https://file.aiquickdraw.com/mathbot/file/1732986779030p8kdsc0f.jpg

Here’s a step-by-step solution for the exercise in the image. Let’s work through each question:

### 1. Express each of the following in exponential form:
- (i) \( 15 \times 15 \times 15 \times 15 \times 15 = 15^5 \)
- (ii) \( y \times y \times y \times y \times y \times y \times y \times y \times y \times y \times y = y^{11} \)
- (iii) \( (-5) \times (-5) \times (-5) = (-5)^3 \)
- (iv) \( 3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 5 \times 5 = 3^3 \times 5^5 \)
- (v) \( (-2) \times (-2) \times (-2) \times (-2) = (-2)^4 \)
- (vi) \( a \times a \times a = a^3 \)

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### 2. Write as a single natural number:
- (i) \( 3^3 = 3 \times 3 \times 3 = 27 \)
- (ii) \( 5^2 = 5 \times 5 = 25 \)
- (iii) \( 11^2 = 11 \times 11 = 121 \)
- (iv) \( 2^8 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 256 \)
- (v) \( 2^2 \times 3^3 = 4 \times 27 = 108 \)
- (vi) \( 3^2 \times 5^3 = 9 \times 125 = 1125 \)
- (vii) \( 3^2 \times 11^2 = 9 \times 121 = 1089 \)
- (viii) \( 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360 \)

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### 3. Which is larger: \( 2^9 \) or \( 9^2 \)?
- \( 2^9 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 512 \)
- \( 9^2 = 9 \times 9 = 81 \)
- **Answer**: \( 2^9 \) is larger.

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### 4. Write the power in words and as a repeated multiplication. Then, evaluate the power:
- (i) \( 8^3 \):  
  **In words**: "Eight raised to the power of three."  
  **Repeated multiplication**: \( 8 \times 8 \times 8 \).  
  **Evaluation**: \( 8 \times 8 \times 8 = 512 \).

- (ii) \( 2^5 \):  
  **In words**: "Two raised to the power of five."  
  **Repeated multiplication**: \( 2 \times 2 \times 2 \times 2 \times 2 \).  
  **Evaluation**: \( 2 \times 2 \times 2 \times 2 \times 2 = 32 \).

- (iii) \( 10^6 \):  
  **In words**: "Ten raised to the power of six."  
  **Repeated multiplication**: \( 10 \times 10 \times 10 \times 10 \times 10 \times 10 \).  
  **Evaluation**: \( 10^6 = 1,000,000 \).

- (iv) \( 12^3 \):  
  **In words**: "Twelve raised to the power of three."  
  **Repeated multiplication**: \( 12 \times 12 \times 12 \).  
  **Evaluation**: \( 12 \times 12 \times 12 = 1728 \).

Let me know if you'd like further explanations or have more questions! Here are five questions to expand your understanding:

1. How do negative bases in exponential forms behave with odd and even powers?
2. Why is \( 10^6 \) commonly used in scientific notation?
3. Can you think of real-world examples where exponents like \( 2^8 \) or \( 10^6 \) are useful?
4. What is the relationship between repeated multiplication and exponents?
5. What happens if the exponent is \( 0 \) (e.g., \( 5^0 \))?

**Tip:** Always simplify powers first before evaluating

Express each of the following in exponential form, evaluate powers, and determine the larger value among given expressions.

This guide explains how to express repeated multiplication in exponential form, evaluate powers, and compare values like 2^9 and 9^2. Ideal for middle school students (Grade 6-8), this exercise covers key exponent rules.

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