To generate the equivalent expression \( 3^{14} \) from \( \frac{3^9}{3^5} \), we can apply the property of negative exponents and the product rule of exponents.
Here’s the breakdown of the process:
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The expression \(\frac{3^9}{3^5}\) can be simplified using the quotient rule of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\). Applying this gives us:
\[ \frac{3^9}{3^5} = 3^{9-5} = 3^4 \]
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Since you want the expression to reach \(3^{14}\), it looks like there's a misunderstanding. In fact, if you have \(3^9 \cdot 3^5\) instead, using the product rule of exponents \(a^m \cdot a^n = a^{m+n}\) would apply, leading to:
\[ 3^9 \cdot 3^5 = 3^{9+5} = 3^{14} \]
So based on your options, if we assume you meant \(3^9 \cdot 3^5\) instead of \(\frac{3^9}{3^5}\), the correct statement would be that we used the product rule of exponents.
If we strictly consider \( \frac{3^9}{3^5} \) leading to \(3^4\), then the relevant rule was only the quotient rule, but that does not lead to \(3^{14}\). Thus, if we refer back to your question, it looks inconsistent.
Therefore, none of the provided answers directly apply to \(3^4\), and the expression \(3^{14}\) cannot be derived from \( \frac{3^9}{3^5} \) without changing what was initially given.
Please verify the expressions you want to analyze. If the original expression you meant was \(3^9 \cdot 3^5\), then the correct answer would be: only the product rule of exponents. If it was \( \frac{3^9}{3^5} \), the answer should be between only the quotient rule of exponents (not provided in options).