$z = \frac{1 - i\sqrt{3}}{2} = -\left( \frac{-1 + i\sqrt{3}}{2} \right) = -\omega$

Now, $S = 21 + \left( z + \frac{1}{z} \right)^3 + \left( z^2 + \frac{1}{z^2} \right)^3 + \left( z^3 + \frac{1}{z^3} \right)^3 + \ldots + \left( z^{21} + \frac{1}{z^{21}} \right)^3$

$= 21 + \left\{ \left[ -\left( \omega + \frac{1}{\omega} \right)^3 \right] + \left( \omega^2 + \frac{1}{\omega^2} \right)^3 - \left( \omega^3 + \frac{1}{\omega^3} \right)^3 + \left( \omega^4 + \frac{1}{\omega^4} \right)^3 - \left( \omega^5 + \frac{1}{\omega^5} \right)^3 + \left( \omega^6 + \frac{1}{\omega^6} \right)^3 + \ldots \right.$

$\left. + \left[ -\left( \omega^{13} + \frac{1}{\omega^{13}} \right)^3 \right] + \left( \omega^{14} + \frac{1}{\omega^{14}} \right)^3 - \left( \omega^{15} + \frac{1}{\omega^{15}} \right)^3 + \left( \omega^{16} + \frac{1}{\omega^{16}} \right)^3 - \left( \omega^{17} + \frac{1}{\omega^{17}} \right)^3 + \left( \omega^{18} + \frac{1}{\omega^{18}} \right)^3 \right\}$

$+ \left[ -\left( \omega^{19} + \frac{1}{\omega^{19}} \right)^3 \right] + \left( \omega^{20} + \frac{1}{\omega^{20}} \right)^3 - \left( \omega^{21} + \frac{1}{\omega^{21}} \right)^3$

Clearly, all the terms present in curly bracket cancels each other.

Hence, $S = 21 + \{0\} + \left[ -\left( \omega + \frac{1}{\omega} \right)^3 \right] + \left( \omega^2 + \frac{1}{\omega^2} \right)^3 - (2^3)$

$= 21 - \left( \frac{\omega^2 + 1}{\omega} \right)^3 + \left( \frac{\omega^4 + 1}{\omega^2} \right)^3 - 8 = 21 - \left( \frac{-\omega}{\omega} \right)^3 + \left( \frac{-\omega^2}{\omega^2} \right)^3 - 8 = 21 + 1 - 1 - 8 = 13$

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