Express the following complex numbers in the standard form a + ib… - Vidyadip

Question

Express the following complex numbers in the standard form a + ib:
$\frac{5+\sqrt{2}\text{i}}{1-\sqrt{2}\text{i}}$

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Answer

We have,
$\frac{5+\sqrt{2}\text{i}}{1-\sqrt{2}\text{i}}=\frac{5+\sqrt{2}\text{i}}{1-\sqrt{2}\text{i}}\times\frac{1+\sqrt{2}\text{i}}{1+\sqrt{2}\text{i}}$
$=\frac{5(1+\sqrt{2}\text{i})+\sqrt{2}\text{i}(1+\sqrt{2}\text{i})}{1+2}$
$=\frac{5+5\sqrt{2}\text{i}+\sqrt{2}\text{i}-2}{3}$
$=\frac{3+6\sqrt{2}\text{i}}{3}$
$=1+2\sqrt{2}\text{i}$
$\therefore \ \frac{5+\sqrt{2}\text{i}}{1-\sqrt{2}\text{i}}=1+2\sqrt{2}\text{i}$

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