Problem 64 - Algebra and Trigonometry

Question

Find all the complex roots. Write your answers in exponential form. The complex fifth roots of \(-i\)

Step-by-Step Solution

Answer

The complex fifth roots of \(-i\) are \(e^{i \frac{3\pi}{10}}, e^{i \frac{7\pi}{10}}, e^{i \frac{11\pi}{10}}, e^{i \frac{15\pi}{10}}, e^{i \frac{19\pi}{10}}\).

Step 1: Express \(-i\) in polar form

First, convert \(-i\) into polar form. \(-i\) is the same as \(-1 \cdot i\). This means the magnitude, or modulus, is 1 and the argument, or angle, is \frac{3\pi}{2}\. So, \(-i = e^{i \frac{3\pi}{2}}\).

Step 2: Apply the general formula for n-th roots

The general formula for finding the n-th roots of a complex number \(z = re^{i\theta}\) is: \(\text{root}_k = r^{1/n} e^{i(\theta+2k\pi)/n}\) where k = 0, 1, ..., n-1. For the 5-th roots of \(-i\), \(n = 5\) and \(r = 1\).

Step 3: Calculate each root

Compute each of the 5 roots using the formula: \(r_k = 1^{1/5}e^{i(\frac{3\pi}{2} + 2k\pi)/5}\) for k = 0, 1, 2, 3, 4. Since the modulus 1 to any power is still 1, focus on the exponents: \( e^{i(\frac{3\pi}{2} + 2\pi k)/5}\)

Step 4: Write out the roots

Substitute k = 0, 1, 2, 3, 4 into the formula: \( \[\[\begin{align*} r_0 &= e^{i \frac{3\pi}{10}} \ r_1 &= e^{i \frac{7\pi}{10}} \ r_2 &= e^{i \frac{11\pi}{10}} \ r_3 &= e^{i \frac{15\pi}{10}} \ r_4 &= e^{i \frac{19\pi}{10}} \end{align*}\]\] \).

Key Concepts

polar form

When dealing with complex numbers, a useful way to represent them is in polar form. This format uses a combination of the magnitude (or modulus) of the number and the angle (or argument) to describe the position of the number on the complex plane. For example, the complex number \( -i \) can be converted to polar form by first determining its modulus and argument. Here:
- The modulus of \( -i \) is 1 since \( | -i | = 1 \).
- The argument of \( -i \) is \( \frac{3\pi}{2} \), as it lies on the negative imaginary axis.
Combining these, the polar form of \( -i \) is \( e^{i \frac{3\pi}{2}} \). This transformation aids in the clear visualization and manipulation of the complex number.

n-th roots

Finding the n-th roots of a complex number involves finding all complex solutions that, when raised to the n-th power, equal the original number. There exists a general formula to compute these roots for any complex number \( z = re^{i\theta} \):
- \[ \text{root}_k = r^{1/n} e^{i(\theta+2k\pi)/n} \]
Here:

• \