Question

Simplify this expression using the quotient rule of integer exponents:: 12 with an exponent of 53 divided by 7 with an exponent of 53. Answer options: 5 with a exponent of 106, 5 with an exponent of 53, 12 with an exponent of 53 divided by 7 with an eponent of 53, 12 with an exponent of 106.

Answer 1

To simplify the expression \( \frac{12^{53}}{7^{53}} \) using the quotient rule of integer exponents, we can apply the rule which states that \( \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m \).

Applying this rule to our expression, we get:

\[ \frac{12^{53}}{7^{53}} = \left(\frac{12}{7}\right)^{53} \]

This means that the expression simplifies to \( \left(\frac{12}{7}\right)^{53} \).

However, looking at the answer options provided:

1. \( 5^{106} \)
2. \( 5^{53} \)
3. \( \frac{12^{53}}{7^{53}} \)
4. \( 12^{106} \)

The most direct representation of our simplified expression \( \frac{12^{53}}{7^{53}} \) is option 3: \( \frac{12^{53}}{7^{53}} \).

Therefore, the correct answer is \( \frac{12^{53}}{7^{53}} \).