To solve $(0 + i)^{923}$, we need to understand the properties of complex numbers.

The number $0 + i$ is a purely imaginary number, and we can represent it as $i$.

Step-by-Step Explanation:

1. Simplify the Expression: $(0 + i)^{923} = i^{923}$

2. Understanding Powers of $i$: The powers of $i$ follow a repeating cycle:

   • $i^1 = i$
   • $i^2 = -1$
   • $i^3 = -i$
   • $i^4 = 1$

   This cycle repeats every four terms.

3. Finding the Equivalent Power Modulo 4: Since the powers of $i$ repeat every 4, we can reduce the exponent by dividing by 4 and finding the remainder. This helps determine the equivalent power: $923 \div 4 = 230 \text{ remainder } 3$ So, $i^{923} = i^3$.

4. Using the Cycle: From the cycle above, we know: $i^3 = -i$

Final Answer:

$i^{923} = -i$

So, $(0 + i)^{923} = -i$.

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Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. What are the powers of other complex numbers like $1 + i$?
2. How would you solve $(0 + i)^n$ for any $n$?
3. What is the general rule for powers of $i$?
4. How do we simplify higher powers of complex numbers like $2 + 3i$?
5. What happens when raising real numbers to powers similar to $i$?

Tip: Powers of $i$ repeat every 4 terms, which makes simplifying them easy by using modulo 4.