To simplify the expression \( 13^{12} \div 13^{-10} \), we can use the properties of exponents. Specifically, when dividing two powers with the same base, we subtract the exponents:
\[ a^m \div a^n = a^{m - n} \]
In this case, we have:
\[ 13^{12} \div 13^{-10} = 13^{12 - (-10)} = 13^{12 + 10} = 13^{22} \]
Therefore, the simplified expression is:
\[ \boxed{13^{22}} \]