The expression in the problem is $(3^4)^2$, and you are asked to express it as a product.

By applying the exponent rule $(a^m)^n = a^{m \cdot n}$, we can simplify $(3^4)^2$ as:

$(3^4)^2 = 3^{4 \cdot 2} = 3^8$

Now, the task is to express $3^8$ as a product of $3^4$'s or other smaller powers of 3. Looking at the options provided:

1. $3^4 \cdot 3^2$
2. $3^4 \cdot 3^4$
3. $3^4 \cdot 3^4 \cdot 3^2 \cdot 3^2$
4. $3^4 \cdot 3^4 \cdot 3^4 \cdot 3^4$

Among these, the correct expression for $3^8$ would be option 2: $3^4 \cdot 3^4$. This is because multiplying $3^4 \cdot 3^4$ gives $3^{4+4} = 3^8$, which matches the expression $(3^4)^2$.

Would you like more details or have any questions? Here are some related questions to explore further:

1. What are other exponent rules besides $(a^m)^n = a^{m \cdot n}$?
2. How would you simplify $(2^3)^4$ using exponent rules?
3. Can you explain the difference between multiplying powers with the same base and raising a power to another power?
4. How would the expression change if it was $(3^5)^2$?
5. Can you express $3^8$ as a product of smaller powers of 3, such as $3^2$?

Tip: When multiplying powers with the same base, simply add the exponents!