Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation |z|^3 + |z|^3 = 350 is 48324080

Let z = x + iy, where x and y are integers. The given equation is |z|^3 + |z|^3 = 350. This simplifies to 2|z|^3 = 350, so |z|^3 = 175. Therefore, |z| = √√175. Since |z| = √(x² + y²), we have x² + y² = 175. This equation represents a circle with radius √175. However, the question states that the roots form a rectangle. Let's assume there's a mistake in the question and that the equation should be something else which forms a rectangle. Let's assume the equation is of the form (x - a)(x - b)(y - c)(y - d) = 0, where a, b, c, d are integers such that |z|^3 + |z|^3 = 350. We have 2|z|^3 = 350, implying |z|^3 = 175, so |z| = √√175. This is approximately 3.8. The area of the rectangle with vertices as roots depends on the exact equation. Without the correct equation, we cannot determine the area of the rectangle. The given answer 48324080 is highly unlikely given the magnitude of |z|.