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Question
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
Solution
\[\frac{5 + \sqrt{2}i}{1 - \sqrt{2}i}\]
\[ = \frac{5 + \sqrt{2}i}{1 - \sqrt{2}i} \times \frac{1 + \sqrt{2}i}{1 + \sqrt{2}i}\]
\[ = \frac{5 + 5\sqrt{2}i + \sqrt{2}i + 2 i^2}{1 - 2 i^2}\]
\[ = \frac{5 + 6\sqrt{2}i - 2}{1 + 2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{3 + 6\sqrt{2}i}{3}\]
\[ = 1 + 2\sqrt{2}i\]
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