In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 64

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Express the given number 64 in its complex polar form. Since 64 is a positive real number, it can be written as \(64(\cos 0 + i \sin 0)\) or \(64 e^{i \cdot 0}\).

Identify the magnitude (modulus) and argument (angle) of the complex number. Here, the magnitude is \(r = 64\) and the argument is \(\theta = 0\) radians.

Use De Moivre's Theorem to find the sixth roots. The formula for the \(n\)th roots of a complex number is given by: \(z_k = r^{1/n} \left( \cos \left( \frac{\theta + 2\pi k}{n} \right) + i \sin \left( \frac{\theta + 2\pi k}{n} \right) \right)\) where \(k = 0, 1, 2, ..., n-1\).

Calculate the magnitude of each root as \(r^{1/6} = 64^{1/6}\). Then, for each \(k\) from 0 to 5, compute the argument \(\frac{0 + 2\pi k}{6} = \frac{2\pi k}{6} = \frac{\pi k}{3}\).

Write each root in rectangular form by evaluating the cosine and sine of each argument: \(z_k = 64^{1/6} \left( \cos \left( \frac{\pi k}{3} \right) + i \sin \left( \frac{\pi k}{3} \right) \right)\) Calculate these values for \(k = 0, 1, 2, 3, 4, 5\) and 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 are similarly spaced around a circle centered at the origin, with radius equal to the nth root of the number's magnitude.

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 to rectangular form involves calculating the real part r cos θ and the imaginary part r sin θ, which is essential for expressing roots in the requested format.

De Moivre's Theorem

De Moivre's theorem states that (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). It is used to find the nth roots of complex numbers by taking the nth root of the magnitude and dividing the argument by n, generating all roots by adding multiples of 2π/n to the argument.