Give me some moderately challenging exponent simplification problems with answers 

Here are five moderately challenging exponent simplification problems, along with their answers:

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1. **Problem:** Simplify \( (3^4 \cdot 3^2) / 3^3 \).

   **Solution:**  
   \[
   = \frac{3^{4+2}}{3^3} = \frac{3^6}{3^3} = 3^{6-3} = 3^3 = 27
   \]

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2. **Problem:** Simplify \( (x^3 y^{-2})^2 \cdot (x^{-1} y^3) \).

   **Solution:**  
   \[
   = x^{3 \cdot 2} y^{-2 \cdot 2} \cdot x^{-1} y^3 
   = x^6 y^{-4} \cdot x^{-1} y^3 
   = x^{6 - 1} y^{-4 + 3} 
   = x^5 y^{-1} 
   = \frac{x^5}{y}
   \]

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3. **Problem:** Simplify \( (2x^{-2} y^3)^3 \).

   **Solution:**  
   \[
   = 2^3 \cdot (x^{-2})^3 \cdot (y^3)^3 
   = 8 \cdot x^{-6} \cdot y^9 
   = \frac{8y^9}{x^6}
   \]

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4. **Problem:** Simplify \( \frac{(a^4 b^2)^3}{a^5 b} \).

   **Solution:**  
   \[
   = \frac{a^{4 \cdot 3} b^{2 \cdot 3}}{a^5 b} 
   = \frac{a^{12} b^6}{a^5 b} 
   = a^{12 - 5} b^{6 - 1} 
   = a^7 b^5
   \]

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5. **Problem:** Simplify \( \left(\frac{m^3}{n^{-2}}\right)^{-2} \).

   **Solution:**  
   \[
   = \frac{n^{2 \cdot -2}}{m^{3 \cdot -2}} 
   = \frac{n^{-4}}{m^{-6}} 
   = \frac{1}{n^4} \cdot m^6 
   = \frac{m^6}{n^4}
   \]

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Let me know if you would like more details on these solutions or have any questions!

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### Related Questions:

1. How can exponent rules simplify complex algebraic expressions?
2. What are some common mistakes made when simplifying exponents?
3. How do negative exponents change the structure of an expression?
4. What is the difference between simplifying and solving exponential expressions?
5. How can exponential functions be represented graphically?

### Tip:
When simplifying exponents, always handle the negative exponents carefully by rewriting them as reciprocals first to avoid common errors.

Give me some moderately challenging exponent simplification problems with answers

Practice simplifying exponent expressions with these moderately challenging problems. Explore exponent rules, such as product and quotient of powers, power of a power, and handling negative exponents, with solutions for Grades 8-10.

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