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