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 (trigonometric) form: \(16 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)\). Here, the modulus \(r = 16\) and the argument \(\theta = \frac{2\pi}{3}\).
2. Recall that the \(n\)th roots of a complex number \(r (\cos \theta + i \sin \theta)\) are given by the formula: \(\sqrt[n]{r} \left( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right)\), where \(k = 0, 1, 2, ..., n-1\).
3. Since we want the fourth roots (\(n=4\)), calculate the modulus of each root as \(\sqrt[4]{16}\) and the arguments as \(\frac{\frac{2\pi}{3} + 2k\pi}{4}\) for \(k = 0, 1, 2, 3\).
4. Write each root in trigonometric form using the calculated modulus and arguments for each \(k\). That is, for each \(k\), the root is: \(\sqrt[4]{16} \left( \cos \left( \frac{2\pi/3 + 2k\pi}{4} \right) + i \sin \left( \frac{2\pi/3 + 2k\pi}{4} \right) \right)\).
5. Convert each root from trigonometric form to rectangular form using the identities: \(x = r \cos \theta\) and \(y = r \sin \theta\), so each root is \(x + iy\).

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 simplifies multiplication, division, and finding roots by working with 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 or extracting nth roots involves raising the magnitude to the nth power or root and multiplying or dividing the angle by n. This theorem is essential for finding complex 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 necessary to express the roots in the standard complex number format.