Which property of exponents was used to generate the equivalent expression 3^14 from 3^5 over 3^−9


options:

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

1 answer

To generate the expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \), we can apply the following properties of exponents:

  1. The Property of Negative Exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). This means that dividing by a negative exponent is equivalent to adding its positive counterpart.
  2. The Product Rule of Exponents is used when multiplying two exponential terms with the same base, which states that \( a^m \cdot a^n = a^{m+n} \). However, this rule isn't directly used in simplifying the expression here.

Starting with \( \frac{3^5}{3^{-9}} \):

  1. Using the Property of Negative Exponents, convert the denominator: \[ \frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \]

Thus, the property used here involves the Property of Negative Exponents primarily.

The correct answer would be: only the Property of Negative Exponents.