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Dr. King’s spellbinding message of hope, love, and the universality of mankind instilled the confidence that even someone like me, a white-looking black girl with no ostensible power, could create words to arouse passion, insight, and social change.
What does the highlighted part mean?
I think the confusion comes from her description of herself as a "white-looking black girl", and the fact that this perhaps looks contradictory. For example a "red-looking green paint" would seem odd, since paint cannot be simultaneously "red" and also "green".
The resolution to this conundrum lies in the fact that "white" and "black" are often in English to describe someone's race (or ethnic origin) as well as to describe the color of skin, and hence a "white-looking black girl" means a girl who is ethnically "black" (for instance, perhaps one or both of her parents are African-American), but who does not have dark skin.
Consequently it may not be obvious to someone merely looking at her that she belongs to (or identifies with) the "black" community compared with the "white" community - bearing in mind also that historically at the time of Dr. King these two communities were very much more separate than they are today.
For this reason, she refers to herself as a "white-looking black girl".
"With no ostensible power" is much easier to understand. A quick glance at a dictionary should tell you that this means she considered herself to have no evident power.
