To simplify the expression \( 11^{-13} \cdot 11^{-7} \) using the properties of exponents, we can use the rule that states \( a^m \cdot a^n = a^{m+n} \).
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Add the exponents: \[ -13 + (-7) = -20 \] So, we can rewrite the expression as: \[ 11^{-20} \]
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To express this with only positive exponents, we use the property that \( a^{-m} = \frac{1}{a^m} \): \[ 11^{-20} = \frac{1}{11^{20}} \]
Now, let's analyze the options:
- \( 12220 \) – This does not correspond to any meaningful expression.
- \( \frac{1}{11^6} \) – This is not equivalent to \( \frac{1}{11^{20}} \).
- \( 1226 \) – This does not correspond to any meaningful expression.
- \( 11120 \) – This seems to be formatted incorrectly but appears to represent \( \frac{1}{11^{20}} \).
Given the analysis, the equivalent expression to \( 11^{-13} \cdot 11^{-7} \) with only positive exponents is:
\[ \frac{1}{11^{20}} \]
Therefore, the correct response is:
11120.