Which property of exponents was used to generate the equivalent expression 314

Question

from 353−9
?(1 point)
Responses

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

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

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

only the Property of Negative Exponents

Answer 1

To simplify the expression \( 35^{3-9} \) to \( 3^{14} \), we can break down the steps as follows:

1. Apply the Power Rule of Exponents: The Power Rule states that \( a^{m-n} = \frac{a^m}{a^n} \). Here, we can interpret \( 35^{3-9} \) as \( 35^3 \div 35^9 \).

2. Apply the Property of Negative Exponents: The expression \( 35^{3-9} \) simplifies to \( \frac{1}{35^{9-3}} = 35^{-6} \). Negative exponents indicate reciprocal.

3. Convert to Prime Base: Since \( 35 = 5 \times 7 \), we can express \( 35^{-6} \) in terms of primes and then manipulate it.

Ultimately, we derive a relationship to the factor \( 3^{14} \) by identifying how \( 35^{3-9} \) can be expressed in terms of its bases after applying the aforementioned properties. However, the exact transition to \( 3^{14} \) from \( 35^{-6} \) involves multiple steps, including prime factorization.

Given the question's focus on what property was specifically used to derive \( 314 \) from \( 353-9 \): The most pertinent answer would be the Property of Negative Exponents and the Product Rule of Exponents.

So the correct response is: the Property of Negative Exponents and the Product Rule of Exponents.