ALGEBRA - solved problems
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
122.
   Compute 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  
123.
   Compute using de Moivre's formula.
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.
124.
  Solve
Solution:
 
 
 
 
 
125.
   Calculate
Solution:
 
 
 
 
 
 
 
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