The continued product of the four values of `[ cos (pi/ 3) + i sin (pi/ 3)]^(3//4)` is

Options

A. 1
B. 2
C. 3
D. none

Answer

To solve the problem of finding the continued product of the four values of \([ \cos(\pi/3) + i \sin(\pi/3)]^{3/4}\), we will follow these steps:

### Step 1: Identify the complex number in polar form
The given expression can be rewritten using Euler's formula:
\[ z = \cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right) = e^{i \frac{\pi}{3}} \]

### Step 2: Raise the complex number to the power of \( \frac{3}{4} \)
Now, we raise \( z \) to the power of \( \frac{3}{4} \):
\[ z^{\frac{3}{4}} = \left(e^{i \frac{\pi}{3}}\right)^{\frac{3}{4}} = e^{i \frac{3\pi}{12}} = e^{i \frac{\pi}{4}} \]

### Step 3: Use the general form for the \( n \)-th roots of a complex number
The \( n \)-th roots of a complex number in polar form \( e^{i\theta} \) can be expressed as:
\[ z_k = e^{i\left(\frac{\pi}{4} + \frac{2k\pi}{n}\right)} \quad \text{for } k = 0, 1, 2, \ldots, n-1 \]
Here, \( n = 4 \), so we have:
\[ z_k = e^{i\left(\frac{\pi}{4} + \frac{2k\pi}{4}\right)} = e^{i\left(\frac{\pi}{4} + \frac{k\pi}{2}\right)} \]

### Step 4: Calculate the four values
For \( k = 0, 1, 2, 3 \):
- For \( k = 0 \): \( z_0 = e^{i \frac{\pi}{4}} \)
- For \( k = 1 \): \( z_1 = e^{i \left(\frac{\pi}{4} + \frac{\pi}{2}\right)} = e^{i \frac{3\pi}{4}} \)
- For \( k = 2 \): \( z_2 = e^{i \left(\frac{\pi}{4} + \pi\right)} = e^{i \frac{5\pi}{4}} \)
- For \( k = 3 \): \( z_3 = e^{i \left(\frac{\pi}{4} + \frac{3\pi}{2}\right)} = e^{i \frac{7\pi}{4}} \)

### Step 5: Find the product of these values
Now we need to find the product:
\[ P = z_0 \cdot z_1 \cdot z_2 \cdot z_3 = e^{i \frac{\pi}{4}} \cdot e^{i \frac{3\pi}{4}} \cdot e^{i \frac{5\pi}{4}} \cdot e^{i \frac{7\pi}{4}} \]
Using the property of exponents:
\[ P = e^{i\left(\frac{\pi}{4} + \frac{3\pi}{4} + \frac{5\pi}{4} + \frac{7\pi}{4}\right)} \]

### Step 6: Simplify the exponent
Calculating the sum of the angles:
\[ \frac{\pi}{4} + \frac{3\pi}{4} + \frac{5\pi}{4} + \frac{7\pi}{4} = \frac{16\pi}{4} = 4\pi \]
Thus, we have:
\[ P = e^{i 4\pi} \]

### Step 7: Evaluate \( e^{i 4\pi} \)
Since \( e^{i 4\pi} = \cos(4\pi) + i \sin(4\pi) = 1 + 0i = 1 \).

### Final Answer
The continued product of the four values is:
\[ \boxed{1} \]