Which of the following is an equivalent expression to 14^-7/9^-13 with only positive exponents generated by applying the Property of Negative Integer Exponents?

Question

1/9^13x14^-7
14^7/9^13
9^13/14^7
14^-7 x 9^13

Answer 1

To simplify the expression \( \frac{14^{-7}}{9^{-13}} \) and express it with only positive exponents, we can use the properties of negative exponents:

1. Recall the property: \( a^{-n} = \frac{1}{a^n} \).
2. Therefore, \( 14^{-7} = \frac{1}{14^7} \) and \( 9^{-13} = \frac{1}{9^{13}} \).
3. Thus, \( \frac{14^{-7}}{9^{-13}} \) can be rewritten as: \[ \frac{\frac{1}{14^7}}{\frac{1}{9^{13}}} \] which simplifies to: \[ \frac{9^{13}}{14^7} \]

Now let's check which option matches this result:

1. \( \frac{1}{9^{13} \cdot 14^{-7}} \) – This does not match.
2. \( \frac{14^{7}}{9^{13}} \) – This does not match (order is reversed).
3. \( \frac{9^{13}}{14^{7}} \) – This matches our result.
4. \( 14^{-7} \cdot 9^{13} \) – This does not match.

Therefore, the equivalent expression is:

\( \frac{9^{13}}{14^{7}} \)