Evaluate the Complex Expression: 3⁹×12⁴×6⁹×4⁹×4⁴×7⁹

$3^9\times12^4\times6^9\times4^9\times4^4\times7^9=\text{ ?}$

To solve this problem, we'll follow these steps:

• Step 1: Group terms with the same exponent to utilize the power of a product rule.
• Step 2: Combine terms under each group using the power of a product property.
• Step 3: Match the result to one of the multiple-choice options provided.

Now, let's work through each step:
Step 1: Observe the expression $3^9 \times 12^4 \times 6^9 \times 4^9 \times 4^4 \times 7^9$. Group the terms with exponents of 9: $3^9, 6^9, 4^9,$ and $7^9$. Group those with exponents of 4: $12^4$ and $4^4$.

Step 2: For the terms with exponent 9, apply the power of a product rule: $(3 \times 6 \times 4 \times 7)^9$

For the terms with exponent 4, apply the power of a product rule: $(12 \times 4)^4$

Step 3: Combine these to form the expression: $(3 \times 6 \times 4 \times 7)^9 \times (12 \times 4)^4$

Therefore, the solution to the problem is $\left(3 \times 6 \times 4 \times 7\right)^9 \times \left(12 \times 4\right)^4$. This corresponds to choice 3.