To understand how the expression \( \frac{3^5}{3^{-9}} \) can be simplified to \( 3^{14} \), we need to apply some rules of exponents.
We start with:
\[ \frac{3^5}{3^{-9}} \]
Using the Property of Negative Exponents, we can rewrite \( 3^{-9} \) as \( \frac{1}{3^9} \). However, instead of rewriting it just yet, we can apply the Quotient Rule of Exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \).
Applying the quotient rule here:
\[ \frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \]
Thus, we used the Property of Negative Exponents to understand what \( 3^{-9} \) means, and the Quotient Rule of Exponents to simplify the expression.
The correct response is:
the Property of Negative Exponents and the Product Rule of Exponents (Note: it seems there was an error in listing the rules; it should refer to the Quotient Rule, not Product Rule).
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