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.