To solve \( 4^{13} \div 3^{-5} \), we can rewrite the division by a negative exponent as multiplication by the positive exponent:
\[ 4^{13} \div 3^{-5} = 4^{13} \times 3^{5} \]
Now we can calculate separately:
- \( 4^{13} = (2^2)^{13} = 2^{26} \)
- \( 3^5 = 243 \)
So, combining these:
\[ 4^{13} \times 3^5 = 2^{26} \times 243 \]
Now, we can see that \( 243 = 3^5 \) is maintained in the expression. Thus the solution is:
\[ 4^{13} \times 3^{5} = 2^{26} \times 3^5 \]
This is the final expression in terms of powers. If you want a numerical approximation, you can calculate \( 4^{13} \) and \( 3^5 \) separately:
- \( 4^{13} = 67108864 \)
- \( 3^5 = 243 \)
Now, multiplying those values gives:
\[ 4^{13} \times 3^{5} = 67108864 \times 243 = 1628415792 \]
So, the final answer is:
\[ 1628415792 \]