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Question
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
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Solution
\[ \frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
\[ = \frac{i^{4 \times 148} + i^{4 \times 147 + 2} + i^{4 \times 147} + i^{4 \times 146 + 2} + i^{4 \times 146}}{i^{4 \times 145 + 2} + i^{4 \times 145} + i^{4 \times 144 + 2} + i^{4 \times 144} + i^{4 \times 143 + 2}}\]
\[ = \frac{\left( i^4 \right)^{148} + \left\{ \left( i^4 \right)^{147} \times i^2 \right\} + \left\{ \left( i^4 \right)^{146} \right\} + \left\{ \left( i^4 \right)^{146} \times i^2 \right\} + \left\{ \left( i^4 \right)^{146} \right\}}{\left\{ \left( i^4 \right)^{145} \times i^2 \right\} + \left\{ \left( i^4 \right)^{145} \right\} + \left\{ \left( i^4 \right)^{144} \times i^2 \right\} + \left\{ \left( i^4 \right)^{144} \right\} + \left\{ \left( i^4 \right)^{143} \times i^2 \right\}}\]
\[ = \frac{1 + i^2 + 1 + i^2 + 1}{i^2 + 1 + i^2 + 1 + i^2} \left[ \because i^4 = 1 \right]\]
\[ = \frac{1 - 1 + 1 - 1 + 1}{- 1 + 1 - 1 + 1 - 1} \left[ \because i^2 = - 1 \right]\]
\[ = - 1\]
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