In Exercises 32–35, find all the complex roots. Write roots in rectangular form.The complex cube roots of 8i

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1. Express the complex number in polar form: \$8i$ can be written as \(8(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}))\).
2. Use De Moivre's Theorem to find the cube roots: If \(z = r(\cos(\theta) + i\sin(\theta))\), then the \(n\)th roots are \(z_k = r^{1/n}(\cos(\frac{\theta + 2k\pi}{n}) + i\sin(\frac{\theta + 2k\pi}{n}))\) for \(k = 0, 1, ..., n-1\).
3. Calculate the magnitude of the roots: \(r^{1/3} = 8^{1/3} = 2\).
4. Determine the angles for each root: For \(k = 0, 1, 2\), calculate \(\frac{\pi/2 + 2k\pi}{3}\).
5. Convert each root back to rectangular form using \(z_k = 2(\cos(\text{angle}) + i\sin(\text{angle}))\).

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' is the real part and 'b' is the imaginary part. In this context, 'i' represents the imaginary unit, defined as the square root of -1. Understanding complex numbers is essential for solving problems involving roots of complex quantities.

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude (r) and angle (θ) with respect to the positive real axis, represented as r(cos θ + i sin θ) or re^(iθ). This form is particularly useful for finding roots of complex numbers, as it simplifies the calculations involved in extracting roots and allows for easier manipulation of angles.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form, the nth roots can be found by taking the nth root of the magnitude and dividing the angle by n. Specifically, the roots are given by r^(1/n)(cos(θ/n + 2kπ/n) + i sin(θ/n + 2kπ/n)), where k is an integer from 0 to n-1. This theorem is crucial for determining the complex cube roots of a number like 8i.