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De Moivre's Theorem is a powerful tool in complex number algebra that helps us find the roots of complex numbers with ease. It states that for any complex number in polar form, given as \( r \text{cis}(\theta) \), the \( n \)-th roots can be calculated by finding the magnitudes and angles for each root.

• Express the complex number in polar form: \( r \) represents the magnitude and \( \theta \) the angle.
• Calculate the magnitude of each root: This is done by taking the \( n \)-th root of \( r \), i.e., \( r^{1/n} \).
• Determine the angle for each root: This involves dividing \( \theta + 2k\pi \) by \( n \), where \( k \) is a number from \( 0 \) to \( n-1 \).

By applying these steps, De Moivre's Theorem simplifies the process of finding complex roots, making it invaluable in solving equations like \( x^3 = 1 \). This theorem is especially helpful for cube roots since splitting a complete circle (\( 2\pi \)) into three equal parts gives the necessary angles needed to find the three cube roots.

The polar form is a way of expressing complex numbers using a combination of magnitude and direction, which is rooted in Euler's formula. A complex number \( z \) can be represented as \( r(\cos(\theta) + i\sin(\theta)) \) or more concisely as \( r \text{cis}(\theta) \). Here, \( r \) is the magnitude or modulus, and \( \theta \) is the angle or argument. To convert a complex number to polar form:

• Calculate the magnitude: This is the distance from the origin to the number, found by \( r = \sqrt{a^2 + b^2} \), for a complex number \( z = a + bi \).
• Find the angle: The angle \( \theta \) is found by taking the arctangent of \( \frac{b}{a} \), but considering the quadrant to ensure the correct angle.

In polar form, calculations like multiplication and division are straightforward as they involve simply adding or subtracting angles and multiplying or dividing magnitudes. This form is particularly suitable for using De Moivre's Theorem to solve for roots, as seen in finding cube roots of \(1\).

Finding the cube roots of a number, especially when that number is expressed in complex form, requires a methodical approach using polar form and De Moivre's Theorem. The cube roots of a number \( x^3 = 1 \), involve finding three distinct solutions that when multiplied together return back to 1.

• Convert the number into its polar form: For \( 1 \), this is \( 1\text{cis}(0) \).
• Apply De Moivre’s Theorem: By solving \( \text{cis}\left(\frac{2k\pi}{3}\right) \) for \( k = 0, 1, 2 \), we find our three roots.
• Compute each root:
  • For \( k=0 \), the root is \( 1 \).
  • For \( k=1 \), calculate \( \text{cis}\left(\frac{2\pi}{3}\right) \), resulting in \( -\frac{1}{2} + i\frac{\sqrt{3}}{2} \).
  • For \( k=2 \), compute \( \text{cis}\left(\frac{4\pi}{3}\right) \), resulting in \( -\frac{1}{2} - i\frac{\sqrt{3}}{2} \).

These three solutions are the cube roots of unity in complex form. They form an equilateral triangle on the complex plane, each separated equally at \( 120^\circ \), perfectly demonstrating symmetry and the beauty of complex roots.