In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _[√2 (cos 5π/6 + i sin 5π/6)]⁴

Verified step by step guidance

1. Step 1: Identify the complex number in polar form. Here, the complex number is given as \( \sqrt{2} (\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6}) \).
2. Step 2: Apply DeMoivre's Theorem, which states that for a complex number in polar form \( r(\cos \theta + i \sin \theta) \), the \( n \)-th power is \( r^n (\cos(n\theta) + i \sin(n\theta)) \).
3. Step 3: Calculate the magnitude raised to the power: \( (\sqrt{2})^4 \).
4. Step 4: Calculate the new angle by multiplying the original angle by the power: \( 4 \times \frac{5\pi}{6} \).
5. Step 5: Convert the result back to rectangular form using \( r^n (\cos(n\theta) + i \sin(n\theta)) \) to find the real and imaginary components.

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 their magnitudes and angles rather than their rectangular coordinates.

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.

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.