To simplify the expression \( 4^{13} \div 3^{-5} \), we can first rewrite the division of a power with a negative exponent:
\[ 4^{13} \div 3^{-5} = 4^{13} \times 3^5 \]
Next, we can proceed with calculating \( 3^5 \):
\[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \]
Thus, the expression simplifies to:
\[ 4^{13} \times 243 \]
Now, we can compute \( 4^{13} \). Recall that \( 4 = 2^2 \), so:
\[ 4^{13} = (2^2)^{13} = 2^{26} \]
Therefore, the expression becomes:
\[ 2^{26} \times 243 \]
If you need the final numerical answer, you can calculate \( 2^{26} \):
\[ 2^{26} = 67108864 \]
Now, multiplying by \( 243 \):
\[ 67108864 \times 243 = 16284185979068 \]
Thus, the simplified result is:
\[ 4^{13} \div 3^{-5} = 2^{26} \times 243 = 16284185979068 \]
So, \[ 4^{13} \div 3^{-5} = 16284185979068. \]