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Question Number 139027 by mohammad17 last updated on 21/Apr/21

$$findRe\left(z\right)andIm\left(z\right)of{[{\left(e^{i(1+2k)\pi }\right)}^{\frac{1}{10}}+1]}^{-1}$$$$$$

Commented by mohammad17 last updated on 21/Apr/21

$$????$$

Answered by MJS_new last updated on 21/Apr/21

$${\mathrm{e}}^{\mathrm{i}(1+2k)\pi }=-1$$$${(-1)}^{\frac{1}{10}}={\mathrm{e}}^{\frac{\mathrm{i}\pi }{10}}=\frac{\sqrt{10+2\sqrt{5}}}{4}-\frac{1-\sqrt{5}}{4}\mathrm{i}$$$$\frac{1}{1+\frac{\sqrt{10+2\sqrt{5}}}{4}-\frac{1-\sqrt{5}}{4}\mathrm{i}}=\frac{1}{2}+\left(\frac{1+\sqrt{5}-\sqrt{5+2\sqrt{5}}}{2}\right)\mathrm{i}$$

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