Powers and de Moivre's formula Problem set A

Work by hand. Give exact values.

1. (a) \(z = 3e^{i\frac{\pi}{12}}\). Find \(z^{4}\) in the form \(x + iy\).
2. (b) \(z = 2e^{i\frac{2\pi}{9}}\). Find \(z^{3}\) in the form \(x + iy\).
3. (c) \(z = \sqrt{2}e^{i\frac{5\pi}{8}}\). Find \(z^{4}\) in the form \(x + iy\).

Find the modulus and argument first, with \(0 \le \arg z < 2\pi\). Give exact values.

1. (a) \(z = \sqrt{3}+i\). Write \(z^{4}\) in the form \(x + iy\).
2. (b) \(z = -1+i\). Write \(z^{6}\) in the form \(x + iy\).
3. (c) \(z = 2-2i\). Write \(z^{4}\) in the form \(x + iy\).

A geometric sequence has first term \(u_1 = 20\) and common ratio \(r = 0.3 + 0.4i\).

1. (a) Find \(u_2\) and \(u_3\) in the form \(x + iy\).
2. (b) Show that \(|u_1|, |u_2|, |u_3|, \dots\) is also a geometric sequence, and write down its common ratio.
3. (c) Find the sum of the infinite sequence \(|u_1|, |u_2|, |u_3|, \dots\).

In a drawing program, a pen starts at the point \(1\) on the complex plane, measured in centimetres. At each step the pen's position is multiplied by \(m = 1.2e^{i\frac{\pi}{12}}\), so the pen spirals outwards. After \(n\) steps its position is \(m^n\).

1. (a) Find the position of the pen after 4 steps in the form \(x + iy\). Give \(x\) and \(y\) to 3 significant figures.
2. (b) Find the distance of the pen from the origin after 4 steps. Give your answer to 3 significant figures.

\(z = 1 + \sqrt{3}i\).

1. (a) Write \(z\) in exponential form.
2. (b) Find the smallest positive integer \(n\) for which \(z^n\) is real and negative, and write down \(z^n\).