To simplify the expression \( 13^{-5} \cdot 13^{-11} \) using the properties of exponents, we can apply the property that states \( a^m \cdot a^n = a^{m+n} \).
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First, add the exponents: \[ -5 + (-11) = -16. \] So, we have: \[ 13^{-5} \cdot 13^{-11} = 13^{-16}. \]
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Next, to express this with a positive exponent, we can use the property \( a^{-n} = \frac{1}{a^n} \): \[ 13^{-16} = \frac{1}{13^{16}}. \]
Thus, the equivalent expression to \( 13^{-5} \cdot 13^{-11} \) with only positive exponents is: \[ \frac{1}{13^{16}}. \]
From the options provided, the correct choice is: \[ \frac{1}{13^{16}}. \]
There may be an error in your response options since \( \frac{1}{13^{16}} \) is what we found, and it resembles \( 11316 \) assuming it means \( \frac{1}{13^{16}} \). Please double-check the format. If those are codes and represent different numerical formats, you would need to clarify.