comes from acts as a source of strength and power that allows Kunta Kinte to endure the many different forms of suffering he has to accept as part of his status as slave. In his narrative, he often compares what he sees and the cultural practices of the whites with those of his own culture, which again allow him to remain true to himself as far as possible. His hope is that he will be able to teach Kizzy about her roots as well so that she will be able to carry the same strength and knowledge about her identity with her through her life, and that it will similarly strengthen and support her.