To generate equivalent expressions using the properties of integer exponents, we start with the expression \( 3^7 \cdot 3^{-9} \).
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Apply the Property of Exponents: According to the property \( a^m \cdot a^n = a^{m+n} \), we can combine the exponents:
\[ 3^7 \cdot 3^{-9} = 3^{7 + (-9)} = 3^{7 - 9} = 3^{-2} \]
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Convert to Positive Exponent: To express \( 3^{-2} \) with a positive exponent, we use the property \( a^{-m} = \frac{1}{a^m} \):
\[ 3^{-2} = \frac{1}{3^2} \]
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Calculate \( 3^2 \):
\[ 3^2 = 9 \]
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Resulting Expression:
\[ 3^{-2} = \frac{1}{9} \]
The simplified fraction with no remaining exponents is:
\[ \frac{1}{9} \]