Question 1043673: Please help me solve (12i)^1/2 to a+bi form
Answer by ikleyn(52776)   (Show Source):
You can put this solution on YOUR website! .
Please help me solve (12i)^1/2 to a+bi form
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The general procedure on how to find the roots of a complex number is explained in the lesson
- How to take a root of a complex number
in this site.
If you just are familiar with complex numbers, operations on them, complex plane, trigonometric form of complex numbers -
- then you will be able to understand it.
If you are not familiar with this material, then you can learn on complex numbers from these lessons
- Complex numbers and arithmetic operations on them
- Complex plane
- Addition and subtraction of complex numbers in complex plane
- Multiplication and division of complex numbers in complex plane
- Raising a complex number to an integer power
- How to take a root of a complex number
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 = =  .
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