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find-x-y-from-the-equation-1-2-x-yi-1-1-i-1-8-1-10-4-




Question Number 25000 by ErickDN last updated on 30/Nov/17
find x,y from the equation:  (1/2)x−yi+(1/(1+i))=((√(1+ω^8 ))+(√(1+ω^(10) )))^4
$${find}\:{x},{y}\:{from}\:{the}\:{equation}: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{x}−{yi}+\frac{\mathrm{1}}{\mathrm{1}+{i}}=\left(\sqrt{\mathrm{1}+\omega^{\mathrm{8}} }+\sqrt{\mathrm{1}+\omega^{\mathrm{10}} }\right)^{\mathrm{4}} \\ $$$$ \\ $$
Answered by ajfour last updated on 01/Dec/17
⇒  (x/2)−iy+((1−i)/2)=((√(−ω))+(√(−ω^2 )))^4   (x/2)−iy=((−1+i)/2)+(e^(−((iπ)/6)) +e^((iπ)/6) )^4   (x/2)−iy=((−1+i)/2)+9  ⇒    x=17  ;  y=−(1/2) .
$$\Rightarrow\:\:\frac{{x}}{\mathrm{2}}−{iy}+\frac{\mathrm{1}−{i}}{\mathrm{2}}=\left(\sqrt{−\omega}+\sqrt{−\omega^{\mathrm{2}} }\right)^{\mathrm{4}} \\ $$$$\frac{{x}}{\mathrm{2}}−{iy}=\frac{−\mathrm{1}+{i}}{\mathrm{2}}+\left({e}^{−\frac{{i}\pi}{\mathrm{6}}} +{e}^{\frac{{i}\pi}{\mathrm{6}}} \right)^{\mathrm{4}} \\ $$$$\frac{{x}}{\mathrm{2}}−{iy}=\frac{−\mathrm{1}+{i}}{\mathrm{2}}+\mathrm{9} \\ $$$$\Rightarrow\:\:\:\:{x}=\mathrm{17}\:\:;\:\:{y}=−\frac{\mathrm{1}}{\mathrm{2}}\:. \\ $$

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