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 (cos 5π/3 + i sin 5π/3)

Verified step by step guidance

1. Recognize that the problem asks for the fifth roots of a complex number given in polar (trigonometric) form: \(32 \left( \cos \frac{5\pi}{3} + i \sin \frac{5\pi}{3} \right)\). 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 \(r\) and the argument \(\theta\) of the complex number. Here, \(r = 32\) and \(\theta = \frac{5\pi}{3}\). The fifth roots will have magnitude \(r^{1/5}\) and arguments given by \(\frac{\theta + 2k\pi}{5}\) for \(k = 0, 1, 2, 3, 4\).
3. Calculate the magnitude of each root as \(\sqrt[5]{32}\), which simplifies to \$2\( because \)2^5 = 32\(. Then, write the argument for each root as \(\theta_k = \frac{5\pi/3 + 2k\pi}{5}\) for \)k = 0, 1, 2, 3, 4$.
4. Express each root in trigonometric form as \(2 \left( \cos \theta_k + i \sin \theta_k \right)\), substituting each \(\theta_k\) value. This gives the five distinct roots evenly spaced around the circle in the complex plane.
5. Convert each root from trigonometric form to rectangular form using the formulas \(x = r \cos \theta_k\) and \(y = r \sin \theta_k\), where \(x\) is the real part and \(y\) is the imaginary part. Round each value 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.

De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form, its nth power is given by r^n (cos nθ + i sin nθ). Conversely, the nth roots can be found by taking the nth root of the magnitude and dividing the angle by n, adding multiples of 2π/n to find all roots.

Conversion Between Polar and Rectangular Forms

To express complex roots in rectangular form (a + bi), convert from polar form using a = r cos θ and b = r sin θ. This step is essential for writing the final answers in the requested format and often involves rounding to a specified decimal place.