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

• A. $i$
• B. $-1$
• C. $-i$
• D. 1

Answer: (C)

Verified step by step guidance

1. Understand that the imaginary unit 'i' is defined as the square root of -1, and powers of 'i' cycle every four terms: i, -1, -i, 1.
2. To simplify i^{1003}, determine the remainder when 1003 is divided by 4, since the powers of 'i' repeat every four terms.
3. Calculate 1003 mod 4. This will give you the position in the cycle of powers of 'i'.
4. Based on the remainder, identify the corresponding power of 'i' from the cycle: i, -1, -i, 1.
5. Use the identified power of 'i' to express i^{1003} in its simplest form.