https://file.aiquickdraw.com/mathbot/file/1726410314305fy1y1vkp.jpg

The problems presented in this image involve powers of the imaginary unit \( i \), where \( i = \sqrt{-1} \). The powers of \( i \) follow a repeating cycle:

\[
i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and then the cycle repeats.}
\]

Let's go through each problem step by step:

### Problem 47: \( i^3 \)
From the cycle, we know that:
\[
i^3 = -i
\]
Thus, the answer is \( -i \).

### Problem 48: \( i^{10} \)
To solve \( i^{10} \), we divide the exponent by 4 and look at the remainder:
\[
10 \div 4 = 2 \text{ remainder } 2
\]
Thus, \( i^{10} = i^2 = -1 \).

### Problem 49: \( (3i)^5 \)
First, calculate \( i^5 \). Since \( i^5 = i \) (from the cycle, \( i^5 = i^1 \)).
Now:
\[
(3i)^5 = 3^5 \cdot i^5 = 243 \cdot i = 243i
\]
So, the answer is \( 243i \).

### Problem 50: \( (2i)^4 \)
First, calculate \( i^4 \). We know from the cycle that \( i^4 = 1 \). Therefore:
\[
(2i)^4 = 2^4 \cdot i^4 = 16 \cdot 1 = 16
\]
So, the answer is \( 16 \).

### Problem 51: \( i^{1000} \)
To solve \( i^{1000} \), divide 1000 by 4 and look at the remainder:
\[
1000 \div 4 = 250 \text{ remainder } 0
\]
Thus, \( i^{1000} = i^0 = 1 \).

### Problem 52: \( i^{1002} \)
To solve \( i^{1002} \), divide 1002 by 4:
\[
1002 \div 4 = 250 \text{ remainder } 2
\]
Thus, \( i^{1002} = i^2 = -1 \).

### Final Answers:
- **47:** \( -i \)
- **48:** \( -1 \)
- **49:** \( 243i \)
- **50:** \( 16 \)
- **51:** \( 1 \)
- **52:** \( -1 \)

Would you like further details or clarification on any of the steps?

Here are five related questions to deepen understanding:
1. What is the cycle pattern for \( i^n \)?
2. How can we generalize the calculation of any power of \( i \)?
3. What happens when you multiply complex numbers like \( (3i)^5 \)?
4. How does multiplying a scalar by an imaginary unit affect the result?
5. Can you explain how to simplify any power of \( i \) modulo 4?

**Tip:** When working with powers of \( i \), remember that the powers repeat every four steps. Always reduce the exponent modulo 4 to simplify.

The image contains mathematical problems involving powers of the imaginary unit (i), including i^3, i^10, (3i)^5, (2i)^4, i^1000, and i^1002.

Learn how to simplify powers of the imaginary unit (i) with a detailed step-by-step solution. This problem covers key mathematical concepts, including the cycle of powers of i and complex numbers. Suitable for high school students in Grades 9-12.

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