To simplify the expression $(4x^2)^3$, follow these steps:

1. Apply the power of a product rule, which states $(ab)^n = a^n \cdot b^n$.

   • Here, $a = 4$ and $b = x^2$, and $n = 3$.

2. Apply the exponent to each part: $(4x^2)^3 = 4^3 \cdot (x^2)^3$

3. Simplify each term: $4^3 = 64 \quad \text{and} \quad (x^2)^3 = x^{2 \cdot 3} = x^6$

4. Combine the results: $(4x^2)^3 = 64x^6$

Thus, the simplified expression is:

$64x^6$

Would you like more details or explanations?

Here are some related questions:

1. How would you simplify $(3x^3)^2$?
2. What is the general rule for simplifying $(a^m)^n$?
3. Can you simplify $(5xy)^4$?
4. How do you simplify expressions with multiple variables, like $(2x^3y)^2$?
5. What is the next step if you had a coefficient in front of $(4x^2)^3$, like $2(4x^2)^3$?

Tip: When simplifying expressions with exponents, always apply the exponent to both the coefficient and the variable separately!