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When dealing with complex numbers, the polar form provides a way to express them using a magnitude and an angle instead of the usual real and imaginary parts. A complex number, typically represented as \( z = a + bi \), can be converted to polar form as \( z = r(\cos \theta + i\sin \theta) \). Here, \( r \) is the magnitude of the complex number, calculated as \( r = \sqrt{a^2 + b^2} \), and \( \theta \) is the argument or angle, found using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).The polar form is particularly useful for simplifying mathematical operations like multiplication and division. In our exercise of finding the fourth roots of \(-16\), we first convert \(-16\) to polar form. The magnitude is \(16\) (distance from the origin on the complex plane), and since \(-16\) lies on the negative real axis, the angle \(\theta\) is \(\pi\) radians. Thus, \(-16\) can be represented as \(16e^{i\pi} \) in polar form.

De Moivre's Theorem is an essential tool for dealing with powers and roots of complex numbers expressed in polar form. It states that for a complex number in polar form \( r(\cos \theta + i \sin \theta) \), its \( n \)-th power can be calculated as \( r^n (\cos(n\theta) + i \sin(n\theta)) \).This theorem can also be reversed to find roots. If we need to find an \( n \)-th root of a complex number, we modify \( \theta \) and \( r \) as follows:

• Compute the \( n \)-th root of the magnitude: \( r^{1/n} \).
• Divide the angle by \( n \) and add \( \frac{2k\pi}{n} \) for each root \( k = 0, 1, 2, ..., n-1 \).

Using De Moivre’s Theorem in the exercise, the fourth roots of \(-16\) are found by computing \(\sqrt[4]{16} = 2 \), followed by calculating the angle \(\theta + \frac{2k\pi}{4} \) for each \( k \).This gives the angles: \( \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \), which are used to find the fourth roots.

To find the fourth roots of a complex number, you essentially divide the angle into four equal parts and find the corresponding magnitudes. The process involves:

• Finding the fourth root of the magnitude.
• Calculating the angles by evenly distributing them in the circle's 360 degrees, represented in radians.

In the provided solution, after converting \(-16\) to polar form \(16e^{i\pi} \), we use De Moivre’s Theorem to find the fourth roots. The magnitude becomes \(2\), as it is the fourth root of \(16\).Using each value of \( k = 0, 1, 2, 3 \), we get different angles for the roots. These lead to:- \( z_0 = \sqrt{2} + i\sqrt{2} \)- \( z_1 = -\sqrt{2} + i\sqrt{2} \)- \( z_2 = -\sqrt{2} - i\sqrt{2} \)- \( z_3 = \sqrt{2} - i\sqrt{2} \)These roots are evenly spread across the complex plane. Each root corresponds to rotating the base angle by 90 degrees, showcasing the beautiful symmetry and simplicity polar coordinates provide.