Z = ((1 + i) ^ 3 * (2i) ^ 4)/((- 1 + i) ^ 4 * (- i) ^ 3)

Sure, let's simplify the expression step by step.

Step 1: Simplify the numerator To simplify the numerator, we can expand the powers of (1 + i) and (2i):

(1 + i) ^ 3 = (1 + i) * (1 + i) * (1 + i) = (1 + 2i + i^2) * (1 + i) = (1 + 2i - 1) * (1 + i) = 2i * (1 + i) = 2i + 2i^2 = 2i - 2

(2i) ^ 4 = (2i) * (2i) * (2i) * (2i) = (4i^2) * (4i^2) = (-4) * (-4) = 16

So, the numerator simplifies to 2i - 2 * 16.

Step 2: Simplify the denominator To simplify the denominator, we can expand the powers of (-1 + i) and (-i):

(-1 + i) ^ 4 = (-1 + i) * (-1 + i) * (-1 + i) * (-1 + i) = (1 - i - i + i^2) * (-1 + i) * (-1 + i) = (1 - 2i + i^2) * (-1 + i) = (-1 + 2i - 1) *