1.

Evaluate `sqrt(24-10i)`.

Answer» Let `sqrt(-24-10i)=(x+iy)." "...(i)`
On squaring both sides of (i), we get
`(-24-10i)=(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)=-24 and 2xy = 10`
`rArr" "x^(2)-y^(2)=-24 and xy = 5`
`rArr" "(x^(2)+y^(2))=sqrt((x^(2)-y^(2))^(2)+4x^(2)y^(2))=sqrt((-24)^(2)+4 xx 24)=sqrt(676)=26`
`rArr" "x^(2)-y^(2)=-24 and x^(2)+y^(2)=26`
`rArr" "2x^(2)=2 and 2y^(2)=50`
`x^(2)=1 and y^(2)=25 rArr x = +-1 and y = +-5`.
Since `xy gt 0`, so x and y are of the same sign.
`therefore" "sqrt(-24-10i)=(1-5i)or(-1+5i)`.


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