Imaginary and Complex Numbers
      Exponentiation and root extraction of complex numbers in the polar form
         Exponentiation and root extraction of complex numbers examples
         Powers and roots of complex numbers, use of de Moivre’s formulas
Exponentiation and root extraction of complex numbers in the polar form - de Moivre's formula
We use the polar form for exponentiation and root extraction of complex numbers that are known as de Moivre's formulas.
  zn = rn · [cos(nj) + isin(nj)]  
 and   
Exponentiation and root extraction of complex numbers examples
Example: Calculate   using de Moivre's formula.
Solution:
 
 
 
 
 
 
since
then
 
 
These complex numbers satisfy the equation z3 = -8 and by the Fundamental theorem of algebra, since this equation is of degree 3, there must be 3 roots.
Thus, for example to check the root zk=2 we cube this solution, 
then  
Example: Calculate    
Solution:
 
 
 
 
r = 64    and     j = p
 
 
thus,
 
These complex numbers satisfy the equation z6 = -64 and by the Fundamental theorem of algebra, since this equation is of degree 6, there must be 6 roots.
Example: Calculate    
Solution:
 
 
 
 
 
 
 
Example: Calculate    
Solution:
 
 
 
 
 
 
 
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