Algebra

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

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

Multiply numerator and denominator of (-(i sqrt(2)) + 2 sqrt(3))/(i sqrt(3)) by -i:
((-(i sqrt(2)) + 2 sqrt(3)) (-i))/(i sqrt(3) (-i))

i×i = -1:
(-(-(i sqrt(2)) + 2 sqrt(3))×i)/(--1 sqrt(3))

((-(i sqrt(2)) + 2 sqrt(3)) (-i))/(-sqrt(3) (-1)) = (-1)/(-1)×((-(i sqrt(2)) + 2 sqrt(3))×i)/(sqrt(3) (-1)) = ((-(i sqrt(2)) + 2 sqrt(3))×i)/(sqrt(3) (-1)):
((-(i sqrt(2)) + 2 sqrt(3))×i)/(-sqrt(3))

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

Multiply numerator and denominator of (sqrt(2) + 2 i sqrt(3))/(sqrt(3) (-1)) by -1:
(-(sqrt(2) + 2 i sqrt(3)))/(sqrt(3))

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

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

Multiply top and bottom by isqrt 3

(isqrt18- i^2 sqrt 6 )  /  i^2  3   = ( i 3 sqrt 2 + sqrt 6 )  /  -3  =  -isqrt2 - (sqrt 6)/3     ???

\(\dfrac{\sqrt6 - i\sqrt2}{i\sqrt3}\\ =\dfrac{\sqrt6}{i\sqrt3} - \dfrac{i\sqrt2}{i\sqrt3}\\ =\dfrac{\sqrt2}{i}-\dfrac{\sqrt2}{\sqrt3}\\ =-i\sqrt{2} - \dfrac{\sqrt6}{3}\)

[√6 - i √2 ]  /  [ i √3 ]       mult. top/ bottom by  i √3

[  i √18 - i^2 √6 ] / [ i^2 *3 ]  =

[3i√2 + √6 ] / [ - 3]  =

- [ 3 i√2 + √6 ] / 3