Which of the following is an equivalent expression to the numerator of 14 to the negative seventh power with a denominator of 9 to the negative thirteenth power with only positive exponents, generated by applying the Property of Negative Integer Exponents?

Answer 1

To simplify the expression \( \frac{14^{-7}}{9^{-13}} \) using the Property of Negative Integer Exponents, we can rewrite the negative exponents as follows:

1. Recall that \( a^{-n} = \frac{1}{a^n} \). Using this property:

   • The numerator \( 14^{-7} \) can be rewritten as \( \frac{1}{14^7} \).
   • The denominator \( 9^{-13} \) can be rewritten as \( \frac{1}{9^{13}} \).

2. Therefore, the expression becomes: \[ \frac{14^{-7}}{9^{-13}} = \frac{1}{14^7} \div \frac{1}{9^{13}} = \frac{1}{14^7} \times 9^{13} \]

3. Dividing by a fraction is equivalent to multiplying by its reciprocal, so: \[ \frac{1}{14^7} \times 9^{13} = \frac{9^{13}}{14^7} \]

Thus, we can express \( \frac{14^{-7}}{9^{-13}} \) with only positive exponents as:

\[ \frac{9^{13}}{14^7} \]

This expression is equivalent to the original expression with only positive exponents.