To identify which expression correctly uses the Product Rule of Exponents to simplify to 2412, we should recall the Product Rule: when multiplying two expressions with the same base, you add their exponents.
The Product Rule states:
\[ a^m \cdot a^n = a^{m+n} \]
Now, let's evaluate each of the provided expressions:
-
\( 24^3 \cdot 24^4 \):
- Applying the Product Rule: \( 24^{3+4} = 24^7 \)
-
\( 12^4 \cdot 12^8 \):
- Applying the Product Rule: \( 12^{4+8} = 12^{12} \)
-
\( 24^4 \cdot 24^8 \):
- Applying the Product Rule: \( 24^{4+8} = 24^{12} \)
-
\( 64 \cdot 4^3 \):
- This one doesn't apply the exponent rule as it is a multiplication of two different bases, 64 and \(4^3\).
From the evaluation above, none of the expressions simplify to \( 24^{12} \), \( 24^7 \), or \( 12^{12} \).
Now, let’s check if \( 2412 \) could match with any of the bases:
Looking at the bases and considering:
- The only base that could potentially yield a result near \( 2412 \) might be with its components – let’s focus on \( 24^7 \) more closely.
But if we are directly looking to express \( 2412 \), we know we’d have to clarify what \( 24^{something} = 2412 \) stands for as it might not match the given exponents exactly.
After thorough examination, it seems I have made an error in clarity. The adjusted conclusion is:
None of the expressions provided directly simplify to yield \( 2412 \) when employing the Product Rule, nor do they seem to derive directly to \( 12^{something}\) or \( 24^{something}\).
In general logic:
You might want to check for hints or statement back regarding \( 24^2 \) for decimals or square roots thereof if none match as \( 24^{something yet remains complex towards base calcs directly conversing with prime factors from readability here for ( 2412 \).
So, could a simpler version or expression of \(2412\) be clearer or laid with dependence on \(12\) in fractions?
Apologies for the confusion; please verify the options leading to solid \(2412\) through algebraic or direct examination related points moving back toward square rooting or simple base cases that shore standard against all \(12, 24\) or even base \(4\) while figuring \(2412\) reinforcing the Product Rule, for easier back-tracking from finish to major going step-wise.