To find the expression that correctly used the Product Rule of Exponents to simplify to \( 24^{12} \), we should apply the Product Rule, which states that when multiplying two powers with the same base, you add their exponents:
\[ a^m \cdot a^n = a^{m+n} \]
Now, let's analyze the provided options:
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\( 24^{3} \cdot 24^{4} \):
- According to the Product Rule: \[ 24^{3} \cdot 24^{4} = 24^{3+4} = 24^{7} \] This does not simplify to \( 24^{12} \).
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\( 24^{4} - 24^{4} \):
- This simplifies to zero: \[ 24^{4} - 24^{4} = 0 \] This is clearly not \( 24^{12} \).
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\( 6^{4} \cdot 4^{3} \):
- To see if this could be equivalent to \( 24^{12} \), we can rewrite \( 24 \): \[ 24 = 6 \cdot 4 \] But multiplying these does not directly apply to the powers correctly to yield \( 24^{12} \).
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\( 12^{4} \cdot 12^{4} \):
- This simplifies as follows using the Product Rule: \[ 12^{4} \cdot 12^{4} = 12^{4+4} = 12^{8} \] This does not yield \( 24^{12} \).
None of the options provided correctly simplifies to \( 24^{12} \) using the Product Rule. However, just for clarity and correction, if you had instead \( 12^{6} \cdot 12^{6} \): \[ 12^{6} \cdot 12^{6} = 12^{6+6} = 12^{12} \] And if \( 12 \) could be expressed in terms of \( 24 \) equivalently, then that would need consideration.
Thus, none of the expressions listed yield \( 24^{12} \) using the Product Rule directly. If you have any additional options or follow-up questions, please provide them!