Question

Which property of exponents was used to generate the equivalent expression 3 to the fourteenth power from 3 to the fifth power divided by 3 to the negative nineth power?

the Property of Negative Exponents and the Product Rule of Exponents
only the Property of Negative Exponents
only the Product Rule of Exponents
only the Power Rule of Exponents

Answer 1

To generate the expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \), we can use the **Quotient Rule of Exponents**, which states that \( \frac{a^m}{a^n} = a^{m-n} \).

When applying the Quotient Rule to your expression:

\[
\frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14}
\]

In this process, we have utilized the concept of negative exponents as well (since \( 3^{-9} \) can be interpreted as \( \frac{1}{3^9} \)), but the key operation was the application of the Quotient Rule which involves subtracting the exponents.

Thus, the correct answer is:

**The Property of Negative Exponents and the Product Rule of Exponents** (given that both were necessary to understand the context, but primarily the Quotient Rule was applied).

If you're asked specifically about the rule used, focusing on the main operation, it would be the **Quotient Rule of Exponents**. However, since that option is not listed and considering your choices, the closest answer that includes the relavant concepts would be:

**The Property of Negative Exponents and the Product Rule of Exponents.**