Explanation

(b)
Let $z = \left( 1 + i\sqrt{3} \right)^{8}$
$= ( - 2)^{8}\left( \frac{1 + i\sqrt{3}}{- 2} \right) = ( - 2)^{8}\left( \omega^{2} \right)^{8}\ \ \ \ \ \ \ \lbrack{\because\omega}^{3} = 1\rbrack$
$= 2^{8}\omega^{16} = 2^{8}\omega$
$= 2^{8}\left( \frac{- 1 + i\sqrt{3}}{2} \right)$
$= 2^{8}\left( \cos{\frac{2\pi}{3} + i\sin\frac{2\pi}{3}} \right)$
$\therefore$ Modulus$= 2^{8} = 256$ and amplitude$= \frac{2\pi}{3}$