In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex fifth roots of −1 − i

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1. Express the complex number \(-1 - i\) in polar (trigonometric) form. To do this, find the modulus \(r\) and the argument \(\theta\). The modulus is given by \(r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}\).
2. Calculate the argument \(\theta\) using \(\tan \theta = \frac{\text{Imaginary part}}{\text{Real part}} = \frac{-1}{-1} = 1\). Since the point \((-1, -1)\) lies in the third quadrant, adjust \(\theta\) accordingly to \(\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\).
3. Use De Moivre's Theorem to find the fifth roots. The general formula for the \(n\)th roots of a complex number \(r(\cos \theta + i \sin \theta)\) is \[ z_k = r^{1/n} \left( \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right) \] where \(k = 0, 1, 2, ..., n-1\).
4. Substitute \(r = \sqrt{2}\), \(\theta = \frac{5\pi}{4}\), and \(n = 5\) into the formula to get each root \(z_k\): \[ z_k = (\sqrt{2})^{1/5} \left( \cos \left( \frac{5\pi/4 + 2k\pi}{5} \right) + i \sin \left( \frac{5\pi/4 + 2k\pi}{5} \right) \right) \] for \(k = 0, 1, 2, 3, 4\).
5. Convert each root from polar form back to rectangular form by calculating \(x = r^{1/5} \cos \left( \frac{5\pi/4 + 2k\pi}{5} \right)\) and \(y = r^{1/5} \sin \left( \frac{5\pi/4 + 2k\pi}{5} \right)\). Write each root as \(x + iy\).

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 of complex numbers.

De Moivre's Theorem

De Moivre's theorem states that for a complex number in polar form, raising it to the nth power corresponds to raising the magnitude to the nth power and multiplying the angle by n. Conversely, to find nth roots, one takes the nth root of the magnitude and divides the angle by n, adding multiples of 2π/n to find all roots.

Finding All nth Roots of a Complex Number

The nth roots of a complex number are found by expressing the number in polar form, then calculating the nth root of its magnitude and dividing its argument by n. All distinct roots are obtained by adding integer multiples of 2π/n to the argument. Finally, roots are converted back to rectangular form for the answer.