Evaluate: `sqrt(1-i)`

1.	Evaluate: `sqrt(1-i)`
Answer» Correct Answer - `{sqrt((sqrt(2)+1)/(2))}-{sqrt((sqrt(2)-1)/(2))}i or -{sqrt((sqrt(2)+1)/(2))}+{sqrt((sqrt(2)-1)/(2))}i` Let `sqrt(1-i) = x -iy`. On squaring both sides, we get `(1-i)=(x-iy)^(2)=(x^(2)-y^(2))-i(2xy)` `therefore" "(x^(2)-y^(2))=1 and 2xy = 1` `rArr" "(x^(2)-y^(2))=sqrt((x^(2)-y^(2))^(2)+4x^(2)y^(2))=sqrt(1^(2)+1^(2))=sqrt(2)` `rArr" "x^(2)-y^(2) = 1 and x^(2) + y^(2) = sqrt(2)` `rArr" "x^(2)=(sqrt(2)+1)/(2) and y^(2)=(sqrt(2)-1)/(2)` `rArr" "x = +- sqrt((sqrt(2)+1)/(2))and y = +- sqrt((sqrt(2)-1)/(2))`. Since xy `gt` 0, so x and y have the same sign. `therefore" "{x=sqrt((sqrt(2)+1)/(2)),y=sqrt((sqrt(2)-1)/(2))}or{x=-sqrt((sqrt(2)+1)/(2)),y=-sqrt((sqrt(2)-1)/(2))}` `therefore" "sqrt(1-i)={sqrt((sqrt(2)+1)/(2))-sqrt((sqrt(2)-1)/(2))}or{-sqrt((sqrt(2)+1)/(2))+ isqrt((sqrt(2)-1)/(2))}`.

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