
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, 
z^{n}_{
} = r^{n}_{ }·_{ }[cos
(nj)
+ isin (nj)] 

and 




Compute


using de Moivre's
formula. 




These complex numbers satisfy the equation z^{3}
= 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 z_{k=}_{2}
we cube this solution, 

then 





Compute 

using de Moivre's
formula. 



r
= 64 and
j =
p 




thus, 






These complex numbers satisfy the equation z^{6}
= 64
and by the Fundamental theorem of algebra, since this equation
is of degree 6, there must be 6 roots. 


Solve 








Calculate 
















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