Question

Which property of exponents was used to generate the equivalent expression 314 from 353−9 ?(1 point)
Responses

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

only the Power Rule of Exponents
only the Power Rule of Exponents

only the Product Rule of Exponents
only the Product Rule of Exponents

only the Property of Negative Exponents
only the Property of Negative Exponents

Answer 1

To generate the expression \( 3^{14} \) from \( 3^{53 - 9} \), we can simplify \( 53 - 9 \) as follows:

\[
53 - 9 = 44
\]

Thus,

\[
3^{53 - 9} = 3^{44}
\]

However, it seems that there's confusion here. If you meant \( 3^{53} \cdot 3^{-9} \) to reach \( 3^{14} \), then we would use:

1. The **Product Rule of Exponents**, which states that \( a^m \cdot a^n = a^{m+n} \).
2. The **Property of Negative Exponents**, which states that \( a^{-n} = \frac{1}{a^n} \).

Using these rules, we would have:

\[
3^{53} \cdot 3^{-9} = 3^{53 + (-9)} = 3^{53 - 9} = 3^{44}
\]

So, if our goal was to reach \( 3^{14} \), we could think of it in the same way where \( 3^{53} \cdot 3^{-39} = 3^{53 + (-39)} = 3^{14} \).

Given the responses you provided, the best possible choice would be:

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

Thus, this would be the most appropriate selection from your options!