Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3−^9 ?(1 point)

Question

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 simplify the expression \( \frac{3^5}{3^{-9}} \) to obtain \( 3^{14} \), we use two properties of exponents:

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

2. The Quotient Rule of Exponents: This states that \( \frac{a^m}{a^n} = a^{m-n} \). So, when we have \( \frac{3^5}{3^{-9}} \), it can be simplified as \( 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \).

Thus, the correct response would be:

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

However, note that technically we used the Quotient Rule, not the Product Rule, for this operation. If "Product Rule" and "Quotient Rule" are distinct in your context, then course context may affect how one should respond. Based on the canonical names for exponent properties, I would recommend checking the definitions provided in your course materials.