Here are three moderately challenging exponent simplification problems, along with step-by-step solutions.

Problem 1: Simplify $(3^4 \cdot 3^2) \div 3^5$

Step 1: Use the property of exponents: $a^m \cdot a^n = a^{m+n}$. $3^4 \cdot 3^2 = 3^{4+2} = 3^6$

Step 2: Substitute this back into the expression: $\frac{3^6}{3^5}$

Step 3: Use the property $\frac{a^m}{a^n} = a^{m-n}$: $3^{6-5} = 3^1$

Final Answer: $3$

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Problem 2: Simplify $(2^3)^4 \cdot 2^{-5}$

Step 1: Use the property of exponents: $(a^m)^n = a^{m \cdot n}$. $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$

Step 2: Substitute this back into the expression: $2^{12} \cdot 2^{-5}$

Step 3: Use the property $a^m \cdot a^n = a^{m+n}$: $2^{12 + (-5)} = 2^{12 - 5} = 2^7$

Final Answer: $128$

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Problem 3: Simplify $\frac{5^2 \cdot 5^{-3}}{5^{-1}}$

Step 1: Simplify the numerator using $a^m \cdot a^n = a^{m+n}$: $5^2 \cdot 5^{-3} = 5^{2 - 3} = 5^{-1}$

Step 2: Substitute this back into the expression: $\frac{5^{-1}}{5^{-1}}$

Step 3: Use $\frac{a^m}{a^n} = a^{m-n}$: $5^{-1 - (-1)} = 5^{0}$

Final Answer: $1$

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Would you like more problems, or do you have any specific questions? Here are five related questions to consider:

1. What are the rules for multiplying and dividing exponents?
2. How do you simplify expressions with negative exponents?
3. Can you provide an example of using exponents in a real-world scenario?
4. How do you handle exponents with fractional bases?
5. What is the significance of $a^0$ for any non-zero $a$?

Tip: When simplifying expressions with exponents, always look to combine like terms first using the properties of exponents.