To find the correct expression that uses the Product Rule of Exponents to simplify to 2412, we need to look at each option and see if it can be simplified using the rule \( a^m \cdot a^n = a^{m+n} \) to reach 2412.
Let's simplify each expression:
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\( 6^4 \cdot 4^3 \): This does not have the same base, so we cannot directly use the Product Rule of Exponents here. It will remain as \( 6^4 \cdot 4^3 \) and does not equal 2412.
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\( 24^3 \cdot 24^4 \): Here, since the bases are the same, we can apply the Product Rule: \[ 24^3 \cdot 24^4 = 24^{3+4} = 24^7 \] \( 24^7 \) is a large number, much greater than 2412.
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\( 12^4 \cdot 12^8 \): The bases are the same here as well: \[ 12^4 \cdot 12^8 = 12^{4+8} = 12^{12} \] \( 12^{12} \) is also very large, exceeding 2412.
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\( 24^4 \cdot 24^8 \): Again, the bases match: \[ 24^4 \cdot 24^8 = 24^{4+8} = 24^{12} \] Similar to the previous two options, \( 24^{12} \) is much larger than 2412.
After evaluating all the options, none of them directly simplify to 2412 through the Product Rule because the Product Rule of Exponents itself doesn't lead to a simple numerical value. However, since you asked for an expression that is simplified correctly using the Product Rule of Exponents, the expression \( 24^3 \cdot 24^4 \) results in \( 24^7 \), and is answerable in terms of product rules even though the resulting number isn't 2412.
So, the most accurate response based on your question regarding using the Product Rule correctly is:
\( 24^3 \cdot 24^4 \).
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