To simplify the expression \( \frac{12^{53}}{7^{53}} \) using the Quotient Rule of Integer Exponents, we apply the rule which states that when you divide two powers with the same exponent, you can divide the bases and keep the exponent:
\[ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m \]
Using this rule, we can rewrite the expression:
\[ \frac{12^{53}}{7^{53}} = \left(\frac{12}{7}\right)^{53} \]
Now looking at the provided options:
- \( 1253 \div 753 \)
- \( 12^{53} \div 7^{53} \)
- \( 553 \)
- \( 12^{106} \)
- \( 5106 \)
None of those options reflect the correct simplification of \( \left(\frac{12}{7}\right)^{53} \). However, option 2 is essentially representing the original expression \( \frac{12^{53}}{7^{53}} \), which, while not fully simplified, is indeed a valid representation of the original expression.
Therefore, the best response based on the given choices is:
12^53 ÷ 7^53