To simplify the expression \( \frac{12^{53}}{7^{53}} \) using the Quotient Rule of Integer Exponents, we can apply the rule that states:
\[ \frac{a^m}{b^m} = \left( \frac{a}{b} \right)^m \]
In this case, we have \( a = 12 \), \( b = 7 \), and \( m = 53 \). Therefore:
\[ \frac{12^{53}}{7^{53}} = \left( \frac{12}{7} \right)^{53} \]
Since none of the answer choices provided (A, B, C, or D) involve this simplified expression directly, we can look at the options more closely:
A. \( 5^{106} \)
B. \( 5^{53} \)
C. \( \frac{12^{53}}{7^{53}} \) (which is equal to our simplified form)
D. \( 12^{106} \)
The correct answer that matches our simplification is:
C. \( \frac{12^{53}}{7^{53}} \)
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