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Question
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Solution
`i^37 + 1/i^67 = i ^(4 xx 9 + 1) + 1/(i^(4 xx 16 +3))`
\[ = \left( i^4 \right)^9 \times i + \frac{1}{\left( i^4 \right)^{16} \times i^3} \]
\[ = i - \frac{1}{i} \left( \because i^3 = - i \right)\]
\[ = i - \frac{1}{i} \times \frac{i}{i}\]
\[ = i - \frac{i}{i^2}\]
\[ = i - \left( - i \right) \left( \because i^2 = - 1 \right) \]
\[ = 2i \]
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