Question

Express the following expression in the form of $a+ib:\frac{(3+i\sqrt{5})(3-i\sqrt{5})}{(\sqrt{3}+\sqrt{2}i)-(\sqrt{3}-i\sqrt{2})}$

Answer 1

Given, $\frac{(3+i\sqrt{5})(3-i\sqrt{5})}{(\sqrt{3}+\sqrt{2}i)-(\sqrt{3}-i\sqrt{2})}$
$=\frac{{\left(3\right)}^2-{\left(i\sqrt{5}\right)}^2}{\sqrt{3}+\sqrt{2}i-\sqrt{3}+\sqrt{2}i}[(a+b)(a-b)=a^2-b^2]$
$=\frac{9-5i^2}{2\sqrt{2}i}$
$=\frac{9-5(-1)}{2\sqrt{2}i}$ ($\because i^2=-1$)
$=\frac{9+5}{2\sqrt{2}i}\times \frac{i}{i}$
$=\frac{14i}{2\sqrt{2}{i}^2}$
$=\frac{14i}{2\sqrt{2}(-1)}$
$=\frac{-7i}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}$
$=\frac{-7\sqrt{2}i}{2}$

$=a+ib$ where $a=0,$ $b=-\frac{7\sqrt{2}}{2}$