Question 1116154: How do I find all the values to the following?
1) i^0.25
2) (1+sqrt(3)i)^(1/3)
3) (i-1)^0.5
4) ((9i)/(1+i))^(1/6)
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The  form of complex numbers is not useful for calculating products, powers, and roots.
For those purposes the polar, or trigonometric, or exponential forms are much more useful.
For example,  can be written using
its absolute value or modulus,  ,
and its argument  such that  .
We could say  to keep it simple, or  if you must use radians.
You can write as  or as  .
All you need to remember is that when multiplying complex numbers,
absolute values multiply, and arguments add up.
2) So, the solutions to 
are numbers of the form  such that
 .
That means that  <--> ,
and  is  or a coterminal angle.
So, the possibilities for between  and  are
 -->  ,
 -->  ,
and  -->  .
Approximate values are
 ,
so we could write the answers as
 
 
 
1) The solutions to 
are four fourth roots of  with
 and  with  .
That means      .
The approximate values of sine and cosine for those angles are:
So, the four complex values of  are
  ,
 
 , and
3)  has 2 values, with
 and  with  .
That means  with , meaning  or  .
Approximate values for sine and cosine of those angles are
 ,
so we could write the answers as
  , and
  .
4)  and  ,
so their ratio can be calculated as
    
Another way to calculate that quotient is using the conjugate of the denominator:
    .
Then,
  ,
where  with  .
That means
  .
  .
Approximate values for sine and cosine of those angles are
So, the six complex values of are
  ,
  ,
  ,
  ,
  , and
  .
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