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Question
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
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Solution
\[ z = \left( 1 + \sqrt{3}i \right)^2 \]
\[ = 1 + 3 i^2 + 2\sqrt{3}i\]
\[ = - 2 + 2\sqrt{3}i\]
\[\text { Then }, \frac{1}{z} = \frac{1}{- 2 + 2\sqrt{3}i} \times \frac{- 2 - 2\sqrt{3}i}{- 2 - 2\sqrt{3}i}\]
\[ = \frac{- 2 - 2\sqrt{3}i}{4 - 12 i^2}\]
\[ = \frac{- 2 - 2\sqrt{3}i}{16}\]
\[ = \frac{- 1}{8} - \frac{\sqrt{3}}{8}i\]
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