To simplify \( 4^{13} \div 3^{-5} \), you can rewrite the division by a negative exponent as multiplication by its positive counterpart:
\[ 4^{13} \div 3^{-5} = 4^{13} \times 3^{5} \]
Now, we can calculate this expression:
\[ 4^{13} \times 3^{5} \]
Calculating \( 4^{13} \):
\[ 4^{13} = (2^2)^{13} = 2^{26} \]
And calculating \( 3^{5} \):
\[ 3^{5} = 243 \]
Now we can multiply these results together:
\[ 2^{26} \times 243 \]
Calculating \( 2^{26} \):
\[ 2^{26} = 67108864 \]
Now, multiply \( 67108864 \) by \( 243 \):
\[ 67108864 \times 243 = 1628410597910449 \]
Thus, the final result for \( 4^{13} \div 3^{-5} \) is
\[ \boxed{1628410597910449} \]