In Exercises 32–35, find all the complex roots. Write roots in rectangular form.The complex fourth roots of 16 (cos 2π/3 + i sin 2π/3)

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1. Identify the given complex number in polar form: \$16(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3})$.
2. Recognize that you need to find the fourth roots of this complex number, which involves using De Moivre's Theorem.
3. Express the complex number in polar form as \(r(\cos \theta + i \sin \theta)\), where \(r = 16\) and \(\theta = \frac{2\pi}{3}\). The fourth roots will have the form \(\sqrt[4]{r}(\cos(\frac{\theta + 2k\pi}{4}) + i \sin(\frac{\theta + 2k\pi}{4}))\) for \(k = 0, 1, 2, 3\).
4. Calculate \(\sqrt[4]{16}\), which is the magnitude of the roots, and it equals 2.
5. For each \(k = 0, 1, 2, 3\), calculate the angles \(\frac{\theta + 2k\pi}{4}\) and express each root in rectangular form using \(\cos\) and \(\sin\) values.

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for solving problems involving roots and trigonometric forms.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ), the nth roots can be found using the formula r^(1/n)(cos(θ/n + 2kπ/n) + i sin(θ/n + 2kπ/n)), where k is an integer from 0 to n-1. This theorem is crucial for finding complex roots, especially when dealing with powers and roots of complex numbers.

Rectangular Form

Rectangular form refers to expressing complex numbers as a + bi, where a and b are real numbers. Converting complex numbers from polar form (r, θ) to rectangular form is necessary for clear representation and further calculations, particularly when identifying and working with the roots of complex numbers.