In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fourth roots of 81 (cos 4π/3 + i sin 4π/3)

Verified step by step guidance

1. Recognize that the problem asks for the fourth roots of a complex number given in polar form: \(81 \left( \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} \right)\). The general form for the \(n\)th roots of a complex number \(r (\cos \theta + i \sin \theta)\) is given by De Moivre's Theorem.
2. Identify the magnitude \(r = 81\) and the argument \(\theta = \frac{4\pi}{3}\). Since we want the fourth roots, set \(n = 4\). The magnitude of each root will be \(r^{1/n} = 81^{1/4}\).
3. Calculate the arguments of the roots using the formula: \(\theta_k = \frac{\theta + 2\pi k}{n}\) for \(k = 0, 1, 2, 3\). This gives four distinct angles for the roots.
4. Write each root in polar form as \(r^{1/n} \left( \cos \theta_k + i \sin \theta_k \right)\) for each \(k\).
5. Convert each root from polar form to rectangular form using \(x = r^{1/n} \cos \theta_k\) and \(y = r^{1/n} \sin \theta_k\), then express each root as \(x + iy\). Round the values to the nearest tenth if necessary.

Key Concepts

Complex Numbers in Polar Form

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). This form is useful for multiplying, dividing, and finding roots of complex numbers by working with their magnitudes and angles separately.

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form, raising it to the nth power corresponds to raising the magnitude to the nth power and multiplying the angle by n. Conversely, finding nth roots involves taking the nth root of the magnitude and dividing the angle by n, adding multiples of 2π/n for all roots.

Conversion Between Polar and Rectangular Forms

After finding roots in polar form, converting them to rectangular form (a + bi) requires using a = r cos θ and b = r sin θ. This step is essential for expressing complex roots in the standard form, which is often required for clarity and further calculations.