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