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Let's solve the two parts of the given question step by step:

### Part 4.a:
Expression: \(\left(\frac{ab}{2a}\right)^3\)

1. **Simplify the fraction inside the parentheses**:
   \[
   \frac{ab}{2a} = \frac{a \cdot b}{2 \cdot a} = \frac{b}{2}
   \]

2. **Now cube the simplified expression**:
   \[
   \left(\frac{b}{2}\right)^3 = \frac{b^3}{2^3} = \frac{b^3}{8}
   \]

Thus, the result for part 4.a is:
\[
\frac{b^3}{8}
\]

---

### Part 4.b:
Expression: \(\frac{(18a^2b^{-2})^2}{12ab}\)

1. **First, simplify the square of the numerator**:
   \[
   (18a^2b^{-2})^2 = 18^2 \cdot (a^2)^2 \cdot (b^{-2})^2 = 324 \cdot a^4 \cdot b^{-4}
   \]
   So the numerator becomes:
   \[
   324a^4b^{-4}
   \]

2. **Now, simplify the denominator**:
   \[
   12ab = 12 \cdot a \cdot b
   \]

3. **Combine the numerator and denominator**:
   \[
   \frac{324a^4b^{-4}}{12ab} = \frac{324}{12} \cdot \frac{a^4}{a} \cdot \frac{b^{-4}}{b}
   \]

4. **Simplify further**:
   \[
   \frac{324}{12} = 27, \quad \frac{a^4}{a} = a^3, \quad \frac{b^{-4}}{b} = b^{-5}
   \]

Thus, the result for part 4.b is:
\[
27a^3b^{-5}
\]

---

### Final Answers:
- **4.a:** \(\frac{b^3}{8}\)
- **4.b:** \(27a^3b^{-5}\)

Do you want further details or have any questions?

---

Here are 5 related questions to expand your understanding:
1. How do you handle negative exponents when simplifying expressions?
2. Can you simplify an expression like \(\left(\frac{2x^2}{3y}\right)^2\)?
3. What happens when you cube a fraction where both the numerator and denominator are polynomials?
4. How do you simplify an expression with a complex denominator, such as \(\frac{a^3b}{4ab^2}\)?
5. What are the rules for combining like terms when simplifying algebraic expressions?

**Tip:** When simplifying expressions, always start by reducing the fraction (if possible) before applying any exponents to make calculations easier.

Solve the expressions: (a) (ab / 2a)^3 and (b) (18a^2b^(-2))^2 / 12ab.

Step-by-step solution for simplifying algebraic expressions involving exponents and fractions. Learn how to solve problems like (ab / 2a)^3 and (18a^2b^(-2))^2 / 12ab using exponent rules and fraction simplification.

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