Question

Evaluate \( \left[ i^{18} + \frac{1}{i} \right]^{25} \):

• A. \( 2(1 - i) \)
• B. \( 7(1 - i) \)
• C. \( 8i + 4 \)
• D. \( 7i - 1 \)

Hint:

When simplifying powers of \( i \), use the periodicity of powers of \( i \) to reduce the exponent.

Answer 1

The Correct Option is A

We are asked to evaluate \( \left[ i^{18} + \frac{1}{i} \right]^{25} \). Step 1: Simplify the powers of \( i \) Recall that powers of \( i \) follow a periodic pattern: \[ i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1 \] Thus, \( i^{18} = i^{4 \times 4 + 2} = i^2 = -1 \). Also, \( \frac{1}{i} = -i \). Step 2: Simplify the expression Substitute these values into the expression: \[ i^{18} + \frac{1}{i} = -1 - i = -(1 + i) \] Step 3: Raise to the power 25 Now, raise the result to the power of 25: \[ \left[ -(1 + i) \right]^{25} = -1 \times (1 + i)^{25} \] Thus, the correct answer is \( 2(1 - i) \).