Question

Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^-9
A. only the Property of Negative Exponents
B. only the Power Rule of Exponents
C. only the Product Rule of Exponents
D. the Property of Negative Exponents and the Product Rule of Exponents

Answer 1

To generate the equivalent expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \), we can use the following properties of exponents:

1. **Property of Negative Exponents**: This states that \( a^{-n} = \frac{1}{a^n} \). Therefore, \( 3^{-9} \) can be rewritten as \( \frac{1}{3^9} \).

2. **Quotient Rule of Exponents**: This states that \( \frac{a^m}{a^n} = a^{m-n} \).

Applying these properties:

\[
\frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14}
\]

From this, we can observe that we used both the Property of Negative Exponents and the Quotient Rule to arrive at the final expression.

Thus, the correct answer is:

**D. the Property of Negative Exponents and the Product Rule of Exponents**