Verified Answer :
- Represent the number \(-64\) in polar form as \(64e^{i(\pi + 2k\pi)}\).
- Calculate the modulus of the roots as \(\sqrt[8]{64} = 2^{3/4} = \sqrt[4]{8}\).
- Find the eight distinct arguments \(\theta_k = \frac{(2k+1)\pi}{8}\) for \(k = 0, 1, \dots, 7\).
- Combine these to define the roots as \(\boxed{z_k = \sqrt[4]{8} \left( \cos\left(\frac{(2k+1)\pi}{8}\right) + i \sin\left(\frac{(2k+1)\pi}{8}\right) \right), k \in \{0, 1, \dots, 7\}}\).
Explanation
- Analysis and Problem Setup
The problem asks for the eighth roots of -64, which is equivalent to solving the equation \(z^8 = -64\) for complex values of \(z\). A negative number, such as -64, can be represented in polar form as \(64e^{i(\pi + 2k\pi)}\) where \(k\) is an integer. To find the eighth roots, we use the formula for the roots of a complex number: \(z_k = \sqrt[8]{|z|} \cdot e^{i\left(\frac{\theta + 2k\pi}{n}\right)}\), where \(n=8\), \(|z|=64\), and \(\theta=\pi\).
- Determining the Modulus
First, we determine the magnitude of the roots. The magnitude of \(z_k\) is given by \(\sqrt[8]{64}\). Since \(64 = 2^6\), we have \(\sqrt[8]{64} = (2^6)^{1/8} = 2^{6/8} = 2^{3/4} = \sqrt[4]{2^3} = \sqrt[4]{8}\). This value, approximately 1.68179, will be the modulus for all eight roots.
- Calculating the Arguments
Next, we determine the angles of the roots. Using the formula \(\theta_k = \frac{\pi + 2k\pi}{8}\) for \(k = 0, 1, \dots, 7\), we find the following angles:
- For \(k=0: \theta_0 = \frac{\pi}{8}\)
- For \(k=1: \theta_1 = \frac{3\pi}{8}\)
- For \(k=2: \theta_2 = \frac{5\pi}{8}\)
- For \(k=3: \theta_3 = \frac{7\pi}{8}\)
- For \(k=4: \theta_4 = \frac{9\pi}{8}\)
- For \(k=5: \theta_5 = \frac{11\pi}{8}\)
- For \(k=6: \theta_6 = \frac{13\pi}{8}\)
- For \(k=7: \theta_7 = \frac{15\pi}{8}\)
- Conclusion and Final Expression
Using De Moivre's theorem, we can write the roots in the form \(z_k = \sqrt[4]{8} \left( \cos\left(\frac{(2k+1)\pi}{8}\right) + i \sin\left(\frac{(2k+1)\pi}{8}\right) \right)\) for \(k = 0, 1, 2, 3, 4, 5, 6, 7\). These eight values represent all the eighth roots of -64. The set of roots is given by \(\{\sqrt[4]{8} e^{i\frac{(2k+1)\pi}{8}} \mid k \in \{0, 1, \dots, 7\}\}\). This can also be written in rectangular form, though the exponential/trigonometric forms are more standard for roots of high order. The final result is the set of these eight distinct complex numbers.
Examples
Finding the roots of complex numbers is a fundamental technique in engineering and physics, especially in fields like electrical circuit analysis and control theory. For example, when analyzing the stability of a system, engineers often look for the roots of characteristic equations (polynomials). Complex roots, such as those found here, provide critical information about oscillating behavior and decay rates in electronic signals or mechanical vibrations. By understanding these roots, one can better design filters or control systems that perform reliably under dynamic conditions.