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Let's go through the problems on the worksheet step by step:

### 1. \( 46^0 \)
- Any non-zero number raised to the power of zero equals **1**.
- **Answer: 1**

### 2. \( w^0 \), where \( w \neq 0 \)
- As in the previous problem, any non-zero number raised to the power of zero equals **1**.
- **Answer: 1**

### 3. \( 8^{-3} \)
- Negative exponents mean reciprocal, so \( 8^{-3} = \frac{1}{8^3} = \frac{1}{512} \).
- **Answer: \( \frac{1}{512} \)**

### 4. \( y^{-9} \)
- Again, negative exponents indicate the reciprocal: \( y^{-9} = \frac{1}{y^9} \).
- **Answer: \( \frac{1}{y^9} \)**

### 5. \( \frac{1}{c^6} \)
- Since \( \frac{1}{c^6} \) is already a fraction, you express it as \( c^{-6} \).
- **Answer: \( c^{-6} \)**

### 6. \( \frac{1}{10^5} \)
- Similarly, \( \frac{1}{10^5} \) can be written as \( 10^{-5} \).
- **Answer: \( 10^{-5} \)**

### 7. \( 9^4 \cdot 9^{-6} \)
- Use the property \( a^m \cdot a^n = a^{m+n} \), so \( 9^4 \cdot 9^{-6} = 9^{4 + (-6)} = 9^{-2} = \frac{1}{9^2} = \frac{1}{81} \).
- **Answer: \( \frac{1}{81} \)**

### 8. \( y^{-9} \cdot y^3 \)
- Similarly, \( y^{-9} \cdot y^3 = y^{-9+3} = y^{-6} = \frac{1}{y^6} \).
- **Answer: \( \frac{1}{y^6} \)**

### 9. \( \frac{x^{-8}}{x^{-12}} \)
- Use the property \( \frac{a^m}{a^n} = a^{m-n} \), so \( \frac{x^{-8}}{x^{-12}} = x^{-8 - (-12)} = x^{4} \).
- **Answer: \( x^4 \)**

### 10. \( \frac{d^{-13}}{d^{-2}} \)
- Similar to the previous one: \( \frac{d^{-13}}{d^{-2}} = d^{-13 - (-2)} = d^{-11} = \frac{1}{d^{11}} \).
- **Answer: \( \frac{1}{d^{11}} \)**

### 11. Simplify \( 8^{-7} \cdot 8^7 \cdot 10^4 \cdot 10^{-4} \)
- First, simplify the powers of 8: \( 8^{-7} \cdot 8^7 = 8^{0} = 1 \).
- Then, simplify the powers of 10: \( 10^4 \cdot 10^{-4} = 10^{4+(-4)} = 10^{0} = 1 \).
- The final result is \( 1 \cdot 1 = 1 \).
- **Answer: 1**

### 12. Multiselect - Select all of the expressions that are simplified:
- **\( n^4 \)** - Yes, this is simplified.
- **\( \frac{1}{n^{-5}} \)** - No, this can be written as \( n^5 \).
- **\( n^6 \cdot n^{-8} \)** - No, this simplifies to \( n^{-2} = \frac{1}{n^2} \).
- **\( n^7 \cdot p^8 \)** - Yes, this is simplified.
- **\( \frac{1}{n^3} \)** - Yes, this is simplified.

- **Answer: \( n^4 \), \( n^7 \cdot p^8 \), and \( \frac{1}{n^3} \)**.

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Would you like more details on any of these problems? Here are five related questions:

1. What is the rule for multiplying numbers with the same base but different exponents?
2. How do you simplify expressions with negative exponents?
3. How can you express \( a^{-n} \) as a fraction?
4. What happens when you raise a number to the power of zero?
5. How do you simplify the division of exponents with the same base?

**Tip:** Always simplify exponents by using the rules of addition or subtraction depending on whether the terms are multiplied or divided.

Simplify expressions with zero, positive, and negative exponents. Questions include powers of 46, y, 8, and c, along with expressions involving multiplication and division of powers.

Learn how to simplify expressions involving zero, positive, and negative exponents. Topics include the zero exponent rule, multiplying and dividing exponents, and converting negative exponents to fractions. Suitable for students in Grades 8-9.

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