Question 1404c

Complex Numbers
Polar Form Arithmetic

Question

The complex number z{z} is given by
z=βα4γ,z=\frac{\beta^*}{\alpha^4 \gamma},
where
α=2e34πi,β=2e56πi,γ=2e13πi.\begin{align*} \alpha &= 2 \mathrm{e}^{\frac{3}{4} \pi \mathrm{i}}, \\ \beta &= 2 \mathrm{e}^{\frac{5}{6} \pi \mathrm{i}}, \\ \gamma &= 2 \mathrm{e}^{- \frac{1}{3} \pi \mathrm{i}}. \end{align*}
Find z{|z|} and k,{k,} where arg(z)=kπ{\arg(z)=k\pi} and π<arg(z)<π.{-\pi < \arg(z) < \pi.}

Attempt

z={|z| = }
k={k = }

Answer

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