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.
 zn = rn · [cos(nj) + isin(nj)]
 and
Exponentiation and root extraction of complex numbers examples
 Example: Calculate 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
 Example: Calculate
 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.
 Example: Calculate
 Solution:
 Example: Calculate
 Solution:
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