Let's break down the expression in the image:

You are asked to simplify:

$(4x^2)^3 + 7x^6$

Step-by-step:

1. Expand $(4x^2)^3$:

   • Use the power of a power rule: $(a^m)^n = a^{m \times n}$.
   • Here, we have $(4x^2)^3$, which means: $4^3 \times (x^2)^3 = 64x^6$

2. Now, simplify the full expression: $64x^6 + 7x^6$

   • Since both terms have $x^6$, factor out $x^6$: $(64 + 7)x^6 = 71x^6$

So the simplified expression is $71x^6$.

Therefore, the correct answer is: 71x^6.

Would you like further details on any of these steps, or do you have any questions?

Here are 5 follow-up questions that may interest you:

1. How do you apply the power of a product rule in algebra?
2. What is the difference between adding and multiplying exponents?
3. Can you simplify expressions like $(2x^3)^4$?
4. How do you factor out common terms in algebraic expressions?
5. What happens if the powers of the variables are different in an expression like $x^2 + x^3$?

Tip: When simplifying powers, always check whether the terms have the same base and exponent to combine them.