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Question
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Solution
\[( 1 + 2i )^{- 3} \]
\[ = \frac{1}{\left( 1 + 2i \right)^3}\]
\[ = \frac{1}{1 + 8 i^3 + 6i + 12 i^2}\]
\[ = \frac{1}{1 - 8i + 6i - 12} \left( \because i^2 = - 1 \text { & } i^3 = - i \right)\]
\[ = \frac{1}{- 2i - 11}\]
\[ = \frac{1}{- 2i - 11} \times \frac{- 2i + 11}{- 2i + 11}\]
\[ = \frac{- 2i + 11}{4 i^2 - 121}\]
\[ = \frac{- 2i + 11}{- 4 - 121}\]
\[ = \frac{- 2i + 11}{- 125}\]
\[ = - \frac{11}{125} + \frac{2i}{125}\]
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