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The worksheet you uploaded involves solving problems using the laws of exponents. Here’s a detailed breakdown of each question along with the solutions:

1. \( \left( \frac{1}{2} \right)^4 \)
   - Solution: \( \left( \frac{1}{2} \right)^4 = \frac{1^4}{2^4} = \frac{1}{16} \)

2. \( 3^1 \)
   - Solution: \( 3^1 = 3 \)

3. \( 3^2 \times 3^3 \)
   - Solution: Using the law \( a^m \times a^n = a^{m+n} \), we get \( 3^5 = 243 \)

4. \( 3^4 \times 3^2 \)
   - Solution: Using the same law \( a^m \times a^n = a^{m+n} \), we get \( 3^6 = 729 \)

5. \( (2^2)^2 \)
   - Solution: Using the power of a power rule \( (a^m)^n = a^{m \times n} \), we get \( 2^4 = 16 \)

6. \( 2^0 \)
   - Solution: Any number raised to the power of 0 equals 1, so \( 2^0 = 1 \)

7. \( 3^2 \times 3^2 \)
   - Solution: Using the multiplication rule for exponents, \( 3^2 \times 3^2 = 3^4 = 81 \)

8. \( (2 \times 3)^3 \)
   - Solution: Apply the exponent to both numbers, \( 2^3 \times 3^3 = 8 \times 27 = 216 \)

9. \( 3^4 \)
   - Solution: \( 3^4 = 81 \)

10. \( 3^3 \times 3^2 \)
   - Solution: Using the same exponent rule, \( 3^5 = 243 \)

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### Related Questions:
1. What is the general rule for multiplying powers with the same base?
2. How does the zero exponent rule apply to different bases?
3. Can you explain the power of a product rule with a different example?
4. What happens if you multiply two powers with different bases but the same exponent?
5. What is the result of \( a^{-n} \), and how does it relate to positive exponents?

#### Tip:
When multiplying powers with the same base, remember to **add the exponents**, and when raising a power to another power, you **multiply the exponents**.

Solve each problem using the laws of exponents.

This worksheet covers problems related to the laws of exponents, including the product of powers, power of a power, and zero exponent rule. Suitable for students in Grades 6-8, it provides practice on simplifying expressions using exponent rules.

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