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Question
Express the following complex number in the standard form a + i b:
\[\frac{(2 + i )^3}{2 + 3i}\]
Solution
\[\frac{\left( 2 + i \right)^3}{2 + 3i}\]
\[ = \frac{\left( 4 + i^2 + 4i \right)\left( 2 + i \right)}{2 + 3i} \left( \because i^2 = - 1 \right)\]
\[ = \frac{8 + 2 i^2 + 8i + 4i + i^3 + 4 i^2}{2 + 3i} \]
\[ = \frac{2 + 11i}{2 + 3i} \times \frac{2 - 3i}{2 - 3i}\]
\[ = \frac{4 - 6i + 22i - 33 i^2}{4 - 9 i^2}\]
\[ = \frac{37 + 16i}{4 + 9}\]
\[ = \frac{37}{13} + \frac{16}{13}i\]
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