Given that ${{z}_{1}}=2{{\mathrm{e}}^{\mathbf{i}\,\left( \frac{\pi }{5} \right)}}$ and ${{z}_{2}}=2{

Question

Given that ${{z}_{1}}=2{{\mathrm{e}}^{\mathbf{i}\,\left( \frac{\pi }{5} \right)}}$ and ${{z}_{2}}=2{{\mathrm{e}}^{\mathbf{i}\,\left( \frac{3\pi }{5} \right)}}$, show that ${{z}_{1}}^{5}={{z}_{2}}^{5}=k$, where $k$ is a real number to be determined. [3] determine the modulus and argument of the complex number ${{z}_{1}}-{{z}_{2}}$ in exact form. [4]

Accepted Answer

$z_1^5 = z_2^5 = -2^5$; $|z_1 - z_2| = 4\sin\frac{\pi}{5}$, $\arg(z_1 - z_2) = -\frac{\pi}{10}$

HCI Complex Numbers Tutorial Q10

HCI Complex Numbers Tutorial Q10

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