Powers and de Moivre's formula Problem set B

Work by hand. Give exact values.

1. (a) \(z = 2e^{i\frac{\pi}{15}}\). Find \(z^{5}\) in the form \(x + iy\).
2. (b) \(z = 4e^{i\frac{\pi}{4}}\). Find \(z^{3}\) in the form \(x + iy\).
3. (c) \(z = \sqrt{3}e^{i\frac{7\pi}{6}}\). 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 \(z^{3}\) in the form \(x + iy\).
2. (b) \(z = -2-2i\). Write \(z^{6}\) in the form \(x + iy\).
3. (c) \(z = 2-2 \sqrt{3}i\). Write \(z^{2}\) in the form \(x + iy\).

A geometric sequence has first term \(u_1 = 16\) and common ratio \(r = -0.45 + 0.6i\).

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 video animation, a point on a shrinking, turning logo starts at \(1\) on the complex plane, measured in centimetres from the centre of the screen. At each frame the point's position is multiplied by \(m = 0.8e^{i\frac{\pi}{9}}\). After \(n\) frames its position is \(m^n\).

1. (a) Find the position of the point after 3 frames in the form \(x + iy\). Give \(x\) and \(y\) to 3 significant figures.
2. (b) Find the distance of the point from the centre of the screen after 3 frames. Give your answer to 3 significant figures.

\(z = 1 - i\).

1. (a) Write \(z\) in exponential form, with \(0 \le \arg z < 2\pi\).
2. (b) Find the smallest positive integer \(n\) for which \(z^n\) is real and negative, and write down \(z^n\).