

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. 

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


and 




Exponentiation
and root extraction of complex numbers examples

Example: 
Calculate 

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 




Example: 
Calculate 




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. 

Example: 
Calculate 







Example: 
Calculate 














Functions
contents A




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