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De Moivre's Theorem is a powerful tool in complex number theory that simplifies working with powers and roots of complex numbers. It states that for any complex number in polar form, \[ z = r (\cos \theta + i \sin \theta) \]where \( n \) is a positive integer, then the \( n \)-th power of \( z \) is given by:\[ z^n = r^n (\cos(n \theta) + i \sin(n \theta)) \]This theorem also extends to roots, allowing us to easily find the \( n \)-th roots of a complex number. Here's how to apply it:

• Write the complex number in polar form.
• Calculate the magnitude of each root by taking the \( n \)-th root of the original magnitude.
• Determine each root's angle by dividing the original angle by \( n \) then adding \( \frac{2\pi k}{n} \) (or \( \frac{360^\circ k}{n} \) in degrees) for each integer \( k \) from 0 to \( n-1 \).

This method simplifies complex calculations, making it easier to find solutions like the fifth roots of a complex number. De Moivre's Theorem is invaluable for handling complex numbers in mathematics.

The polar form of complex numbers is a way to express complex numbers using polar coordinates instead of rectangular coordinates. A complex number \( z \) is usually written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In polar form, the same number is represented as:\[ z = r (\cos \theta + i \sin \theta) \text{ or } z = r \text{ cis } \theta \]Here, \( r \) is the magnitude (or modulus) of the complex number, and \( \theta \) is the argument (or angle). To convert a complex number from rectangular form to polar form, use:

• The magnitude \( r = \sqrt{a^2 + b^2} \).
• The angle \( \theta = \arctan\left(\frac{b}{a}\right) \) measured from the positive x-axis.

The polar form is especially beneficial when multiplying, dividing, or finding the roots of complex numbers, as operations become more straightforward with angles and magnitudes. For example, finding the fifth roots becomes easier with this form utilizing De Moivre's Theorem.

Finding roots of complex numbers involves calculating values that, when raised to a specific power, give the original complex number. To find the \( n \)-th roots of a complex number in polar form, leverage both the magnitude and angle:

• Determine the root magnitude as \( r^{1/n} \), being the \( n \)-th root of the original magnitude.
• Calculate the angles \( \theta_k \) for each root using: \( \theta_k = \frac{\theta}{n} + \frac{k360^\circ}{n} \) for each integer \( k \) from 0 to \( n-1 \).

For example, for finding the fifth roots of a complex number given as \( 10^5 \text{ cis } 15^\circ \):

• Discover the magnitude of each root, which is simply: \( 10^5 \) raised to the power of \( \frac{1}{5} \), yielding 10.
• Calculate different angles for \( k = 0, 1, 2, 3, 4 \) to find each unique root's angle.

Then, each of these roots can be expressed in trigonometric form using the root's magnitude and calculated angle. Finally, convert them into their rectangular form \( a + bi \) for further comprehension. This approach simplifies handling roots within complex numbers, leveraging the polar form and angular properties to find precise solutions.