Here are the essential concepts you must grasp in order to answer the question correctly.
DeMoivre's Theorem
DeMoivre's Theorem states that for any complex number expressed in polar form as r(cos θ + i sin θ), the nth power of this complex number can be calculated as r^n (cos(nθ) + i sin(nθ)). This theorem simplifies the process of raising complex numbers to powers by allowing us to work with their magnitudes and angles rather than their rectangular coordinates.
Recommended video:
Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)
Polar and Rectangular Forms of Complex Numbers
Complex numbers can be represented in two forms: rectangular form (a + bi, where a and b are real numbers) and polar form (r(cos θ + i sin θ), where r is the magnitude and θ is the angle). Understanding how to convert between these forms is essential for applying DeMoivre's Theorem and for expressing the final result in the required rectangular form.
Recommended video:
Converting Complex Numbers from Polar to Rectangular Form
Magnitude and Argument of Complex Numbers
The magnitude of a complex number is its distance from the origin in the complex plane, calculated as √(a² + b²) for a complex number a + bi. The argument is the angle formed with the positive real axis, typically measured in radians. These two components are crucial for using DeMoivre's Theorem, as they allow us to manipulate the complex number effectively when raising it to a power.
Recommended video: