wich property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^-9?

Question

only the product rule of exponents
only the power rule of exponents
the property of negative exponents and the the power rule of exponents
only the property of negative exponents

Answer 1

To simplify \( \frac{3^5}{3^{-9}} \) and show that it can be expressed as \( 3^{14} \), we can use the property of exponents that states:

\[ \frac{a^m}{a^n} = a^{m-n} \]

Applying this rule:

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

In this case, we utilized:

1. The property of negative exponents (to understand that \( 3^{-9} \) is the same as \( \frac{1}{3^9} \)).
2. The product rule of exponents in the context of subtraction (when combining exponents).

Thus, the correct answer is:

the property of negative exponents and the power rule of exponents.