Sure! I'll solve the odd-numbered problems from the image. I'll start with problems that I can clearly see. Let's get started:

### Problem 27:
Simplify:  
\[
\left( \frac{2^2 \cdot 3^3}{6^2} \right)^3
\]

**Solution:**
First, let's simplify the expression inside the parentheses.

- \( 2^2 \cdot 3^3 = 4 \cdot 27 = 108 \)
- \( 6^2 = 36 \)

So the expression becomes:
\[
\left( \frac{108}{36} \right)^3
\]
Now, simplify \( \frac{108}{36} \):
\[
\frac{108}{36} = 3
\]
Therefore, the expression simplifies to:
\[
3^3 = 27
\]
**Answer:** \( 27 \)

---

### Problem 29:
Simplify:  
\[
\frac{\left( -2x^2 \right)^2}{4x^3}
\]

**Solution:**
Expand the numerator:
\[
\left( -2x^2 \right)^2 = (-2)^2 \cdot (x^2)^2 = 4x^4
\]
So the expression becomes:
\[
\frac{4x^4}{4x^3}
\]
Simplify by dividing both the coefficient and the exponent of \( x \):
\[
= x^{4-3} = x^1 = x
\]

**Answer:** \( x \)

---

### Problem 31:
Simplify:  
\[
\frac{6a^{-3}}{2b^{-2}}
\]

**Solution:**
Simplify the coefficients:
\[
\frac{6}{2} = 3
\]
Now handle the exponents:
\[
a^{-3} = \frac{1}{a^3}, \quad b^{-2} = \frac{1}{b^2}
\]
So the expression simplifies to:
\[
3 \cdot \frac{1}{a^3} \cdot b^2 = \frac{3b^2}{a^3}
\]

**Answer:** \( \frac{3b^2}{a^3} \)

---

### Problem 33:
Simplify:  
\[
\left( \frac{a^{-2} b^3}{a^{-3} b^2} \right)^2
\]

**Solution:**
First, simplify the expression inside the parentheses:
\[
\frac{a^{-2} b^3}{a^{-3} b^2} = a^{-2 - (-3)} \cdot b^{3-2} = a^{1} \cdot b^{1} = ab
\]
Now, square the result:
\[
(ab)^2 = a^2 b^2
\]

**Answer:** \( a^2b^2 \)

---

### Problem 35:
Simplify:
\[
\frac{x^{-2}y^4}{z^{-3}} \cdot \frac{z}{y^3}
\]

**Solution:**
First, simplify the expression:
\[
\frac{x^{-2}y^4}{z^{-3}} \cdot \frac{z}{y^3} = \frac{x^{-2} y^4 \cdot z}{z^{-3} \cdot y^3} 
\]
This simplifies to:
\[
= \frac{x^{-2} y^{4-3} \cdot z^{1+3}}{1} = \frac{x^{-2} y^1 z^4}{1} = \frac{y z^4}{x^2}
\]

**Answer:** \( \frac{yz^4}{x^2} \)

---

### Problem 37:
Simplify:  
\[
\frac{(x^{-1} y)^2}{y^{-3}}
\]

**Solution:**
First, expand the numerator:
\[
(x^{-1} y)^2 = x^{-2} y^2
\]
So the expression becomes:
\[
\frac{x^{-2} y^2}{y^{-3}} = x^{-2} y^{2 - (-3)} = x^{-2} y^{5}
\]

**Answer:** \( \frac{y^5}{x^2} \)

---

These are the simplified results for the odd-numbered problems visible from the image.

Would you like more details on any of these steps, or do you have any further questions?

Here are 5 related questions to deepen your understanding:
1. How do you simplify expressions with both negative and fractional exponents?
2. Can you solve a similar expression involving fractional exponents?
3. What happens when you multiply two expressions with different bases but the same exponents?
4. How do you simplify a complex fraction involving exponents?
5. Can you apply these exponent rules to solve equations rather than just simplifying?

**Tip:** When simplifying expressions involving exponents, always pay careful attention to the properties of exponents, such as the power of a power rule and the product of powers rule, to ensure accuracy.

Explore detailed solutions to odd-numbered problems involving exponents and algebra. Learn how to simplify expressions with negative and fractional exponents, apply exponent rules, and solve complex algebraic fractions. Suitable for high school students.

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