After this introduction, let me briefly explain you how to solve your problem.

So, you need to find .

Write 12*i in the trigonometric form z = ,

where "r" is the modulus and  is the argument (polar angle).

In your case,  z = 12*i  in trigonometric form is z = , so the modulus is r = 12 and the polar angle is  = .

Now, to find the square root of this complex number, you have

  1.  to take a square root of the modulus:  =  = .

  2.  to divide the argument (polar angle) by 2:   =  = .

  3.  to consider the complex number  = , which is in your case 

       =  =  =  =  = .

      It is one of the two complex roots.  // Notice that the modulus of  is  and the argument is  = .

                                          // Also notice that the final expression for  is just a + bi form.

  4.  to get the second root  in trigonometric form, you have to use the same modulus as  has, namely ,  but use another 
      argument, which this time is  = .

      Then your  =  =  =  = 

                     =  =  =  =  = .

      // Notice that  = .
      // All this long way with  lead us to the opposite number to .
      // But now you know all the procedure, how it works for square roots of complex numbers.
      // Surely, it may seem too complex, at the first glance.
      // But there is a powerful symmetry in it, which work nicely for all n > 2.
      // If you read the lessons I recommended you, you will be able to learn its real power and beauty.


Answer.   has two values:  =  and  =  = .