Question 1404c

Complex Numbers

Polar Form Arithmetic

Question

The complex number

{z}

is given by

z=\frac{\alpha^3}{\beta^* \gamma},

where

\begin{align*} \alpha &= 2 \mathrm{e}^{- \frac{1}{3} \pi \mathrm{i}}, \\ \beta &= 3 \mathrm{e}^{\frac{3}{4} \pi \mathrm{i}}, \\ \gamma &= \mathrm{e}^{- \frac{5}{6} \pi \mathrm{i}}. \end{align*}

Find

{|z|}

and

{k,}

where

{\arg(z)=k\pi}

and

{-\pi < \arg(z) < \pi.}

Attempt

{|z| = }

{k = }

Answer

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