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Question Number 146521 by mathdanisur last updated on 13/Jul/21

$${(\mathit{i}-1)}^{-100}=?$$

Answered by mathmax by abdo last updated on 13/Jul/21

$${(\mathrm{i}-1)}^{-100}={(-1+\mathrm{i})}^{-100}={\left(\sqrt{2}{\mathrm{e}}^{\frac{\mathrm{i}3\pi }{4}}\right)}^{-100}={\left(\sqrt{2}\right)}^{-100}{\mathrm{e}}^{-\frac{\mathrm{i}3\pi }{4}\times 100}$$$$=\frac{1}{2^{50}}{\mathrm{e}}^{-\left(3\mathrm{i}\pi \right)\times 25}=\frac{1}{2^{50}}{\mathrm{e}}^{-75\mathrm{i}\pi }=\frac{1}{2^{50}}{\mathrm{e}}^{-74\mathrm{i}\pi }{\mathrm{e}}^{-\mathrm{i}\pi }=-\frac{1}{2^{50}}$$

Commented by mathdanisur last updated on 13/Jul/21

$$thankyouSer$$

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