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Question Number 204510 by Ghisom last updated on 20/Feb/24

$$\mathrm{solve}\mathrm{for}x\ne y\wedge y\ne z\wedge z\ne x$$$$\left(\mathrm{exact}\mathrm{solutions}\mathrm{required}\right)$$$$\sqrt{(-3+4\mathrm{i})x}=y$$$$\sqrt{(-3+4\mathrm{i})y}=z$$$$\sqrt{(-3+4\mathrm{i})z}=x$$

Answered by Frix last updated on 20/Feb/24

$$\mathrm{Obviously}x=y=z=0\mathrm{but}\mathrm{this}\mathrm{is}\mathrm{excluded}.$$$$\mathrm{Btw}x=y=z=(-3+4\mathrm{i}){\mathrm{doesn}}^{\prime }\mathrm{t}\mathrm{work}\mathrm{anyway}$$$$\mathrm{because}\sqrt{{(-3+4\mathrm{i})}^2}=3-4\mathrm{i}\ne -3+4\mathrm{i}$$$$$$$$\sqrt{(-3+4\mathrm{i})z}=x\Rightarrow z=-\frac{(3+4\mathrm{i})x^2}{25}$$$$\sqrt{(-3+4\mathrm{i})y}=-\frac{(3+4\mathrm{i})x^2}{25}\Rightarrow y=\frac{(117-44\mathrm{i})x^4}{15625}$$$$\sqrt{(-3+4\mathrm{i})x}=\frac{(117-44\mathrm{i})x^4}{15625}$$$$x=-\frac{(76443-16124\mathrm{i})x^8}{6103515625};x\ne 0$$$$x^7=-76443+16124\mathrm{i}$$$$x^7=5^7{\mathrm{e}}^{\mathrm{i}(\pi -{\tan }^{-1}\frac{16124}{76443})}$$$$\mathrm{The}\mathrm{rest}\mathrm{is}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{fitting}\mathrm{solution}...$$$$\mathrm{Sorted}\mathrm{by}\mathrm{the}\mathrm{angle}(\mathrm{circular}\mathrm{solutions}$$$$x\to y\to z\to x)$$$$x={5\mathrm{e}}^{\mathrm{i}\left(\frac{\pi -{\tan }^{-1}\frac{16124}{76443}}{7}\right)}\approx 4.56727+2.03470\mathrm{i}$$$$y={5\mathrm{e}}^{\mathrm{i}\left(\frac{3\pi -{\tan }^{-1}\frac{16124}{76443}}{7}\right)}\approx 1.25686+4.83945\mathrm{i}$$$$z={5\mathrm{e}}^{-\mathrm{i}\left(\frac{3\pi +{\tan }^{-1}\frac{16124}{76443}}{7}\right)}\approx .967372-4.90553\mathrm{i}$$

Commented by Ghisom last updated on 20/Feb/24

thank you

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