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

Find: (i − 1)^(−100) = ?

$$\mathrm{Find}:\:\:\:\left(\boldsymbol{\mathrm{i}}\:−\:\mathrm{1}\right)^{−\mathrm{100}} \:\:=\:\:? \\ $$

Answered by Frix last updated on 10/May/24

(−1+i)^(−100) =((−1+i)^(−1) )^(100) = =(−(1/2)−(1/2)i)^(100) =(((√2)/2)e^(−((3π)/4)i) )^(100) = =(((√2)/2))^(100) e^(−((300π)/4)i) =(1/2^(50) )e^(75πi) =−(1/2^(50) )=−2^(−50)

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

Commented by hardmath last updated on 10/May/24

thank you dear professor

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

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

=((i−1)^2 )^(−50) =(−2i)^(−50) ={(−2i)^2 }^(−25) =(−4)^(−25) =(1/(−4^(25) ))=−(1/2^(50) )

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

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