Q) What is the square root of the complex number 5+12i ?

A)

z = 5 + 12 i

z = 13 (5/13 + 12/13 i)

z = 13 (cosQ + i.sinQ)

cosQ = 5/13 ; sinQ = 12/13

Using DeMoivre's theorem:

z^(1/2) = +- sqrt (13) * (cos(Q/2)+ i.sin(Q/2) )

z^(1/2) = +- sqrt (13) * [ rt ( (cosQ + 1)/2) + i.rt ( (1 - cosQ)/2) ]

z^(1/2) = +- sqrt (13) * [ rt (9/13) + i.rt ( 4/13) ]

z^(1/2) = +- sqrt (13) * [ 3/rt(13) + i.2/rt(13) ]

z^(1/2) = +- ( 3 + 2i)

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## Monday, August 11, 2008

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