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Question Number 139023 by mohammad17 last updated on 21/Apr/21
find Re(z) and Im(z) of z=(−2i)^(−(3/2))
$${find}\:{Re}\left({z}\right)\:{and}\:{Im}\left({z}\right)\:{of}\:{z}=\left(−\mathrm{2}{i}\right)^{−\frac{\mathrm{3}}{\mathrm{2}}} \\ $$
Answered by MJS_new last updated on 21/Apr/21
−2i=2e^(−(π/2)i)   (2e^(−(π/2)i) )^(−(3/2)) =2^(−(3/2)) e^(((3π)/4)i) =((√2)/4)(cos ((3π)/4) +i sin ((3π)/4))=  =−(1/4)+(1/4)i
$$−\mathrm{2i}=\mathrm{2e}^{−\frac{\pi}{\mathrm{2}}\mathrm{i}} \\ $$$$\left(\mathrm{2e}^{−\frac{\pi}{\mathrm{2}}\mathrm{i}} \right)^{−\frac{\mathrm{3}}{\mathrm{2}}} =\mathrm{2}^{−\frac{\mathrm{3}}{\mathrm{2}}} \mathrm{e}^{\frac{\mathrm{3}\pi}{\mathrm{4}}\mathrm{i}} =\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\left(\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{4}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{4}}\right)= \\ $$$$=−\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{4}}\mathrm{i} \\ $$

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