To simplify the expression \( 13^{-9} \cdot 13 \cdot 13^{-5} \), we can use the properties of exponents. Specifically, we can use the rule that states \( a^m \cdot a^n = a^{m+n} \).
First, we rewrite \( 13 \) as \( 13^1 \):
\[ 13^{-9} \cdot 13^1 \cdot 13^{-5} \]
Now we can add the exponents together:
\[ -9 + 1 - 5 \]
Calculating this:
\[ -9 + 1 = -8 \] \[ -8 - 5 = -13 \]
This gives us:
\[ 13^{-13} \]
To write this in exponential form with a positive exponent, we can use the property \( a^{-m} = \frac{1}{a^m} \):
\[ 13^{-13} = \frac{1}{13^{13}} \]
Thus, the fully simplified expression in exponential form with positive exponents is:
\[ \frac{1}{13^{13}} \]
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