(i) The complex number \(u = \sqrt{3} + i\) can be expressed in polar form as \(re^{i\theta}\). The modulus \(r\) is given by \(r = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{4} = 2\).

The argument \(\theta\) is \(\theta = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}\).

Thus, \(u = 2e^{i\frac{\pi}{6}}\).

The modulus of \(u^4\) is \((2)^4 = 16\), and the argument of \(u^4\) is \(4 \times \frac{\pi}{6} = \frac{2\pi}{3}\).

(ii) Substitute \(u = \sqrt{3} + i\) into the equation \(z^3 - 8z + 8\sqrt{3} = 0\) and verify that it satisfies the equation. The other root is \(\sqrt{3} - i\).

(iii) On the Argand diagram, plot the point \(u = \sqrt{3} + i\). Draw a circle centered at \(u\) with radius 2. Draw the line \(y = 2\). Shade the region where \(|z - u| \leq 2\) and \(\text{Im } z \geq 2\).