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Question

The complex number z is equal to 9 + bi, where b is a positive real number and i^2 = -1. Given that the imaginary parts of z^2 and z^3 are equal, find b.

Accepted Answer

We compute z^2 = (9+bi)^2 = 81 + 18bi - b^2and z^3 = 729 + 243bi - 27b^2 - b^3i^3.Therefore, setting the imaginary parts equal, we get 18b = 243b - b^3,or b^3 = 225b. Since b > 0, we can divide by b to get b^2 = 225, and so b = 15.