91Ó°ÊÓ

Complex numbers, like the one in our exercise, can be expressed in two different ways: in rectangular form, which uses real and imaginary parts, and in polar form, which uses a magnitude and an angle. Transitioning from rectangular to polar form can make complex number operations much easier, such as multiplication, division, and exponentiation.
To express a complex number in polar form, you write it as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude, and \( \theta \) is the argument or angle. Alternatively, it can be expressed in exponential form as \( z = re^{i\theta} \).
For the given complex number \( 4(\sqrt{3} + i) \), we convert it into polar form to compute powers or roots easily. First, we calculate the magnitude, then the argument, and then combine these to achieve the polar representation. This way, the number becomes more manageable in operations like taking cube roots, as required in the exercise.

The magnitude of a complex number measures its size or length when plotted on a complex plane. Imagine a right triangle with legs corresponding to the real and imaginary parts of a complex number and the hypotenuse as the magnitude.
For a complex number \( a + bi \), the magnitude \( r \) is given by the formula \( r = \sqrt{a^2 + b^2} \). This formula is derived from the Pythagorean theorem.
In our exercise, the complex number is \( 4\sqrt{3} + 4i \). To find its magnitude, we calculate \( r = \sqrt{(4\sqrt{3})^2 + 4^2} \). Breaking it down step-by-step:

• Calculate \((4\sqrt{3})^2\) which is \(48\).
• Calculate \(4^2\) which is \(16\).
• Sum these results to get \(64\).
• Finally, take the square root to find the magnitude, \( r = 8 \).

Thus, the length of our complex number vector is 8 on the complex plane.

The argument of a complex number, usually represented by \( \theta \), is the angle the line connecting the complex number to the origin makes with the positive real axis. This angle helps in telling the direction of the complex number on the complex plane.
For a complex number \( a + bi \), the argument \( \theta \) can be calculated using the inverse tangent function as \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). This gives the angle in radians, typically between \( -\pi \) and \( \pi \).
In the exercise, we are given the components \( \sqrt{3} \) and \( 1 \) for \( a \) and \( b \) respectively (when factoring out the 4). The argument is calculated using \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \). Solving this gives us an angle of \( \frac{\pi}{6} \) radians.
Understanding the argument is essential for expressing a number in polar form and performing operations on it efficiently, such as calculating powers as needed in the original exercise.