Use De Moivre’s theorem for integral powers. 𝑧𝑛 = 𝑟𝑛 cis (𝑛𝜃)

A telecommunications engineer models signal interference as a complex number \(z = 2\text{cis}\left(\frac{\pi}{8}\right)\). a) Use De Moivre's theorem to determine \(z^4\) in modulus-argument form. (1 mark) b) Express your result from part a) in Cartesian form \(a + bi\). (1 mark) c) Explain whether \(z^4\) is a real number, a purely imaginary number, or neither. Justify your answer. (1 mark)

Consider the complex number \(z = 1 + i\sqrt{3}\) where \(i = \sqrt{-1}\). Use De Moivre's theorem to determine the value of \(z^5\) in the form \(a + bi\), where \(a, b \in \mathbb{R}\).

Consider the complex number \(z = 2\,\text{cis}\left(\frac{\pi}{8}\right)\). a) Use De Moivre's theorem to determine \(z^4\) in modulus-argument form. (1 mark) b) Hence express \(z^4\) in Cartesian form \(a + bi\). (1 mark) c) Use your result from part (b) to calculate \((z^4)^2\) in Cartesian form. (1 mark)