Question

Which property of exponents was used to generate the equivalent expression 3 to the fourteenth power from 3 to the fifth power divided by 3 to the negative nineth power?

Answer 1

To determine which property of exponents was used to generate the expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \), we can apply the **quotient of powers property**, which states that:

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

In this case, we can identify \( a = 3 \), \( m = 5 \), and \( n = -9 \). Thus, we can rewrite the expression as follows:

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

So, the property of exponents used in this case is the **quotient of powers property**.