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

$(a)\;-\large\frac{5\sqrt{2}}{2} \normalsize i\qquad(b)\;-\large\frac{7\sqrt{2}}{2} \normalsize i\qquad(c)\;-\large\frac{9\sqrt{2}}{2} \normalsize i\qquad(d)\;-\large\frac{3\sqrt{2}}{2} \normalsize i$

Answer : $\;-\large\frac{7\sqrt{2}}{2} \normalsize i$
$\large\frac{(\normalsize 3 + \normalsize i \sqrt{5})(\normalsize 3 - \normalsize i \sqrt{5})}{(\normalsize \sqrt{3} + \normalsize i \sqrt{2})-(\normalsize \sqrt{3} - \normalsize i \sqrt{2})}=$
$=\large\frac{(3)^{2}-(\sqrt{5})^{2}}{\sqrt{3}+\sqrt{2} i -\sqrt{3}+\sqrt{2}i}\qquad [(a+b)(a-b)=a^2-b^2]$
$= \large\frac{9-5i^{2}}{2\sqrt{2}i}$
$= \large\frac{9+5}{2 \sqrt{2} i} \times \large\frac{i}{i}\qquad [i^{2}=-1]$
$= \large\frac{14}{2 \sqrt{2} i^2}$
$=- \large\frac{7i}{\sqrt{2}} \times \large\frac{\sqrt{2}}{\sqrt{2}}$
$=-\large\frac{7\sqrt{2}}{2} \normalsize i$