The given expression is $(4ab^2)^3$. To simplify this expression, apply the power of a product rule, which states that:

$(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n$

Steps:

1. First, distribute the exponent to each term inside the parentheses: $(4ab^2)^3 = 4^3 \cdot a^3 \cdot (b^2)^3$
2. Now simplify each part: $4^3 = 64$ $a^3 = a^3$ $(b^2)^3 = b^{6} \quad \text{(by multiplying the exponents)}$

Thus, the simplified expression is: $64a^3b^6$

Would you like more details or have any questions?

Here are five related questions you might find helpful:

1. How do you simplify expressions with more complex powers?
2. What happens if the base includes negative terms?
3. How do you handle fractional exponents when simplifying?
4. What are the key differences between powers of sums and powers of products?
5. How do you simplify expressions with variables raised to negative exponents?

Tip: When distributing exponents over variables, remember to apply the exponent to each factor inside the parentheses.