Question

Express the mentioned below complex number in the standard form $a+ib$
(xii) $\frac{5+\sqrt{2}i}{1-\sqrt{2}i}$

Answer 1

$\because \frac{5+\sqrt{2}i}{1-\sqrt{2}i}$

$\frac{5+\sqrt{2}i}{1-\sqrt{2}i}\times \frac{1+\sqrt{2}i}{1+\sqrt{2}i}$

$=\frac{5+5\sqrt{2}i+\sqrt{2}i+2i^2}{1^2-(\sqrt{2}i{)}^2}$

$=\frac{5+6\sqrt{2}i+2\times (-1)}{1-2i^2}$

$=\frac{3+6\sqrt{2}i}{3}$

$=1+2\sqrt{2}i$

$\therefore \frac{5+\sqrt{2}i}{1-\sqrt{2}i}=1+2\sqrt{2}i$