Give me some moderately challenging exponent problems 

Here are some moderately challenging exponent problems that require applying various exponent rules, such as the product of powers, power of a power, and division of powers:

1. **Simplify:**  
   \[
   \frac{(3^4 \cdot 3^2)^3}{3^9}
   \]

2. **Solve for \(x\):**  
   \[
   5^{2x + 1} = 125
   \]

3. **Simplify:**  
   \[
   (2^3 \cdot 4^2)^{\frac{1}{2}}
   \]

4. **Expand and simplify:**  
   \[
   (x^2 y^{-3})^4 \cdot (x^{-1} y^2)^2
   \]

5. **Solve for \(x\):**  
   \[
   9^{x+1} = \frac{1}{27}
   \]

Would you like detailed solutions or hints for any of these problems?

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Here are five additional related questions to deepen your understanding:

1. Simplify \(\left(\frac{4^3 \cdot 2^{-4}}{8}\right)^2\).
2. Solve for \(x\): \(2^{2x} \cdot 8^{x+1} = 512\).
3. Simplify \((x^3 y^{-2})^{\frac{1}{3}} \cdot (x^{-1} y)^{2}\).
4. If \(3^{2x - 1} = 81\), what is the value of \(x\)?
5. Simplify and express with positive exponents: \(\frac{a^{-2} b^{3} c^4}{a b^{-1} c^{-2}}\).

**Tip:** When working with exponents, rewrite any base with a common power if possible. For instance, \(8 = 2^3\) can simplify many problems involving base 2.

Give me some moderately challenging exponent problems

Explore moderately challenging exponent problems designed for high school students. Solve problems using various exponent rules, including product, power, and division of powers. Suitable for Grades 9-10, these exercises cover simplification techniques and equations involving exponents.

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