Simplify the power of $i$.
$i^{1003}$

• A. $i$
• B. $-1$
• C. $-i$
• D. 1

Answer: (C)

Verified step by step guidance

1. Recall that the imaginary unit \(i\) is defined such that \(i^2 = -1\).
2. Recognize the cyclic pattern of powers of \(i\): \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). After \(i^4\), the powers repeat in this cycle every 4 powers.
3. To simplify \(i^{1003}\), find the remainder when 1003 is divided by 4, since the powers of \(i\) repeat every 4 steps.
4. Calculate \$1003 \mod 4\( to determine which power in the cycle \)i^{1003}$ corresponds to.
5. Use the remainder to identify the equivalent power of \(i\) from the cycle and write the simplified form accordingly.