Which property of exponents was used to generate the equivalent expression 3 to the fourteenth power14 from 3 to the fifth power 5/3 to the ninth power−9?(1 point)

Question

Responses

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 Property of Negative Exponents
only the Property of Negative Exponents

only the Power Rule of Exponents

Answer 1

To analyze how the expression \(3^5 \div 3^{-9}\) can be simplified to \(3^{14}\), we can apply the following properties of exponents:

1. The Property of Negative Exponents: This property states that \(a^{-n} = \frac{1}{a^n}\). Therefore, \(3^{-9} = \frac{1}{3^9}\), which can be understood as dividing by that term.

2. The Product Rule of Exponents: This rule states that \(a^m \div a^n = a^{m-n}\). In this case, when simplifying \(3^5 \div 3^{-9}\), you can rewrite it as \(3^{5 - (-9)}\), which simplifies to \(3^{5 + 9} = 3^{14}\).

Given this analysis, the correct response is:

the Property of Negative Exponents and the Product Rule of Exponents