Express the following expression in the form of a + ib.
Question:

Express the following expression in the form of a + ib.

$\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-i \sqrt{2})}$

Solution:

$\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-i \sqrt{2})}$

$=\frac{(3)^{2}-(i \sqrt{5})^{2}}{\sqrt{3}+\sqrt{2} i-\sqrt{3}+\sqrt{2} i} \quad\left[(a+b)(a-b)=a^{2}-b^{2}\right]$

$=\frac{9-5 i^{2}}{2 \sqrt{2} i}$

$=\frac{9-5(-1)}{2 \sqrt{2} i} \quad\left[i^{2}=-1\right]$

$=\frac{9+5}{2 \sqrt{2} i} \times \frac{i}{i}$

$=\frac{14 i}{2 \sqrt{2} i^{2}}$

$=\frac{14 i}{2 \sqrt{2}(-1)}$

$=\frac{-7 i}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$

$=\frac{-7 \sqrt{2} i}{2}$

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