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 \(i\) has the property \(i^2 = -1\). This is the fundamental definition that helps simplify powers of \(i\).
2. Recognize the pattern of powers of \(i\): \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). After \(i^4\), the powers repeat every 4 steps.
3. To simplify \(i^{1003}\), find the remainder when 1003 is divided by 4, because the powers of \(i\) cycle every 4.
4. Calculate \(1003 \mod 4\) to determine which power in the cycle \(i^{1003}\) corresponds to.
5. Use the remainder from the previous step to identify the equivalent power of \(i\) from the cycle and write the simplified form accordingly.