The problem asks which expression is equivalent to $(a^8)^4$.

Solution:

Using the power of a power rule in exponents, which states: $(a^m)^n = a^{m \cdot n}$

Here: $(a^8)^4 = a^{8 \cdot 4} = a^{32}$

Thus, the correct answer is: $\boxed{a^{32}}$

Would you like a more detailed explanation or have any other questions? Here are some related questions to explore:

1. What is the difference between $(a^m)^n$ and $(ab)^n$?
2. How do you simplify expressions with multiple layers of exponents, such as $((a^m)^n)^p$?
3. What happens when the base of the exponent is negative, such as $(-a^m)^n$?
4. How does the rule change if the exponent is a fraction, e.g., $(a^m)^{1/n}$?
5. Can you simplify expressions involving products of powers with the same base, like $a^m \cdot a^n$?

Tip:

Always apply the power of a power rule carefully by multiplying the exponents, not adding them!