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Question
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
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Solution
\[ x = \frac{1 + i}{\sqrt{2}}\]
\[ \Rightarrow x^2 = \left( \frac{1 + i}{\sqrt{2}} \right)^2 \]
\[ = \left( \frac{1 + i^2 + 2i}{2} \right)\]
\[ = \frac{2i}{2}\]
\[ = i\]
\[ \Rightarrow x^6 = \left( x^2 \right)^3 \]
\[ = i^3 \]
\[ = - i\]
\[ \Rightarrow x^2 = i\]
\[ \Rightarrow x^4 = \left( x^2 \right)^2 \]
\[ = i^2 \]
\[ = - 1\]
\[\text { Now }, x^6 + x^4 + x^2 + 1 = - i - 1 + i + 1\]
\[ = 0\]
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