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Question
The value of \[(1 + i )^4 + (1 - i )^4\] is
Options
8
4
-8
-4
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Solution
-8
\[\text { Using } a^4 + b^4 = \left( a^2 + b^2 \right)^2 - 2 a^2 b^2 \]
\[(1 + i )^4 + (1 - i )^4 \]
\[ = \left( \left( 1 + i \right)^2 + \left( 1 - i \right)^2 \right)^2 - 2 \left( 1 + i \right)^2 \left( 1 - i \right)^2 \]
\[ = \left( 1 + i^2 + 2i + 1 + i^2 - 2i \right)^2 - 2\left( 1 + i^2 + 2i \right)\left( 1 + i^2 - 2i \right) \]
\[ = \left( 1 - 1 + 2i + 1 - 1 - 2i \right)^2 - 2\left( 1 - 1 + 2i \right)\left( 1 - 1 - 2i \right)\]
\[ = \left( 0 \right) - 2\left( 2i \right)\left( - 2i \right) \left( \because i^2 = - 1 \right)\]
\[ = 8 i^2 \]
\[ = - 8\]
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