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Understanding De Moivre's Theorem is key when working with powers and roots of complex numbers. It is a formula that connects complex numbers and trigonometry, enabling us to raise complex numbers to any power with ease.

The theorem states that for a complex number in polar form, that is, in the format of \( r(\cos \theta + i \sin \theta) \), and any integer n, the n-th power of the complex number can be expressed as:
\[ (r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \]
In the exercise provided, De Moivre's Theorem helps in finding the roots of a complex number. Specifically, when we look for the n-th roots, this theorem can be modified as:
\[ \sqrt[n]{r} ( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} ) \]
Here, the value of \( k \) iterates from 0 to \( n-1 \), giving us the distinct n-th roots of the original complex number.

Complex numbers can be expressed in a form that uses a combination of trigonometric functions, known as the polar form. This is incredibly useful when multiplying, dividing, or dealing with powers and roots of complex numbers.

The polar form of a complex number is given by:
\[ r (\cos \theta + i \sin \theta) \]
where \( r \) is the magnitude (or modulus) of the complex number, calculated as \( r = \sqrt{a^2 + b^2} \), and \( \theta \) is the argument (or angle), found by \( \theta = \arctan(\frac{b}{a}) \). However, if the complex number lies along the axes, the angle is determined based on the sign of the imaginary part. In the given exercise, \(-8i\) is purely imaginary and negative, hence \( \theta \) is \( -\frac{\pi}{2} \).

By converting to polar form, complex calculations become simpler as we can directly apply trigonometric identities and properties, as well as De Moivre's Theorem.

The trigonometric functions, sine (\( \sin \)) and cosine (\( \cos \)), are fundamental when we deal with the polar form of complex numbers and apply De Moivre's Theorem. These functions relate the angles of a right triangle to its sides and have a natural extension to functions on a unit circle.

For any angle \( \theta \), we define:
\[ \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \quad \text{and} \quad \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \]
Using these functions, we can analyze and interpret the movement around a circle which is especially useful in expressing the angular component of a complex number in polar form. During the simplification step in the provided exercise, it becomes necessary to evaluate \( \cos \) and \( \sin \) for specific angles to obtain the real and imaginary parts of the roots.