jmc

Question

Let a and b be nonzero complex numbers such that a^2 + ab + b^2 = 0. Evaluate a^9 + b^9(a + b)^9.

Accepted Answer

Since a^2 + ab + b^2 = 0, (a - b)(a^2 + ab + b^2) = 0. This simplifies to a^3 - b^3 = 0, so a^3 = b^3. Then b^9 = a^9. Also, (a + b)^2 = a^2 + 2ab + b^2 = (a^2 + ab + b^2) + ab = ab,so (a + b)^3 = ab(a + b) = a(ab + b^2) = a(-a^2) = -a^3.Then (a + b)^9 = (-a^3)^3 = -a^9, so a^9 + b^9(a + b)^9 = 2a^9-a^9 = -2.