To determine which property of exponents was used to generate the expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \), let's analyze the transformation step by step.
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Negative Exponent Property: The expression \( 3^{-9} \) can be rewritten using the property of negative exponents: \[ 3^{-9} = \frac{1}{3^9} \] This gives: \[ \frac{3^5}{3^{-9}} = 3^5 \cdot 3^{9} \]
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Product Rule of Exponents: Now, applying the product rule of exponents, which states that \( a^m \cdot a^n = a^{m+n} \), we combine the exponents: \[ 3^5 \cdot 3^9 = 3^{5 + 9} = 3^{14} \]
Thus, we used both the property of negative exponents (to rewrite \( 3^{-9} \)) and the product rule of exponents (to combine the powers of \( 3 \)). Therefore, the correct answer is:
the property of negative exponents and the product rule of exponents.