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Question Number 66431 by hmamarques1994@gmail.com last updated on 15/Aug/19

$$$$$$\mathbf{Determine}\mathbf{x}\mathbf{e}\mathbf{y}:$$$$$$$$\{\begin{array}{l}{\mathbf{x}}^{\frac{1}{\sqrt{\mathbf{i}}}}+\frac{1}{{\mathbf{y}}^{\mathbf{i}\sqrt{\mathbf{i}}}}=10\\ \frac{1}{{\left(\mathbf{xy}\right)}^{\mathbf{i}\sqrt{\mathbf{i}}}}=21\end{array}$$$$$$

Answered by MJS last updated on 15/Aug/19

$$x^{\frac{1}{\sqrt{\mathrm{i}}}}=x^{\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\mathrm{i}}$$$$\frac{1}{y^{\mathrm{i}\sqrt{\mathrm{i}}}}=y^{\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\mathrm{i}}$$$$a+b=10$$$$ab=21$$$$\Rightarrow a=3\wedge b=7\vee a=7\wedge b=3$$$$$$$$\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{solve}$$$$$$$$z^{\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\mathrm{i}}=3\Rightarrow z=3^{\frac{\sqrt{2}}{2}}{\mathrm{e}}^{\mathrm{i}\frac{\sqrt{2}\ln 3}{2}}\approx 1.55077+1.52444\mathrm{i}$$$$$$$$z^{\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\mathrm{i}}=7\Rightarrow z=7^{\frac{\sqrt{2}}{2}}{\mathrm{e}}^{\mathrm{i}\frac{\sqrt{2}\ln 7}{2}}\approx .766442+3.88400\mathrm{i}$$$$\vee z=7^{\frac{\sqrt{2}}{2}}{\mathrm{e}}^{\pi \sqrt{2}}{\mathrm{e}}^{\mathrm{i}\frac{\sqrt{2}(\ln 7-2\pi )}{2}}\approx -335.646-25.1114\mathrm{i}$$$$$$$$\mathrm{solutions}:$$$$$$$$x=3^{\frac{\sqrt{2}}{2}}{\mathrm{e}}^{\mathrm{i}\frac{\sqrt{2}\ln 3}{2}}\wedge y=7^{\frac{\sqrt{2}}{2}}{\mathrm{e}}^{\mathrm{i}\frac{\sqrt{2}\ln 7}{2}}$$$$x=3^{\frac{\sqrt{2}}{2}}{\mathrm{e}}^{\mathrm{i}\frac{\sqrt{2}\ln 3}{2}}\wedge y=7^{\frac{\sqrt{2}}{2}}{\mathrm{e}}^{\pi \sqrt{2}}{\mathrm{e}}^{\mathrm{i}\frac{\sqrt{2}(\ln 7-2\pi )}{2}}$$$$\mathrm{and}\mathrm{exchange}x\rightleftarrows y$$

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