Complex Numbers

Step 1: Recognize the properties of complex numbers.
We are dealing with complex conjugates.
Let: The complex number is a ratio of a complex number and its conjugate.
We can rewrite as: where is the argument of the complex number .
From the given form of , we know that:
Step 2: Use properties of powers of complex numbers.
Now, we raise to the power of 6: Similarly, the complex conjugate of , which is: is equal to .
Raising this to the power of 6 gives:
Step 3: Add the results.
Now, we add and :
Step 4: Conclusion. The value of the given expression is 0, which is option (d).

Step 1: Simplifying the Complex Expression
First, simplify the given complex expression: Since , we have: Now the expression becomes: This simplifies to:
Step 2: Conjugate of a Complex Number
The conjugate of a complex number is .
Therefore, the conjugate of is: But we need to convert it into the form where the denominator is 25.
Step 3: Final Calculation
Multiply both the numerator and denominator by 25 to normalize the expression:
Step 4: Conclusion Thus, the conjugate complex number is: