Complex number

\((\sqrt2 + i\sqrt3)\times i\sqrt6 - \sqrt2\times(2i\sqrt2 - 3) + i\times (\dfrac{6}{\sqrt6}-\dfrac{6}{\sqrt3})\\ = x - y + z\)

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)

(√2 + i√3) * i√6 - √2 * (2i√2 - 3) + i * (6/√6 - 6/√3) =

Simplify the following:
(sqrt(2)+i sqrt(3))×i sqrt(6)-sqrt(2) (2 i sqrt(2)-3)+i (6/sqrt(6)-(6)/sqrt(3))

Rationalize the denominator. 6/sqrt(6) = 6/sqrt(6)×(sqrt(6))/(sqrt(6)) = (6 sqrt(6))/(6):
(sqrt(2)+i sqrt(3))×i sqrt(6)-sqrt(2) (2 i sqrt(2)-3)+i ((6 sqrt(6))/(6)-(6)/sqrt(3))

(6 sqrt(6))/(6) = 6/6×sqrt(6) = sqrt(6):
(sqrt(2)+i sqrt(3))×i sqrt(6)-sqrt(2) (2 i sqrt(2)-3)+i (sqrt(6)-(6)/sqrt(3))

Rationalize the denominator. (-6)/sqrt(3) = (-6)/sqrt(3)×(sqrt(3))/(sqrt(3)) = (-6 sqrt(3))/(3):
(sqrt(2)+i sqrt(3))×i sqrt(6)-sqrt(2) (2 i sqrt(2)-3)+i (sqrt(6)+(-6 sqrt(3))/(3))

(-6)/3 = (3 (-2))/3 = -2:
(sqrt(2)+i sqrt(3))×i sqrt(6)-sqrt(2) (2 i sqrt(2)-3)+i (sqrt(6)+-2 sqrt(3))

sqrt(6) (sqrt(2)+i sqrt(3)) = 3 i sqrt(2)+2 sqrt(3):
3 i sqrt(2)+2 sqrt(3) i-sqrt(2) (2 i sqrt(2)-3)+i (sqrt(6)-2 sqrt(3))

i (3 i sqrt(2)+2 sqrt(3)) = 2 i sqrt(3)-3 sqrt(2):
2 i sqrt(3)-3 sqrt(2)-sqrt(2) (2 i sqrt(2)-3)+i (sqrt(6)-2 sqrt(3))

-(2 i sqrt(2)-3) = 3+-2 i sqrt(2):
-3 sqrt(2)+2 i sqrt(3)+sqrt(2) 3-2 i sqrt(2)+i (sqrt(6)-2 sqrt(3))

sqrt(2) (3+-2 i sqrt(2)) = -4 i+3 sqrt(2):
-3 sqrt(2)+2 i sqrt(3)+-4 i+3 sqrt(2)+i (sqrt(6)-2 sqrt(3))

Add like terms. -3 sqrt(2)+2 i sqrt(3)-4 i+3 sqrt(2)+i (sqrt(6)-2 sqrt(3)) = -4 i+2 i sqrt(3)+i (sqrt(6)-2 sqrt(3)):
-4 i+2 i sqrt(3)+i (sqrt(6)-2 sqrt(3))

Factor i from -4 i+2 i sqrt(3)+i (sqrt(6)-2 sqrt(3)):
i (-4+2 sqrt(3)-2 sqrt(3)+sqrt(6))

Add like terms. -4+2 sqrt(3)-2 sqrt(3)+sqrt(6) = sqrt(6)-4:
Answer: |i (sqrt(6) - 4)