Give me some moderately challenging exponent simplification problems with step by step answers 

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:**
\[
3
\]

---

### 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:**
\[
128
\]

---

### 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:**
\[
1
\]

---

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.

Give me some moderately challenging exponent simplification problems with step by step answers

Explore three moderately challenging exponent simplification problems complete with step-by-step solutions. This resource covers essential exponent rules, suitable for students in Grades 7-9.

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