Simplify the following:
(1 - sqrt(2) + 1 - sqrt(2 i))/(1 - sqrt(2 i))
2 i = 1 + 2 i - 1 = 1 + 2 i + i^2 = (1 + i)^2:
(1 - sqrt(2) + 1 - sqrt((1 + i)^2 ) )/(1 - sqrt(2 i))
Cancel exponents. sqrt((1 + i)^2) = 1 + i:
(1 - sqrt(2) + 1 - 1 + i)/(1 - sqrt(2 i))
1 - sqrt(2) + 1 - (1 + i) = 1 - i - sqrt(2):
(1 - i - sqrt(2))/(1 - sqrt(2 i))
2 i = 1 + 2 i - 1 = 1 + 2 i + i^2 = (1 + i)^2:
(1 - i - sqrt(2))/(1 - sqrt((1 + i)^2 ) )
Cancel exponents. sqrt((1 + i)^2) = 1 + i:
(1 - i - sqrt(2))/(1 - 1 + i)
-(i + 1) = -1 - i:
(1 - i - sqrt(2))/(1 + -i - 1)
1 - 1 - i = (1 - 1) - i = -i:
(1 - i - sqrt(2))/(-i)
Multiply numerator and denominator of (1 - i - sqrt(2))/(-i) by -1:
(-(1 - i - sqrt(2)))/(i)
-((1 - i) - sqrt(2)) = (-1 + i) + sqrt(2):
(-1 + i + sqrt(2))/(i)
Multiply numerator and denominator of (-1 + i + sqrt(2))/(i) by -i:
((-1 + i + sqrt(2)) (-i))/(i (-i))
i×i = -1:
((-1 + i + sqrt(2)) (-i))/(--1)
((-1 + i + sqrt(2)) (-i))/(-(-1)) = (-1)/(-1)×((-1 + i + sqrt(2))×i)/(-1) = ((-1 + i + sqrt(2))×i)/(-1):
((-1 + i + sqrt(2))×i)/(-1)
Multiply numerator and denominator of ((-1 + i + sqrt(2))×i)/(-1) by -1:
-(-1 + i + sqrt(2))×i
i ((-1 + i) + sqrt(2)) = i sqrt(2) - (1 + i):
--1 - i + i sqrt(2)
-(i sqrt(2) - (1 + i)) = -(-1 - i) - i sqrt(2):
-(i sqrt(2)) - (-i - 1)
-(-1 - i) = i + 1:
Answer: |1 + i - i sqrt(2)