In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form.[12 (cos π/14 + i sin π/14)]⁷

Verified step by step guidance

1. Identify the complex number in polar form: \( z = 12 (\cos \frac{\pi}{14} + i \sin \frac{\pi}{14}) \).
2. Apply DeMoivre's Theorem, which states \( [r (\cos \theta + i \sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta)) \), where \( r = 12 \), \( \theta = \frac{\pi}{14} \), and \( n = 7 \).
3. Calculate the new modulus: \( r^n = 12^7 \).
4. Calculate the new argument: \( n\theta = 7 \times \frac{\pi}{14} \).
5. Convert the result back to rectangular form using \( r^n (\cos(n\theta) + i \sin(n\theta)) = a + bi \), where \( a = r^n \cos(n\theta) \) and \( b = r^n \sin(n\theta) \).

Key Concepts

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 angles and magnitudes rather than directly multiplying complex numbers.

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 expressing the final result in the required 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 concepts are crucial for using DeMoivre's Theorem, as they help determine the values of r and θ needed for calculations.