Question:
Express $\frac{(3+i \sqrt{5})(3-\sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-\sqrt{2} i)}$ in the form $(a+i b)$
Solution:
We have, $\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2 i})-(\sqrt{3}-\sqrt{2 i})}$
$=\frac{(3)^{2}-(i \sqrt{5})^{2}}{\sqrt{3}+\sqrt{2 i}-\sqrt{3}+\sqrt{2 i}}\left[\because(a+b)(a-b)=a^{2}-b^{2}\right]$
$=\frac{9+5}{2 \sqrt{2 i}} \times \frac{\sqrt{2 i}}{\sqrt{2 i}}$
$=\frac{14 \sqrt{2 i}}{2(\sqrt{2 i})^{2}}$
$=\frac{7 \sqrt{2 i}}{-2}$
$=\frac{-7 \sqrt{2 i}}{2}$