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Question Number 123607 by mohammad17 last updated on 26/Nov/20

$$ifa=\frac{1+i}{\sqrt{2}}find{a}^{1943}?$$

Answered by Dwaipayan Shikari last updated on 26/Nov/20

$$a=\frac{1+i}{\sqrt{2}}=\sqrt{i}$$$$a^{1943}={\left(i\right)}^{\frac{1943}{2}}=i^{970}.i.\sqrt{i}=-i\sqrt{i}$$$$Anotherway$$$$a=\frac{1+i}{\sqrt{2}}=e^{\frac{i\pi }{4}}\Rightarrow a^{1943}=e^{\frac{1943\pi i}{4}}=e^{435\pi i}{e}^{\frac{3\pi }{4}i}=-1.(-\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}})$$$$=\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}$$

Answered by malwan last updated on 26/Nov/20

$$a=[1,\frac{\pi }{4}]$$$$\Rightarrow a^{1943}=[1^{1943},1943\times \frac{\pi }{4}]=[1,\pi ][1,\frac{3\pi }{4}]$$$$=-1(cos\frac{3\pi }{4}+isin\frac{3\pi }{4})=\frac{1-i}{\sqrt{2}}$$

Answered by TANMAY PANACEA last updated on 26/Nov/20

$$a=\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}=cos\frac{\pi }{4}+isin\frac{\pi }{4}=e^{i\times \frac{\pi }{4}}$$$$1943=4\times 485+3$$$$a=e^{i\times \frac{\pi }{4}}$$$$a^{1943}={\left(e^{i\times \frac{\pi }{4}}\right)}^{4\times 485+3}=e^{i\times 485\pi +i\times \frac{3\pi }{4}}$$$$$$$$e^{i\times 485\pi }\times e^{i\times \frac{3\pi }{4}}$$$$=(cos485\pi +isin485\pi )(cos{135}^o+isin{135}^o)$$$$=\{cos(242\times 2\pi +\pi )+i\times 0\}(-\frac{1}{\sqrt{2}}+i\times \frac{1}{\sqrt{2}})$$$$=(cos\pi +0)(\frac{-1}{\sqrt{2}}+i\times \frac{1}{\sqrt{2}})$$$$=(-1)(\frac{-1}{\sqrt{2}}+i\times \frac{1}{\sqrt{2}})=\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}$$

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