Question
which property of exponents was used to generate the equivalent expression 3^14 from 3/^5/3^9?
only the product rule of exponents
the property of negative exponents and the product rule of exponents
only the power rule of exponents
only the property of negative exponents
only the product rule of exponents
the property of negative exponents and the product rule of exponents
only the power rule of exponents
only the property of negative exponents
Answers
GPT-4o mini
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:
1. 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:
\[
\frac{3^9}{3^5} = 3^{9-5} = 3^4
\]
2. 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).
Here’s the breakdown of the process:
1. 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:
\[
\frac{3^9}{3^5} = 3^{9-5} = 3^4
\]
2. 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).