Give me some moderately challenging exponent simplifications  with step by answers 

Here are some moderately challenging exponent simplification problems along with step-by-step answers:

### Problem 1
**Simplify:**  
\[
\frac{(x^3 \cdot y^4)^2}{x^4 \cdot y^3}
\]

**Solution:**

1. Start by applying the power rule \((a^m)^n = a^{m \cdot n}\):
   \[
   (x^3 \cdot y^4)^2 = x^{3 \cdot 2} \cdot y^{4 \cdot 2} = x^6 \cdot y^8
   \]

2. Substitute back into the expression:
   \[
   \frac{x^6 \cdot y^8}{x^4 \cdot y^3}
   \]

3. Apply the division rule \(a^m / a^n = a^{m - n}\):
   \[
   = x^{6 - 4} \cdot y^{8 - 3} = x^2 \cdot y^5
   \]

**Answer:**  
\[
x^2 \cdot y^5
\]

---

### Problem 2
**Simplify:**  
\[
(3x^2 y^{-1})^3 \cdot (x^3 y^4)^{-1}
\]

**Solution:**

1. Start with the first part \((3x^2 y^{-1})^3\) and apply the power rule:
   \[
   3^3 \cdot x^{2 \cdot 3} \cdot y^{-1 \cdot 3} = 27x^6 y^{-3}
   \]

2. Now simplify the second part \((x^3 y^4)^{-1}\) by applying the power rule:
   \[
   x^{3 \cdot -1} \cdot y^{4 \cdot -1} = x^{-3} y^{-4}
   \]

3. Combine both parts:
   \[
   27x^6 y^{-3} \cdot x^{-3} y^{-4}
   \]

4. Apply the product rule \(a^m \cdot a^n = a^{m + n}\):
   \[
   = 27x^{6 + (-3)} y^{-3 + (-4)} = 27x^3 y^{-7}
   \]

5. Write the final answer with positive exponents:
   \[
   = \frac{27x^3}{y^7}
   \]

**Answer:**  
\[
\frac{27x^3}{y^7}
\]

---

### Problem 3
**Simplify:**  
\[
\left(\frac{x^2}{y^{-3}}\right)^4 \cdot y^5
\]

**Solution:**

1. Start by applying the power rule to the fraction \(\left(\frac{x^2}{y^{-3}}\right)^4\):
   \[
   = \frac{x^{2 \cdot 4}}{y^{-3 \cdot 4}} = \frac{x^8}{y^{-12}}
   \]

2. Rewrite the fraction with positive exponents:
   \[
   = x^8 \cdot y^{12}
   \]

3. Multiply by \(y^5\):
   \[
   = x^8 \cdot y^{12 + 5} = x^8 \cdot y^{17}
   \]

**Answer:**  
\[
x^8 \cdot y^{17}
\]

---

### Problem 4
**Simplify:**  
\[
\frac{a^5 \cdot b^{-2}}{a^{-3} \cdot b^4}
\]

**Solution:**

1. Apply the division rule to each part of the fraction:
   \[
   = a^{5 - (-3)} \cdot b^{-2 - 4}
   \]

2. Simplify the exponents:
   \[
   = a^{5 + 3} \cdot b^{-2 - 4} = a^8 \cdot b^{-6}
   \]

3. Rewrite with positive exponents:
   \[
   = \frac{a^8}{b^6}
   \]

**Answer:**  
\[
\frac{a^8}{b^6}
\]

---

### Problem 5
**Simplify:**  
\[
(x^{-2} \cdot y^3)^{-2} \cdot x^5 \cdot y^{-4}
\]

**Solution:**

1. Begin by simplifying \((x^{-2} \cdot y^3)^{-2}\) using the power rule:
   \[
   = x^{-2 \cdot -2} \cdot y^{3 \cdot -2} = x^4 \cdot y^{-6}
   \]

2. Substitute back into the expression:
   \[
   x^4 \cdot y^{-6} \cdot x^5 \cdot y^{-4}
   \]

3. Apply the product rule for each variable:
   \[
   = x^{4 + 5} \cdot y^{-6 + (-4)} = x^9 \cdot y^{-10}
   \]

4. Rewrite with positive exponents:
   \[
   = \frac{x^9}{y^{10}}
   \]

**Answer:**  
\[
\frac{x^9}{y^{10}}
\]

---

Let me know if you need any additional explanations on these solutions. 

---

### Related Questions

1. Simplify: \(\frac{(2a^3 b^{-2})^2}{a^4 b^{-3}}\)
2. Simplify: \((5x^{-2} y^4)^{-3} \cdot (x^3 y)^{-2}\)
3. Simplify: \(\frac{(x^4 y^{-3})^2}{x^5 y^4}\)
4. Simplify: \(\left(\frac{3x^2}{y^{-3}}\right)^{-2} \cdot y^6\)
5. Simplify: \(\frac{(a^{-1} b^2)^{-3}}{a^4 b^{-3}}\)

---

### Tip
When simplifying exponents, remember that **negative exponents indicate reciprocals** (e.g., \(a^{-n} = \frac{1}{a^n}\)), and be cautious to keep signs of exponents correct during each step.

Give me some moderately challenging exponent simplifications with step-by-step answers.

Explore moderately challenging exponent simplification problems with step-by-step answers. This guide covers exponent rules including power, division, product, and negative exponent rules, ideal for students in Grades 9-10.

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