Apply All Exponent Rules Quick Check

Question

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

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
only the Property of Negative Exponents

only the Product Rule of Exponents

Answer 1

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

1. The Property of Negative Exponents, which states that \( a^{-n} = \frac{1}{a^n} \) or, rephrased in this context, \( \frac{1}{3^{-9}} = 3^{9} \).
2. The Product Rule of Exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \).

Putting this together, we can rewrite the expression: \[ \frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \]

Based on this analysis, the correct response is: the Property of Negative Exponents and the Product Rule of Exponents.