\((i) The complex number w is given by w = -1 + i. First, find w²:\)

\(w² = (-1 + i)² = 1 - 2i - 1 = -2i.\)

\(The modulus of w² is |w²| = |-2i| = 2.\)

\(The argument of w² is arg(w²) = -\frac{\pi}{2}.\)

Next, find w³:

\(w³ = w \cdot w² = (-1 + i)(-2i) = 2 - 2i.\)

\(The modulus of w³ is |w³| = \sqrt{2^2 + (-2)^2} = 2\sqrt{2}.\)

\(The argument of w³ is arg(w³) = \frac{\pi}{4}.\)

(ii) The center of the circle is the midpoint of w and w²:

\(Center = \left(\frac{-1 + 0}{2}, \frac{1 - 2}{2}\right) = \left(-\frac{1}{2}, -\frac{1}{2}\right).\)

The diameter is |w - w²| = |-1 + i + 2i| = |-1 + 3i| = \sqrt{(-1)^2 + 3^2} = \sqrt{10}.

\(The radius is \frac{\sqrt{10}}{2}.\)

\(The equation of the circle is |z + \frac{1}{2} + \frac{1}{2}i| = \frac{1}{2}\sqrt{10}.\)