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Question
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
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Solution
\[\frac{1}{\left( 2 + i \right)^2}\]
\[ = \frac{1}{4 + i^2 + 4i} \left( \because i^2 = - 1 \right)\]
\[ = \frac{1}{3 + 4i}\]
\[ = \frac{1}{3 + 4i} \times \frac{3 - 4i}{3 - 4i}\]
\[ = \frac{3 - 4i}{9 - 16 i^2}\]
\[ = \frac{3 - 4i}{9 + 16}\]
\[ = \frac{3}{25} - \frac{4}{25}i\]
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