We need to simplify the expression $(4x^2)^3$.

Step-by-step simplification:

1. Apply the exponent rule: $(a \cdot b)^n = a^n \cdot b^n$, which means the exponent applies to both 4 and $x^2$.

   $(4x^2)^3 = 4^3 \cdot (x^2)^3$

2. Simplify each part:

   • $4^3 = 64$
   • $(x^2)^3 = x^{2 \cdot 3} = x^6$

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

Now, compare this with the answer choices:

• 12x⁵: Not equivalent.
• 64x⁶: Equivalent.
• 82 • x⁶: Not equivalent (the coefficient is incorrect).
• 64x⁵: Not equivalent (the power of $x$ is incorrect).
• 16x⁶: Not equivalent (the coefficient is incorrect).

Correct answer:

• 64x⁶

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you simplify expressions with multiple variables raised to powers?
2. What are the basic exponent rules for products and powers?
3. How do you apply the distributive property to expressions with exponents?
4. Can you explain how to multiply two terms with exponents?
5. How does the power of a product differ from the power of a sum?

Tip: Always distribute the exponent to both the coefficient and the variable when simplifying expressions of the form $(ax^n)^m$.