Complex numbers：練習問題

Express the complex number \(z = 4\left(\cos \frac{2\pi}{3} + i\sin \frac{2\pi}{3}\right)\) in the Cartesian form \(x + iy\).

The complex number \( z \) satisfies the equation \( |z - 4i| = 2|z - i| \). Find the Cartesian equation of the locus of \( z \) in the Argand diagram.

Given that the complex number \(z\) satisfies the equation \(3z - z^* = 4 + 8i\), where \(z^*\) denotes the complex conjugate of \(z\), find the value of \(z\).

The locus of points \(z\) in the Argand diagram is defined by the equation \(|z - 3| = |z + i|\). Find the Cartesian equation of this locus in terms of \(x\) and \(y\).

The quadratic equation \( 2z^2 + (p + i)z + q = 0 \), where \( p \) and \( q \) are real constants, has a root \( z = 1 - i \). Find the values of \( p \) and \( q \).

Given that \(z = 1 + i\sqrt{3}\), express \(z^6\) in the form \(a + bi\), where \(a\) and \(b\) are real constants.