1.

Find square root of `8-15 i`

Answer» Let `sqrt(8-15i)=(x+iy)." "...(i)`
On squaring both sides of (i), we get
`(8-15i)=(x-iy)^(2) rArr (8-15i)=(x^(2)-y^(2))-i(2xy)." "...(ii)`
On comparing real parts and imaginary parts on both sides of (ii), we get `x^(2)-y^(2)=8 and 2xy = 15`
`rArr" "x^(2)-y^(2)=8 and xy = (15)/(2)`
`rArr" "(x^(2)+y^(2))=sqrt((x^(2)-y^(2))^(2)+4x^(2)y^(2))=sqrt(64+225)=sqrt(289)=17`
`rArr" "x^(2)-y^(2)=8 and x^(2)+y^(2)=17`
`rArr" "2x^(2)=25 and 2y^(2)=9`
`x^(2)=(25)/(2) and y^(2)=(9)/(2)`
`rArr" "x = +- (5)/(sqrt(2)) and y = +- (3)/(sqrt(2))`.
Since `xy gt 0`, so x and y are of the same sign.
Hence, `sqrt(8-15i) = ((5)/(sqrt(2))-(3)/(2)i)or((-5)/(sqrt(2))+(3)/(sqrt(2))i)`.


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