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Consider $f=(a+bi)^{3}$, where $a,b\in\mathbb R$.

(a) In terms of $a$ and $b$, find

(i) the real part of $f$;
(ii) the imaginary part of $f$.

(b) Hence, or otherwise, show that $(1+\sqrt{3}\,i)^{3}=-8$.

The roots of the equation $z^{3}=-8$ are $u,\;v,\;w$, where $u=1+\sqrt{3}\,i$ and $v\in\mathbb R$.

(c) Write down $v$ and $w$, giving your answers in Cartesian form.

On an Argand diagram, $u,\;v,\;w$ are represented by the points $U,\;V,\;W$ respectively.

(d) Find the area of triangle $UVW$.

Each of the points $U,\;V,\;W$ is rotated counter‑clockwise about the origin through an angle $\pi/4$ to form new points $U',\;V',\;W'$ representing complex numbers $u',\;v',\;w'$ respectively.

(e) Find $u',\;v',\;w'$, giving your answers in the form $r\,e^{i\theta}$, where $-\pi<\theta\le\pi$.

(f) Given that $u',\;v',\;w'$ are the solutions of $z^{3}=c+di$, where $c,d\in\mathbb R$, find the values of $c$ and $d$.

It is given that $u,\;v,\;w,\;u',\;v',\;w'$ are all solutions of $z^{n}=a$ for some $a\in\mathbb C$, where $n\in\mathbb N$.

(g) Find the smallest positive value of $n$.