Question

What is the value of \( i^{4k+2} \) where \( k \) is an integer?

• A. \(1 \)
• B. \(-1 \)
• C. \(i \)
• D. \(-i \)

Hint:

Memorize the power cycle of \(i\): \(i,-1,-i,1\). The pattern repeats every 4 powers, making higher powers easy to simplify.

Answer 1

The Correct Option is B

Concept: The imaginary unit \(i\) is defined by the property: \[ i=\sqrt{-1} \] The powers of \(i\) follow a repeating cyclic pattern after every four powers. Understanding this cycle helps in simplifying higher powers of complex numbers. The cycle is: \[ i^1=i \] \[ i^2=-1 \] \[ i^3=-i \] \[ i^4=1 \] Then the pattern repeats again.

Step 1: Breaking the exponent into separate powers.
The given expression is: \[ i^{4k+2} \] Using exponent rule: \[ a^{m+n}=a^m\times a^n \] we write: \[ i^{4k+2}=i^{4k}\times i^2 \]

Step 2: Simplifying the term involving multiples of four.
Since powers of \(i\) repeat after every four powers: \[ i^4=1 \] Therefore: \[ i^{4k}=(i^4)^k \] Substituting: \[ i^{4k}=1^k \] Since any power of 1 remains 1: \[ i^{4k}=1 \]

Step 3: Substituting the remaining value.
Now substitute in the original expression: \[ i^{4k+2}=1\times i^2 \] We know: \[ i^2=-1 \] Therefore: \[ i^{4k+2}=-1 \] Hence the final value becomes: \[ \boxed{-1} \] Therefore option (B) is the correct answer.