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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!

Which expression shows how to write (3^4)^2 as a product?

Understand how to simplify (3^4)^2 using exponent rules. This problem focuses on applying the power of a power rule in exponents, resulting in the expression 3^4 * 3^4, suitable for students in grades 8-9.

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