To simplify the expression \( 13^{-5} \cdot 13^{-11} \) using the properties of exponents, we can use the property that states \( a^m \cdot a^n = a^{m+n} \). Here \( a = 13 \), \( m = -5 \), and \( n = -11 \).
Calculating the exponents:
\[ m + n = -5 + (-11) = -16 \]
So, we have:
\[ 13^{-5} \cdot 13^{-11} = 13^{-16} \]
Now, to express this with positive exponents, we can apply the property \( a^{-n} = \frac{1}{a^n} \). Therefore:
\[ 13^{-16} = \frac{1}{13^{16}} \]
Among the options provided, the equivalent expression with only positive exponents is:
\(\frac{1}{13^{16}}\)
None of the options directly state \(\frac{1}{13^{16}}\), but if we were to assume the options are not listed correctly or if we need to find a valid option, we can check the options provided:
- \(\frac{1}{13^6}\)
- \(\frac{1}{26^6}\)
- \(\frac{1}{13^{16}}\) (not listed)
- \(\frac{1}{26^{16}}\)
The answer is \(\frac{1}{13^{16}}\), which corresponds to other calculations as shown, but if strictly based on the options you mentioned, none match. The option to choose theoretically would be:
None of the options listed correctly reflect the simplification of \( 13^{-5} \cdot 13^{-11} \), which should be \(\frac{1}{13^{16}}\). If there was a typo or missing option for \(\frac{1}{13^{16}}\), that would be the correct equivalent expression with only positive exponents.