Question

If \(z=4 \angle \frac{\pi}{6}\) find \(z^{6}\) in polar form.

Short answer

Answer: The sixth power of the given complex number in polar form is \(z^6 = 4096\angle \pi\).

Step-by-step solution

Identify the given complex number in polar form

The given complex number is \(z=4\angle\frac{\pi}{6}\). Here, \(r=4\) and \(\theta=\frac{\pi}{6}\).

Apply De Moivre's theorem

According to De Moivre's theorem, \((r\angle\theta)^n = r^n\angle n\theta\). In this case, we need to find \(z^6\), so we have \(n=6\). Plug in the values of \(r\), \(\theta\), and \(n\) to find the sixth power of \(z\): \((4\angle\frac{\pi}{6})^6 = 4^6\angle 6\frac{\pi}{6}\)

Calculate the result

Using the values we found in the previous step, calculate the result for \(z^6\): \(4^6\angle 6\frac{\pi}{6} = 4096\angle \pi\)

Write the final answer in polar form

So, the sixth power of the given complex number in polar form is: \(z^6 = 4096\angle \pi\)

Key concepts

Polar Form

The polar form of a complex number makes it easier to visualize and manage rotations and magnitudes. It expresses a complex number in terms of its distance from the origin, called the modulus, and the angle it forms with the positive real axis, called the argument or phase angle. The polar form of a complex number is written as \( r \angle \theta \), where:

• \( r \): This is the modulus or the absolute value of the complex number, which represents the distance of the point from the origin in the complex plane.
• \( \theta \): This is the argument or phase angle, measured in radians, that gives the direction of the point from the origin in the complex plane.

This representation is particularly useful because multiplying powers of a complex number or raising it to a power becomes straightforward using De Moivre's Theorem. With this form, any complex number \( z = r(\cos \theta + i\sin \theta) \) can be easily converted into polar form \( z = r \angle \theta \).
Switching between rectangular (algebraic) form \( z = a + bi \) and polar form requires a bit of trigonometry, where \( r = \sqrt{a^2 + b^2} \) and \( \theta = \tan^{-1}(\frac{b}{a}) \).

Complex Numbers

Complex numbers extend our familiar set of real numbers and encompass solutions to truly exist where real numbers alone fail.Each complex number consists of a real part and an imaginary part, typically expressed as \( z = a + bi \), where:

• \( a \): The real part of the complex number.
• \( b \): The imaginary part, which is a real number multiplied by the unit imaginary component \( i \), where \( i^2 = -1 \).

Complex numbers live in a two-dimensional plane, often called the complex plane, where the horizontal axis represents the real numbers, and the vertical axis represents the imaginary numbers. This ingenious system allows for operations like addition, subtraction, and even rotation of numbers through multiplication.
Understanding complex numbers is crucial for solving quadratic equations and other higher-degree polynomials with real coefficients that lack real roots.

Complex Number Multiplication

Multiplying complex numbers can at first seem daunting due to their imaginary component. Yet, multiplying two complex numbers involves more straightforward trigonometric adjustments when they're expressed in polar form. This is where the beauty of the polar form shines.Consider two complex numbers represented in polar form as \( r_1 \angle \theta_1 \) and \( r_2 \angle \theta_2 \). Their product can be found simply by:

• Multiplying their moduli: Multiply \( r_1 \) and \( r_2 \) to get the system's resulting modulus.
• Adding their angles: Add \( \theta_1 \) and \( \theta_2 \) to find the resultant angle.

Therefore, the product is \( r_1r_2 \angle (\theta_1 + \theta_2) \).
This method is not only convenient for straightforward multiplication but also serves as the foundation for De Moivre's Theorem, which raises complex numbers to powers efficiently by adjusting their radius and rotating their angles systematically.