Simplify the following:
(1 + (sqrt(3))/(3))/(1 - (sqrt(3))/(3))
Put each term in 1 - (sqrt(3))/(3) over the common denominator 3: 1 - (sqrt(3))/(3) = 3/3 + (-sqrt(3))/(3):
(1 + (sqrt(3))/(3))/(3/3 + (-sqrt(3))/(3))
3/3 - (sqrt(3))/(3) = (3 - sqrt(3))/(3):
(1 + (sqrt(3))/(3))/((3 - sqrt(3))/(3))
Put each term in 1 + (sqrt(3))/(3) over the common denominator 3: 1 + (sqrt(3))/(3) = 3/3 + (sqrt(3))/(3):
(3/3 + (sqrt(3))/(3))/((3 - sqrt(3))/(3))
3/3 + (sqrt(3))/(3) = (3 + sqrt(3))/(3):
((3 + sqrt(3))/(3))/((3 - sqrt(3))/(3))
Multiply the numerator by the reciprocal of the denominator, (3 + sqrt(3))/(3 (3 - sqrt(3))/(3)) = (3 + sqrt(3))/(3)×3/(3 - sqrt(3)):
((3 + sqrt(3))×3)/(3 (3 - sqrt(3)))
((3 + sqrt(3))×3)/(3 (3 - sqrt(3))) = 3/3×(3 + sqrt(3))/(3 - sqrt(3)) = (3 + sqrt(3))/(3 - sqrt(3)):
(3 + sqrt(3))/(3 - sqrt(3))
Multiply numerator and denominator of (3 + sqrt(3))/(3 - sqrt(3)) by 3 + sqrt(3):
((3 + sqrt(3)) (3 + sqrt(3)))/((3 - sqrt(3)) (3 + sqrt(3)))
(3 - sqrt(3)) (3 + sqrt(3)) = 3×3 + 3 sqrt(3) - sqrt(3)×3 - sqrt(3) sqrt(3) = 9 + 3 sqrt(3) - 3 sqrt(3) - 3 = 6:
((3 + sqrt(3)) (3 + sqrt(3)))/(6)
Combine powers. ((3 + sqrt(3)) (3 + sqrt(3)))/(6) = ((3 + sqrt(3))^(1 + 1))/(6):
((3 + sqrt(3))^1 + 1)/(6)
1 + 1 = 2:
((3 + sqrt(3))^2)/(6)
(3 + sqrt(3))^2 = 9 + 3 sqrt(3) + 3 sqrt(3) + 3 = 12 + 6 sqrt(3):
(12 + 6 sqrt(3))/(6)
Factor 6 out of 12 + 6 sqrt(3) giving 6 (2 + sqrt(3)):
(6 (2 + sqrt(3)))/(6)
(6 (2 + sqrt(3)))/(6) = 6/6×(2 + sqrt(3)) = 2 + sqrt(3):
Answer: |2 + sqrt(3)
