To simplify the expression \( \frac{3^5}{3^{-9}} \) and generate the equivalent expression \( 3^{14} \), the following properties of exponents are used:
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Property of Negative Exponents: This property states that \( a^{-n} = \frac{1}{a^n} \). Therefore, \( 3^{-9} \) can be rewritten as \( \frac{1}{3^9} \), which means that dividing by \( 3^{-9} \) is equivalent to multiplying by \( 3^9 \).
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Product Rule of Exponents: This rule states that when you multiply two powers with the same base, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
Combining these, the calculation would look like this:
\[ \frac{3^5}{3^{-9}} = 3^5 \cdot 3^9 = 3^{5+9} = 3^{14} \]
Therefore, the correct response is:
the Property of Negative Exponents and the Product Rule of Exponents.
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