In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _(√3 − i)⁶

Verified step by step guidance

1. Step 1: Express the complex number \( \sqrt{3} - i \) in polar form. To do this, find the modulus \( r \) using \( r = \sqrt{(\sqrt{3})^2 + (-1)^2} \).
2. Step 2: Determine the argument \( \theta \) of the complex number using \( \theta = \tan^{-1}\left(\frac{-1}{\sqrt{3}}\right) \).
3. Step 3: Write the complex number in polar form as \( r(\cos \theta + i \sin \theta) \).
4. Step 4: Apply DeMoivre's Theorem, which states \( (r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \), to find \( (\sqrt{3} - i)^6 \).
5. Step 5: Convert the result back to rectangular form by calculating \( r^6 \cos(6\theta) \) for the real part and \( r^6 \sin(6\theta) \) for the imaginary part.

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 the 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 converting them to polar coordinates, making calculations more manageable.

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 modulus and θ is the argument). Understanding how to convert between these forms is essential for applying DeMoivre's Theorem effectively, as the theorem operates in polar coordinates.

Modulus and Argument of Complex Numbers

The modulus 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, found using the arctan function. These two components are crucial for converting a complex number to polar form, which is necessary for applying DeMoivre's Theorem.