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Let's solve the given problem by simplifying the given expression.

The expression to simplify is:

\[
\left(2mn^8\right)^2
\]

### Step-by-Step Simplification:

1. **Apply the Power to Each Term:**
   The expression inside the parentheses is \((2mn^8)\), and it is raised to the power of 2.

   We use the rule that \((abc)^n = a^n b^n c^n\), which means we distribute the exponent to each factor inside:

   \[
   \left(2mn^8\right)^2 = 2^2 \cdot m^2 \cdot (n^8)^2
   \]

2. **Calculate Each Power:**
   - \(2^2 = 4\)
   - \(m^2\) remains as \(m^2\)
   - \((n^8)^2 = n^{8 \times 2} = n^{16}\)

3. **Combine the Results:**

   \[
   4m^2n^{16}
   \]

### Final Answer:
The correct answer is:

**A. \(4m^2n^{16}\)**

Would you like me to explain any part of this process in more detail or answer any questions?

Here are some related questions that might help expand your understanding:

1. How do you apply the power rule to products inside parentheses?
2. What is the general rule for multiplying exponents with the same base?
3. Why does \( (n^8)^2 \) result in \( n^{16} \)?
4. How do you handle coefficients when raising a product to a power?
5. What would the result be if the original expression had a negative power?

**Tip:** When dealing with an expression that has multiple terms inside parentheses raised to a power, remember to distribute the power to each term separately.

This problem involves simplifying an algebraic expression with exponents. Learn how to apply the power of a product rule and the exponent multiplication rule to get the solution 4m^2n^16. Ideal for students in Grades 8-10.

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