To simplify the expression \((13^{-7})^0 \cdot \frac{7^{-11}}{7^{-9}}\), we can apply the properties of exponents step by step.
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Applying the Zero Exponent Rule: Any non-zero number raised to the power of 0 is equal to 1. Therefore, we simplify \((13^{-7})^0\) to: \[ (13^{-7})^0 = 1 \]
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Simplifying the Fraction: Next, we'll simplify the fraction \(\frac{7^{-11}}{7^{-9}}\) using the quotient rule of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\): \[ \frac{7^{-11}}{7^{-9}} = 7^{-11 - (-9)} = 7^{-11 + 9} = 7^{-2} \]
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Combining the Results: Now, substitute back into the expression: \[ 1 \cdot 7^{-2} = 7^{-2} \]
Thus, the equivalent expression is: \[ \boxed{7^{-2}} \]