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Q. $ {{(1+i)}^{3}}+{{(1-i)}^{3}} $

J & K CETJ & K CET 2014Complex Numbers and Quadratic Equations

Solution:

$ {{(1+i)}^{3}}+{{(1-i)}^{3}}={{(1+i)}^{2}}(1+i)+{{(1-i)}^{2}}(1-i) $ $ =(1+{{i}^{2}}+2i)(1+i)+(1+{{i}^{2}}-2i)\,(1-i) $ $ \left[ \begin{align} & \because \,\,\,\,\,{{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab \\ & {{(a-b)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab \\ \end{align} \right] $ $ =2i(1+i)+(-2i)(1-i) $ $ (\because \,\,{{i}^{2}}=-1) $ $ =2i+2{{i}^{2}}-2i+2{{i}^{2}} $ $ =4{{i}^{2}}=-4 $ $ (\because \,\,{{i}^{2}}=-1) $