Question

which property of exponents was used to generate the equivalent expression 3 to the power of 14 from 3 to the power of 5 over 3 to the power of negative 9

Answer 1

To generate the equivalent expression \(3^{14}\) from \(\frac{3^{5}}{3^{-9}}\), the property of exponents used is the **quotient of powers** property.

This property states that when you divide two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. Mathematically, it can be expressed as:

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

In your case:

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

So, the property applied here is:

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