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 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.
1. **Negative Exponent Property:** The expression \( 3^{-9} \) can be rewritten using the property of negative exponents:
\[
3^{-9} = \frac{1}{3^9}
\]
This gives:
\[
\frac{3^5}{3^{-9}} = 3^5 \cdot 3^{9}
\]
2. **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.**
1. **Negative Exponent Property:** The expression \( 3^{-9} \) can be rewritten using the property of negative exponents:
\[
3^{-9} = \frac{1}{3^9}
\]
This gives:
\[
\frac{3^5}{3^{-9}} = 3^5 \cdot 3^{9}
\]
2. **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.**