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 in polar form, z = r(cos θ + i sin θ), the nth power of z can be expressed as z^n = r^n (cos(nθ) + i sin(nθ)). This theorem simplifies the process of raising complex numbers to a power by converting them to polar coordinates, making calculations more manageable.
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Polar and Rectangular Forms of Complex Numbers
Complex numbers can be represented in two forms: rectangular form, a + bi, where a is the real part and b is the imaginary part, 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 effectively.
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Magnitude and Argument of a Complex Number
The magnitude of a complex number z = a + bi is given by |z| = √(a² + b²), representing its distance from the origin in the complex plane. The argument, θ, is the angle formed with the positive real axis, calculated using θ = arctan(b/a). These values are crucial for converting a complex number to polar form before applying DeMoivre's Theorem.
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