7 The complex number \(u\) is defined by \(u = \frac { \sqrt { 2 } - a \sqrt { 2 } \mathrm { i } } { 1 + 2 \mathrm { i } }\), where \(a\) is a positive integer.
- Express \(u\) in terms of \(a\), in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
It is now given that \(a = 3\). - Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).
- Using your answer to part (b), find the two square roots of \(u\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\).