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ALGEBRA
- solved problems |
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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 |
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Compute
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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, |
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then |
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Compute |
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using de Moivre's
formula. |
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r
= 64 and
j =
p |
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thus, |
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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 |
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Calculate |
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