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Question
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
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Solution
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9 = \left( i^{4 \times 10 + 1} + \frac{1}{i^{4 \times 64 + 1}} \right)^9 \]
\[ = \left[ \left( i^4 \right)^{10} \times i + \frac{1}{\left( i^4 \right)^{64} \times i} \right]^9 \]
\[ = \left( i + \frac{1}{i} \right)^9 \left(\because i^4 = 1\right)\]
\[= \left( i + \frac{i}{i^2} \right)^9 \]
\[ = \left( i - i \right)^9 \left( \because i^2 = - 1 \right)\]
\[ = 0\]
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