We are given the following algebraic expression to simplify:

$(4a^2)^2 \times (3b^4)^3 \div (2ab^2)^3$

Step-by-Step Simplification:

1. Apply the power rule $(x^m)^n = x^{m \cdot n}$ to each term:

   • $(4a^2)^2 = 4^2 \cdot a^{2 \cdot 2} = 16a^4$
   • $(3b^4)^3 = 3^3 \cdot b^{4 \cdot 3} = 27b^{12}$
   • $(2ab^2)^3 = 2^3 \cdot a^3 \cdot b^{2 \cdot 3} = 8a^3b^6$

2. Rewrite the expression with the simplified terms: $\frac{(16a^4 \times 27b^{12})}{8a^3b^6}$

3. Simplify the numerical coefficients: $\frac{16 \times 27}{8} = \frac{432}{8} = 54$

4. Simplify the powers of $a$: $\frac{a^4}{a^3} = a^{4-3} = a$

5. Simplify the powers of $b$: $\frac{b^{12}}{b^6} = b^{12-6} = b^6$

6. Final simplified expression: $54ab^6$

Final Answer:

$54ab^6$

Would you like a deeper explanation on any step?

Here are 5 related questions:

1. How does the power rule work with variables?
2. Can you explain simplifying fractional powers more deeply?
3. What happens when negative exponents appear?
4. How do you handle products of powers with different bases?
5. How do you simplify expressions with roots?

Tip: Always simplify the coefficients and variables separately when dealing with expressions involving multiplication and division of powers.