Let's break down the problem. You are asked to simplify the expression $(4a^2b)^3 (ab^{1/2})^2$.

Step 1: Simplify each part of the expression

• $(4a^2b)^3$: Use the power of a product rule $(xy)^n = x^n y^n$, and distribute the exponent: $(4a^2b)^3 = 4^3 \cdot (a^2)^3 \cdot b^3 = 64a^6b^3$

• $(ab^{1/2})^2$: Again, apply the power of a product rule: $(ab^{1/2})^2 = a^2 \cdot (b^{1/2})^2 = a^2b$

Step 2: Combine both simplified terms

Now, multiply the two simplified expressions: $64a^6b^3 \cdot a^2b = 64a^{6+2}b^{3+1} = 64a^8b^4$

Step 3: Compare with the options

We now have the simplified expression $64a^8b^4$, which matches option B.

Final Answer:

The correct expression is B. 64a^8b^4.

Would you like further clarification on any step, or have any other questions?

Here are 5 related questions to explore:

1. How do the properties of exponents help simplify expressions like this one?
2. What is the distributive property of exponents, and how does it apply to multiplication?
3. Can we apply the same method to expressions with negative exponents?
4. How would the solution change if the exponents were fractions or negative?
5. What are other examples of simplifying expressions with powers?

Tip: Always start by simplifying each part of the expression separately, then combine the results. This approach reduces the complexity of the problem.