Squaring the equation z + 1/z = 1 gives z^2 + 2 + 1/z^2 = 1. Then multiplying by z^2 gives z^4 + 2z^2 + 1 = z^2. Rearranging, we get z^4 + 2z^2 + 1 - z^2 = 0, which factors as (z^2 + 1)(z^2 + 1) = 0. Since z^2 + 1 = 0 has no solutions, z^2 = -1. Taking the square root of both sides, we find z = i or z = -i.
If z = i, then z^3 = i^3 = \boxed{-i}.
Alternatively, if z = -i, then z^3 = (-i)^3 = -i^3 = -(-i) = \boxed{i}.
Let z be a complex number such that z + 1/z = 1. What is z^3?
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