Determine-x-e-y-x-1-i-1-y-i-i-10-1-xy-i-i-21-

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

Determine x e y :

{\begin{cases} x^{\frac{1}{\sqrt{i}}} + \frac{1}{y^{i \sqrt{i}}} = 10 \\ \frac{1}{{(xy)}^{i \sqrt{i}}} = 21 \end{cases}

Answered by MJS last updated on 15/Aug/19

x^{\frac{1}{\sqrt{i}}} = x^{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i}

\frac{1}{y^{i \sqrt{i}}} = y^{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i}

a + b = 10

a b = 21

\Rightarrow a = 3 \land b = 7 \lor a = 7 \land b = 3

we have to solve

z^{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i} = 3 \Rightarrow z = 3^{\frac{\sqrt{2}}{2}} e^{i \frac{\sqrt{2} \ln 3}{2}} \approx 1.55077 + 1 . 52444 i

z^{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i} = 7 \Rightarrow z = 7^{\frac{\sqrt{2}}{2}} e^{i \frac{\sqrt{2} \ln 7}{2}} \approx . 766442 + 3 . 88400 i

\lor z = 7^{\frac{\sqrt{2}}{2}} e^{π \sqrt{2}} e^{i \frac{\sqrt{2} (\ln 7 - 2 π)}{2}} \approx - 335.646 - 25 . 1114 i

solutions :

x = 3^{\frac{\sqrt{2}}{2}} e^{i \frac{\sqrt{2} \ln 3}{2}} \land y = 7^{\frac{\sqrt{2}}{2}} e^{i \frac{\sqrt{2} \ln 7}{2}}

x = 3^{\frac{\sqrt{2}}{2}} e^{i \frac{\sqrt{2} \ln 3}{2}} \land y = 7^{\frac{\sqrt{2}}{2}} e^{π \sqrt{2}} e^{i \frac{\sqrt{2} (\ln 7 - 2 π)}{2}}

and exchange x ⇄ y

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