Simplify the power of $i$.
$i^{85}$

• A. $i$
• B. -1
• C. $-i$
• D. 1

Answer: (A)

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 that powers of \( i \) repeat in a cycle of 4: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and then the pattern repeats.
3. To simplify \( i^{85} \), find the remainder when 85 is divided by 4, since the powers cycle every 4 steps. Calculate \( 85 \mod 4 \).
4. Use the remainder from the previous step to determine which value in the cycle \( i^{85} \) corresponds to. For example, if the remainder is 1, then \( i^{85} = i \); if 2, then \( i^{85} = -1 \), and so on.
5. Write the simplified form of \( i^{85} \) based on the remainder and the cyclic pattern of powers of \( i \).