It is given that ${{z}^{3}}=4\sqrt{2}\left( -1-\mathrm{i} \right)$ and $\alpha ={{z}^{3}}+\frac{1}{{

Question

It is given that ${{z}^{3}}=4\sqrt{2}\left( -1-\mathrm{i} ight)$ and $\alpha ={{z}^{3}}+\frac{1}{{{\left( {{z}^{*}} ight)}^{3}}}$. (iii) Without using a calculator, find an exact expression for $\alpha .$ Give your answer in the form $r{{\mathrm{e}}^{\mathrm{i}	heta }}$, where $r>0$ and $-\pi (iv) Without using a calculator, find the three smallest positive whole number values of $n$ for which $\frac{{{\alpha }^{n}}}{{{\alpha }^{*}}}$ is a real number.

Accepted Answer

(iii) $ |\alpha| = \frac{65}{8} $, $ \arg\alpha = -\frac{3\pi}{4} $; (iv) $ n = 3, 7, 11 $

2015 TPJC P2 Q3 [Modified]

2015 TPJC P2 Q3 [Modified]

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