Step 1: Understanding the Concept:
This problem involves operations with complex numbers, specifically simplifying a ratio of complex numbers and then solving an equation involving powers of a complex number. The key is to first simplify the base of the power and then use the properties of powers of .
Step 2: Key Formula or Approach:
1. To simplify a fraction of complex numbers like , multiply the numerator and denominator by the complex conjugate of the denominator, .
2. Use Euler's formula or standard powers of to solve the final equation. We know that . Also, .
Step 3: Detailed Explanation:
1. Simplify the base of the power: Let's simplify the complex number . Multiply the numerator and denominator by the conjugate of the denominator, which is : The numerator is . The denominator is .
2. Solve the equation: Substitute the simplified base back into the original equation:
3. Find the required power: We need to find an exponent such that . Let's check the powers of :
We see that the smallest positive integer power that gives is 2. So, we must have the exponent equal to 2 (or for any integer ). Let's take the simplest case:
4. Check the options: The value is one of the given options. For n=2, we have . For n=6, we have . For n=8, we have . Therefore, the only possible value among the options is . Step 4: Final Answer:
By simplifying the complex fraction to and solving the equation , we find that a possible value for is 4. This corresponds to option (B).