Algebra.Com's Answer #658825 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Please help me solve (12i)^1/2 to a+bi form
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document.write( "The general procedure on how to find the roots of a complex number is explained in the lesson\r
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document.write( " - How to take a root of a complex number\r
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document.write( "If you just are familiar with complex numbers, operations on them, complex plane, trigonometric form of complex numbers -
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document.write( " - then you will be able to understand it.\r
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document.write( "If you are not familiar with this material, then you can learn on complex numbers from these lessons\r
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document.write( " - Complex numbers and arithmetic operations on them\r
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document.write( " - Complex plane\r
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document.write( " - Addition and subtraction of complex numbers in complex plane\r
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document.write( " - Multiplication and division of complex numbers in complex plane\r
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document.write( " - Raising a complex number to an integer power\r
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document.write( " - How to take a root of a complex number\r
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document.write( "After this introduction, let me briefly explain you how to solve your problem.\r\n" );
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document.write( "So, you need to find  .\r\n" );
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document.write( "Write 12*i in the trigonometric form z =  ,\r\n" );
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document.write( "where \"r\" is the modulus and  is the argument (polar angle).\r\n" );
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document.write( "In your case, z = 12*i in trigonometric form is z =  , so the modulus is r = 12 and the polar angle is =  .\r\n" );
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document.write( "Now, to find the square root of this complex number, you have\r\n" );
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document.write( " 1. to take a square root of the modulus:  =  =  .\r\n" );
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document.write( " 2. to divide the argument (polar angle) by 2:  =  =  .\r\n" );
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document.write( " 3. to consider the complex number  =  , which is in your case \r\n" );
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document.write( " =  =  =  =  =  .\r\n" );
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document.write( " It is one of the two complex roots. // Notice that the modulus of is and the argument is = .\r\n" );
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document.write( " // Also notice that the final expression for is just a + bi form.\r\n" );
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document.write( " 4. to get the second root  in trigonometric form, you have to use the same modulus as has, namely , but use another \r\n" );
document.write( " argument, which this time is  =  .\r\n" );
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document.write( " Then your =  =  =  = \r\n" );
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document.write( " =  =  =  =  =  .\r\n" );
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document.write( " // Notice that =  .\r\n" );
document.write( " // All this long way with lead us to the opposite number to .\r\n" );
document.write( " // But now you know all the procedure, how it works for square roots of complex numbers.\r\n" );
document.write( " // Surely, it may seem too complex, at the first glance.\r\n" );
document.write( " // But there is a powerful symmetry in it, which work nicely for all n > 2.\r\n" );
document.write( " // If you read the lessons I recommended you, you will be able to learn its real power and beauty.\r\n" );
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document.write( "Answer. has two values: = and = =  .\r\n" );
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