SOLUTION: Write the complex number of square root of 3 + i

  can be written   ∠   ("m angle theta"),

 where:



  m = 



and

   = 





For 3+i:  m =  = 

         =  or approx. 18.435 degrees.



Now, to find the square root, just take the square root of m and divide the angle  by two:

 

  sqrt(3+i) =   ∠    



or approx.   =  ∠ 



If you must express this as 'a + bi' form (rectangular form):

     a = m * cos() = 1.778*cos(9.218) = 1.755  

     b = m * sin() = 1.778*sin(9.218) = 0.285



so   ≈  





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EDIT:  the notation I used really comes from Electrical Engineering, but keep in mind it is just shorthand for  a+bi =