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Question Number 207220 by hardmath last updated on 09/May/24

$$\mathrm{Find}:{(\mathbf{i}-1)}^{-100}=?$$

Answered by Frix last updated on 10/May/24

$${(-1+\mathrm{i})}^{-100}={\left({(-1+\mathrm{i})}^{-1}\right)}^{100}=$$$$={(-\frac{1}{2}-\frac{1}{2}\mathrm{i})}^{100}={\left(\frac{\sqrt{2}}{2}{\mathrm{e}}^{-\frac{3\pi }{4}\mathrm{i}}\right)}^{100}=$$$$={\left(\frac{\sqrt{2}}{2}\right)}^{100}{\mathrm{e}}^{-\frac{300\pi }{4}\mathrm{i}}=\frac{1}{2^{50}}{\mathrm{e}}^{75\pi \mathrm{i}}=-\frac{1}{2^{50}}=-2^{-50}$$

Commented by hardmath last updated on 10/May/24

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor}$$

Answered by Rasheed.Sindhi last updated on 10/May/24

$$={\left({(i-1)}^2\right)}^{-50}={(-2i)}^{-50}$$$$={\left\{{(-2i)}^2\right\}}^{-25}$$$$={(-4)}^{-25}$$$$=\frac{1}{-4^{25}}=-\frac{1}{2^{50}}$$

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