Convert to exponential/polar form

5 The complex number \(z\) is defined by \(z = \frac { 9 \sqrt { } 3 + 9 i } { \sqrt { } 3 - i }\). Find, showing all your working,

1. an expression for \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\),
2. the two square roots of \(z\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

4 The complex number \(u\) is given by \(u = - 1 - \mathrm { i } \sqrt { 3 }\).

1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the exact values of \(r\) and \(\theta\).
   The complex number \(v\) is given by \(v = 5 \left( \cos \frac { 1 } { 6 } \pi + \mathrm { i } \sin \frac { 1 } { 6 } \pi \right)\).
2. Express the complex number \(\frac { \mathrm { v } } { \mathrm { u } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

Express the complex number \(1 - i\sqrt{3}\) in modulus-argument form. Tick \((\checkmark)\) one box. [1 mark] \(2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\) \(2\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)\) \(2\left(\cos\left(-\frac{\pi}{3}\right) + i\sin\left(-\frac{\pi}{3}\right)\right)\) \(2\left(\cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right)\right)\)

A complex number is defined by \(z = -3 + 4\mathrm{i}\).

1. 1. Express \(z\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\), where \(-\pi \leqslant \theta \leqslant \pi\).
   2. Express \(\bar{z}\), the complex conjugate of \(z\), in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [4]

Another complex number is defined as \(w = \sqrt{5}(\cos 2.68 + \mathrm{i}\sin 2.68)\).

1. Express \(zw\) in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\). [3]