Which property of exponents was used to generate the equivalent expression 3 to the power of 14

Question

from 3 to the power of 5/3 to the power of −9
?(1 point)
Responses

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

only the Power Rule of Exponents
only the Power 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

Answer 1

To generate the expression \( 3^{14} \) from \( 3^5 / 3^{-9} \), you would use the following rules of exponents:

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

2. Product Rule of Exponents: This states that \( \frac{a^m}{a^n} = a^{m-n} \). Applying this to \( 3^5 / 3^{-9} \) allows you to combine the exponents:

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

Therefore, the correct response is:

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