Which property of exponents was used to generate the equivalent expression 3 baseline 14 from 3baseline5/3baseline−9?(1 point)

Question

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 Product Rule of Exponents
only the Product Rule of Exponents

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

only the Property of Negative Exponents

Answer 1

To simplify the expression \( \frac{3^{5}}{3^{-9}} \), you can use the Property of Negative Exponents and the Product Rule of Exponents.

Breakdown:

1. Property of Negative Exponents:

   • Recall that \( a^{-b} = \frac{1}{a^b} \). Thus, \( 3^{-9} = \frac{1}{3^9} \).
   • However, you can also rewrite \( \frac{1}{3^{-9}} \) as \( 3^{9} \) which helps in the next step.

2. Product Rule of Exponents:

   • The rule states that \( \frac{a^m}{a^n} = a^{m-n} \).
   • Applying this to our expression \( \frac{3^{5}}{3^{-9}} \): \[ = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14}. \]

Conclusion:

Both the Property of Negative Exponents (which allows the conversion from a negative exponent) and the Product Rule of Exponents (which handles the division of exponentials) were used.

Thus, the correct answer is:

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