Let $ z_1,z_2$ $\in$ $\mathbb{C}$ and A,D their respective images in the complex plane. let B be the image of $z_1³$ in the complex plane.
A is in the first quadrant and B is in the second quadrant.
Let [OABD] be a rectangle with 256 units of area. O is the origin of the complex plane.
The task is to determine $z_1$ in its algebric form (a+bi)
Hint: can you draw rectangle OABD? if not, which point (A,B,or D) that you have difficulty to specify?
Let $z_1=\rho e^{i \phi}$ then $z_1^3=\rho^3 e^{3i \phi}$.
The area of rectangle OABD is twice of the area for triangle OAB. So $S(OAB)=256/2=128$.
$$S(OAB)=(1/2)|OA||OB|\sin(2\phi)=(1/2)\sin(2\phi)\rho^4=128......(1)$$
Since A is in the first quadrant and B is in the second quadrant, $\pi/6<\phi<\pi/3$.
And $$\cos(2\phi)=|OA|/|OB|=\rho/\rho^3=\rho^{-2}......(2)$$
From (1) and (2) we obtain:
$$\rho^4=(1/2)(1 + 5\sqrt{41})$$
$$\phi=(1/2)\arcsin\left(\frac{256}{\rho^4}\right)$$
Finally we have:
$$z_1=\rho e^{i \phi}=\rho \cos\phi+i\rho \sin\phi$$