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Question
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Solution
\[\left( i^{77} + i^{70} + i^{87} + i^{414} \right)^3 = \left( i^{4 \times 19 + 1} + i^{4 \times 17 + 2} + i^{4 \times 21 + 3} + i^{4 \times 103 + 2} \right)^3 \]
\[ = \left[ \left\{ \left( i^4 \right)^{19} \times i \right\} + \left\{ \left( i^4 \right)^{17} \times i^2 \right\} + \left\{ \left( i^4 \right)^{21} \times i^3 \right\} + \left\{ \left( i^4 \right)^{103} \times i^2 \right\} \right]\]
\[ = \left( i - 1 - i - 1 \right)^3 \left( \because i^4 = 1, i^3 = - i and i^2 = - 1 \right)\]
\[ = \left( - 2 \right)^3 \]
\[ = - 8\]
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