Express (1 – 2i)^–3 in the form (a + ib).

If (x + iy)³ = (u + iv) then prove that = 4 (x² – y²).

If (x + iy)^1/3 = (a + ib) then prove that = 4 (a² – b²).

Find real values of x and y for which

(x⁴ + 2xi) – (3x² + iy) = (3 – 5i) + (1 + 2iy).

If z² + |z|² = 0, show that z is purely imaginary.