In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form.(−2 − 2i)⁵

Verified step by step guidance

1. Express the complex number \((-2 - 2i)\) in polar form. To do this, find the magnitude \(r\) using \(r = \sqrt{(-2)^2 + (-2)^2}\) and the angle \(\theta\) using \(\theta = \tan^{-1}\left(\frac{-2}{-2}\right)\).
2. Convert the complex number to polar form: \(r(\cos \theta + i\sin \theta)\).
3. Apply DeMoivre's Theorem: \((r(\cos \theta + i\sin \theta))^n = r^n (\cos(n\theta) + i\sin(n\theta))\), where \(n = 5\).
4. Calculate \(r^5\) and \(5\theta\) to find the new magnitude and angle.
5. Convert the result back to rectangular form using \(r^5(\cos(5\theta) + i\sin(5\theta))\) to get the final expression in the form \(a + bi\).

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 multiplication and exponentiation more manageable.

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

Magnitude and Argument of a Complex Number

The magnitude of a complex number z = a + bi is calculated as |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, found using θ = arctan(b/a). These two components are crucial for converting a complex number to polar form, which is necessary for applying DeMoivre's Theorem.