1.

Evaluate `sqrt(6 + 8i)`.

Answer» Let `sqrt(6+8i)=(x+iy)." "...(i)`
On squaring both sides of (i), we get
`6+8i=(x+iy)^(2) rArr 6+8i=(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)=6 and 2xy = 8`
`rArr" "x^(2)-y^(2)=6 and xy = 4`
`rArr" "(x^(2)+y^(2))=sqrt((x^(2)-y^(2))^(2)+4x^(2)y^(2))=sqrt(6^(2)+4 xx 16)=sqrt(100)=10`
`rArr" "x^(2)-y^(2)=6 and x^(2)+y^(2)=10`
`rArr" "2x^(2)=16 and 2y^(2)=4`
`x^(2)=8 and y^(2)=2`
`rArr" "x = +- 2 sqrt(2) and y = +- sqrt(2)`.
Since `xy gt 0`, so x and y are of the same sign.
`therefore" "(x=2 sqrt(2) and y = sqrt(2)) or (x = -2 sqrt(2) and y = -sqrt(2))`.
Hence, `sqrt(6+8i) = (2 sqrt(2) + sqrt(2)i) or (-2 sqrt(2)-sqrt(2)i)`.


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