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

Verified step by step guidance

1. Identify the problem as finding the sixth roots of a complex number, which is 64 in this case.
2. Express 64 in polar form. Since 64 is a real number, it can be written as \$64 = 64 \text{cis} 0^\circ\(, where \)\text{cis} \theta = \cos \theta + i \sin \theta$.
3. Use De Moivre's Theorem to find the sixth roots. According to the theorem, the \(n\)th roots of a complex number \(r \text{cis} \theta\) are given by \(r^{1/n} \text{cis} \left(\frac{\theta + 360k}{n}\right)\) for \(k = 0, 1, 2, \ldots, n-1\).
4. Calculate \(r^{1/6}\), which is \$64^{1/6}\(. Since \)64 = 2^6\(, \)64^{1/6} = 2$.
5. Find the angles for each root by substituting \(k = 0, 1, 2, 3, 4, 5\) into the formula \(\frac{0 + 360k}{6}\), and express each root in rectangular form using \$2 \text{cis} \theta = 2(\cos \theta + i \sin \theta)$.

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 and b are real numbers, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for solving problems involving roots, especially when dealing with non-real solutions.

Roots of Complex Numbers

Finding the roots of complex numbers involves determining the values that, when raised to a certain power, yield the original complex number. For sixth roots, we apply De Moivre's Theorem, which states that the nth roots of a complex number can be found by converting the number to polar form and then dividing the angle by n while adjusting for the periodic nature of trigonometric functions.

Rectangular Form

Rectangular form refers to expressing complex numbers in the standard a + bi format. When finding roots, it is often necessary to convert from polar form (r(cos θ + i sin θ)) back to rectangular form to provide a clear and usable representation of the roots, especially when rounding to specific decimal places.