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 8i in polar form. Recall that any complex number can be written as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the argument (angle). For 8i, find \(r = |8i|\) and \(\theta = \arg(8i)\).
2. Calculate the magnitude \(r\) of 8i using \(r = \sqrt{x^2 + y^2}\), where \(x\) and \(y\) are the real and imaginary parts respectively. Since 8i has real part 0 and imaginary part 8, find \(r\) accordingly.
3. Determine the argument \(\theta\) of 8i. Since it lies on the positive imaginary axis, identify the angle \(\theta\) in radians.
4. Use De Moivre's Theorem to find the cube roots. The \(n\)th roots of a complex number \(r(\cos \theta + i \sin \theta)\) are given by: \(\sqrt[n]{r} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right)\) for \(k = 0, 1, ..., n-1\). Here, \(n=3\).
5. Calculate each root by substituting \(k=0, 1, 2\) into the formula, then convert each root from polar form back to rectangular form using \(x = r \cos \theta\) and \(y = r \sin \theta\).

Key Concepts

Complex Numbers in Rectangular and Polar Form

Complex numbers can be expressed in rectangular form as a + bi, where a and b are real numbers, and i is the imaginary unit. Alternatively, they can be represented in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). Converting between these forms is essential for finding roots.

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form, raising it to a power n corresponds to raising the magnitude to n and multiplying the angle by n. Conversely, finding nth roots involves taking the nth root of the magnitude and dividing the angle by n, considering all possible angles by adding multiples of 2π.

Finding Complex Roots of a Number

To find the nth roots of a complex number, first convert it to polar form, then apply the nth root to the magnitude and divide the argument by n. The roots are spaced evenly around the circle in the complex plane, differing by 2π/n in angle. Finally, convert each root back to rectangular form for the answer.