Question

Which property of exponents was used to generate the equivalent expression 3to the power of14
from 3to the power of5/3to the power of−9
?(1 point)
Responses

only the Property of Negative Exponents
only the Property of Negative Exponents

only the Product Rule of Exponents
only the Product Rule of Exponents

the Property of Negative Exponents and the Product Rule of Exponents
the Property of Negative Exponents and the Product Rule of Exponents

only the Power Rule of Exponents
only the Power Rule of Exponents

Answer 1

To generate the expression \(3^{14}\) from the expression \(\frac{3^5}{3^{-9}}\), we can use the Product Rule of Exponents along with the Property of Negative Exponents.

1. **Property of Negative Exponents**: It states that \(a^{-n} = \frac{1}{a^n}\). In this case, \(3^{-9}\) can be rewritten as \(\frac{1}{3^9}\), or equivalently, when dividing \(3^5\) by \(3^{-9}\), it becomes \(3^5 \cdot 3^9\).

2. **Product Rule of Exponents**: The Product Rule states that \(a^m \cdot a^n = a^{m+n}\). Thus, we combine \(3^5\) and \(3^9\) to get \(3^{5+9} = 3^{14}\).

Since both of these properties were utilized, the correct response is:

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