https://file.aiquickdraw.com/mathbot/file/1730231551445on8dpy7i.jpg

Let's analyze each of the statements to determine whether they are true or false.

1. **Option A: \( i^2 = i^6 \)**
   - We know that \( i^2 = -1 \).
   - Calculating \( i^6 \): \( i^6 = (i^2)^3 = (-1)^3 = -1 \).
   - Since \( i^2 = -1 \) and \( i^6 = -1 \), this statement is **true**.

2. **Option B: \( i^2 = -1 \)**
   - By definition, \( i \) is the imaginary unit, where \( i^2 = -1 \).
   - This statement is **true**.

3. **Option C: \( \sqrt{-100} = 10i^2 \)**
   - \( \sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i \).
   - \( 10i^2 = 10 \cdot (-1) = -10 \).
   - The two sides are not equal (\( 10i \neq -10 \)), so this statement is **false**.

4. **Option D: \( \sqrt{-100} = 10i \)**
   - As calculated above, \( \sqrt{-100} = 10i \).
   - This statement is **true**.

5. **Option E: \( (10i)^2 = -100i \)**
   - Calculating \( (10i)^2 \): \( (10i)^2 = 100i^2 = 100 \cdot (-1) = -100 \).
   - \( -100 \neq -100i \), so this statement is **false**.

### Summary of Correct Options:
- **A, B, and D** are true.

Would you like further details on any of these explanations or have additional questions?

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Here are five related questions:
1. What is the significance of the imaginary unit \( i \)?
2. How do powers of \( i \) cycle, and what is the pattern?
3. How does complex number multiplication affect phase and magnitude?
4. How do we handle square roots of negative numbers in complex analysis?
5. What is the difference between real and imaginary components in complex numbers?

**Tip:** When working with powers of \( i \), remember the cycle: \( i, -1, -i, 1 \), which repeats every four powers.

Select all of the following equations that are true:
A. i^2 = i^6
B. i^2 = -1
C. √-100 = 10i^2
D. √-100 = 10i
E. (10i)^2 = -100i

Explore which equations involving the imaginary unit i are true, including calculations with i^2, square roots of negative numbers, and properties of powers of i. This problem is suitable for students in Grades 10-12 and covers complex number concepts.

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