Roots of complex numbers
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How do we use de Moivre;s Theorem and the Fundamental Theorem of Algebra to find the nth roots of a complex number?

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o find the nth roots of a complex number using De Moivre’s Theorem and the Fundamental Theorem of Algebra, you can follow these steps:

1. First, represent the complex number in its polar form, which is in the form of r(cos θ + i sin θ), where r is the magnitude and θ is the argument of the complex number.
2. Apply De Moivre’s Theorem to find the nth roots of the magnitude r. De Moivre’s Theorem states that for any complex number z = r(cos θ + i sin θ), the nth roots are given by r^(1/n) * [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)], where k ranges from 0 to n-1.
3. These values represent the nth roots of the magnitude. To find the nth roots of the original complex number, pair each of these magnitude roots with the original argument θ and convert them back to Cartesian form.

By utilizing De Moivre’s Theorem and the Fundamental Theorem of Algebra, you can easily compute the nth roots of a complex number in both polar and Cartesian forms.