Question

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

Answer 1

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

$=\frac{9+5}{3\sqrt{3}-3\sqrt{2}i+\sqrt{6}i+2}$

$=\frac{14}{\left(2+3\sqrt{3}\right)+\left(-3\sqrt{2}+\sqrt{6}\right)i}\times \frac{\left(-3\sqrt{2}-\sqrt{6}\right)i}{\left(-3\sqrt{2}-\sqrt{6}\right)i}$

$=\frac{14\left(-3\sqrt{2}-\sqrt{6}\right)i}{{\left(2+3\sqrt{3}\right)}^2+{\left(-3\sqrt{2}+\sqrt{6}\right)}^2}$

$=\frac{14\left(3\sqrt{2}+\sqrt{6}\right)i}{55}$

$=\frac{42\sqrt{2}}{55}+\frac{14\sqrt{6}i}{55}$