Answer:
-sqrt(2) + 3 sqrt(3) - sqrt(6)
Explanation:
Simplify the following:
(6 sqrt(3))/2 - sqrt(6) + (sqrt(6))/(sqrt(3)) + sqrt(2) - (4 sqrt(3))/(sqrt(6)) - sqrt(2)
6/2 = (2×3)/2 = 3:
3 sqrt(3) - sqrt(6) + (sqrt(6))/(sqrt(3)) + sqrt(2) - (4 sqrt(3))/(sqrt(6)) - sqrt(2)
Rationalize the denominator. (sqrt(6))/(sqrt(3)) = (sqrt(6))/(sqrt(3))×(sqrt(3))/(sqrt(3)) = (sqrt(6) sqrt(3))/3:
3 sqrt(3) - sqrt(6) + (sqrt(6) sqrt(3))/3 + sqrt(2) - (4 sqrt(3))/(sqrt(6)) - sqrt(2)
sqrt(6) sqrt(3) = sqrt(6×3):
3 sqrt(3) - sqrt(6) + (sqrt(6×3))/3 + sqrt(2) - (4 sqrt(3))/(sqrt(6)) - sqrt(2)
6×3 = 18:
3 sqrt(3) - sqrt(6) + (sqrt(18))/3 + sqrt(2) - (4 sqrt(3))/(sqrt(6)) - sqrt(2)
sqrt(18) = sqrt(2×3^2) = 3 sqrt(2):
3 sqrt(3) - sqrt(6) + (3 sqrt(2))/3 + sqrt(2) - (4 sqrt(3))/(sqrt(6)) - sqrt(2)
(3 sqrt(2))/3 = 3/3×sqrt(2) = sqrt(2):
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) - (4 sqrt(3))/(sqrt(6)) - sqrt(2)
Rationalize the denominator. (-4 sqrt(3))/(sqrt(6)) = (-4 sqrt(3))/(sqrt(6))×(sqrt(6))/(sqrt(6)) = (-4 sqrt(3) sqrt(6))/6:
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) + (-4 sqrt(3) sqrt(6))/6 - sqrt(2)
The gcd of -4 and 6 is 2, so (-4 sqrt(3) sqrt(6))/6 = ((2 (-2)) sqrt(3) sqrt(6))/(2×3) = 2/2×(-2 sqrt(3) sqrt(6))/3 = (-2 sqrt(3) sqrt(6))/3:
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) + (-2 sqrt(3) sqrt(6))/3 - sqrt(2)
sqrt(3) sqrt(6) = sqrt(3×6):
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) - 2/3 sqrt(3×6) - sqrt(2)
3×6 = 18:
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) - (2 sqrt(18))/3 - sqrt(2)
sqrt(18) = sqrt(2×3^2) = 3 sqrt(2):
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) - 2/3 3 sqrt(2) - sqrt(2)
Combine powers. (-2×3 sqrt(2))/3 = 2 sqrt(2)×3^(1 - 1) (-1):
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) - 2 sqrt(2)×3^(1 - 1) - sqrt(2)
1 - 1 = 0:
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) - 2 sqrt(2)×3^0 - sqrt(2)
3^0 = 1:
3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) - 2 sqrt(2)×1 - sqrt(2)
Add like terms. 3 sqrt(3) - sqrt(6) + sqrt(2) + sqrt(2) - 2 sqrt(2) - sqrt(2) = -sqrt(2) + 3 sqrt(3) - sqrt(6):
Answer: -sqrt(2) + 3 sqrt(3) - sqrt(6)