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.