In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fifth roots of 32

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Recognize that the problem asks for the complex fifth roots of 32, which means we want to find all complex numbers \(z\) such that \(z^5 = 32\).

Express 32 in polar (trigonometric) form. Since 32 is a positive real number, it can be written as \(32(\cos 0 + i \sin 0)\), where the magnitude \(r = 32\) and the argument \(\theta = 0\) radians.

Use De Moivre's Theorem to find the fifth roots. The magnitude of each root is \(r^{1/5} = 32^{1/5}\), and the arguments are given by \(\frac{\theta + 2k\pi}{5}\) for \(k = 0, 1, 2, 3, 4\).

Calculate each root in polar form as \(z_k = r^{1/5} \left( \cos \left( \frac{2k\pi}{5} \right) + i \sin \left( \frac{2k\pi}{5} \right) \right)\) for \(k = 0, 1, 2, 3, 4\).

Convert each root from polar form to rectangular form using \(x = r^{1/5} \cos \left( \frac{2k\pi}{5} \right)\) and \(y = r^{1/5} \sin \left( \frac{2k\pi}{5} \right)\), then write each root as \(x + iy\). Round to the nearest tenth if necessary.

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주요 개념

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Complex Roots of Unity

Complex roots of unity are solutions to the equation z^n = 1, evenly spaced on the unit circle in the complex plane. For any complex number, its nth roots can be found by considering these roots of unity scaled and rotated appropriately.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ the argument. Converting between polar and rectangular form (a + bi) involves using trigonometric functions: a = r cos θ and b = r sin θ.

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). This theorem is essential for finding nth roots by dividing the argument θ by n and taking the nth root of the magnitude r.