To generate equivalent expressions for \( 3^7 \) and \( 3^{-9} \) using the properties of integer exponents, we can rewrite the negative exponent in terms of positive exponents.
First, remember the property of negative exponents: \[ a^{-n} = \frac{1}{a^n} \]
Using this, we can rewrite \( 3^{-9} \): \[ 3^{-9} = \frac{1}{3^9} \]
Now, we can express both numbers with positive exponents. The expression we want to simplify is: \[ \frac{3^7}{3^9} \]
Using the property of exponents which states that when you divide two powers with the same base, you subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n} \]
We apply that here: \[ \frac{3^7}{3^9} = 3^{7-9} = 3^{-2} \]
Now, rewriting \( 3^{-2} \) using positive exponents gives us: \[ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \]
Thus, the simplified fraction with no remaining exponents is: \[ \boxed{\frac{1}{9}} \]