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4-1.Complex numbers
medium
Express the following expression in the form of $a+i b$.
$\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2 i})-(\sqrt{3}-i \sqrt{2})}$
A
$\frac{-7 \sqrt{2} i}{2}$
B
$\frac{-7 \sqrt{2} i}{2}$
C
$\frac{-7 \sqrt{2} i}{2}$
D
$\frac{-7 \sqrt{2} i}{2}$
Solution
$\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-i \sqrt{2})}$
$=\frac{(3)^{2}-(i \sqrt{5})^{2}}{\sqrt{3}+\sqrt{2} i-\sqrt{3}+\sqrt{2} i}\left[(a+b)(a-b)=a^{2}-b^{2}\right]$
$=\frac{9-5 i^{2}}{2 \sqrt{2} i}$
$=\frac{9-5(-1)}{2 \sqrt{2} i} \quad\left[i^{2}=-1\right]$
$=\frac{9+5}{2 \sqrt{2} i} \times \frac{i}{i}$
$=\frac{14 i}{2 \sqrt{2} i^{2}}$
$=\frac{14 i}{2 \sqrt{2}(-1)}$
$=\frac{-7 i}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}$
$=\frac{-7 \sqrt{2} i}{2}$
Standard 11
Mathematics
