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Potentiation and rooting of complex numbers in trigonometric form


Potentiation

Being and no an integer greater than 1we have:

Like this:

This formula is known as Moivre's 1st Formula.

Example

Calculate :

Let's consider for later we calculate . To apply Moivre's 1st formula, we need to calculate the modulus and the argument of z.

Module:

Argument:

Calculating :

Rooting

If , its nth roots are given by:

This expression is known as Moivre's 2nd Formula.

Example

Determine the cubic roots of z = 8.

Resolution

Let's calculate the modulus and the argument of z for the application of the 2nd Moivre formula:

Module:

Argument:

The cubic roots of 8 are given by:


The number k can assume the values 0, 1 and 2:

Geometrically, note that the three roots are over a radius circumference. 2 and are vertices of an equilateral triangle; his arguments form a PAN, whose first term is 0 and the reason é.

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