\((\sqrt2 + i\sqrt3)\times i\sqrt6 - \sqrt2\times(2i\sqrt2 - 3) + i\times (\dfrac{6}{\sqrt6}-\dfrac{6}{\sqrt3})\\ = x - y + z\)
I am going to divide it into 3 parts.
x = (sqrt2 + i sqrt3) * i sqrt6
y = sqrt2 *(2i sqrt2 - 3)
z = i * (6/sqrt6 - 6/sqrt3)
x = \((\sqrt2 + i\sqrt3) \times i\sqrt6 = i\sqrt{12} + i^2\sqrt{18}=2i\sqrt3 - 3\sqrt2\)
y = \(\sqrt2 \times (2i\sqrt2 - 3)=4i - 3\sqrt2\)
z = \(i\times (\dfrac{6}{\sqrt6}-\dfrac{6}{\sqrt3})=i\times (\dfrac{6-6\sqrt2}{\sqrt6})=i\times (\sqrt{6}-\sqrt{12})=i\sqrt6-2i\sqrt3\)
x - y + z
\(=(2i\sqrt3 - 2\sqrt2) -(4i-3\sqrt2)+(i\sqrt6 -2i\sqrt3)\\ =2i\sqrt3 - 2\sqrt2 - 4i + 3\sqrt2 + i\sqrt6 - 2i\sqrt3\\ = \sqrt2 +i(\sqrt6-4)\)
Therefore the original equation is sqrt2 + i(sqrt6 - 4)
