Question

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

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

Answer 1

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

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

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

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

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

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

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

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