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ALGEBRA
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Exponentiation
and root extraction of complex numbers in the polar form - de
Moivre's formula
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We use the polar form
for exponentiation and root extraction of complex numbers that
are known as
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de Moivre's formulas, |
zn
= rn · [cos
(nj)
+ isin (nj)] |
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and |
![](MoivresF.gif) |
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Compute
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![](CNEx6.gif) |
using de Moivre's
formula. |
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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, |
![](CNEx6e.gif) |
then |
![](CNEx6f.gif) |
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Compute |
![](CNEx7.gif) |
using de Moivre's
formula. |
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r
= 64 and
j =
p |
![](CNEx7a.gif) |
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thus, |
![](CNEx7b.gif) |
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![](CNEx7d.gif) |
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![](CNEx7c.gif) |
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. |
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Solve |
![](CNEx8.gif) |
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Calculate |
![](CNEx9.gif) |
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