To simplify the expression \( 4^3 \cdot 7^{-2} \cdot 7^{9} \cdot 4^{-13} \), we will first use the properties of integer exponents.
1. **Combine the bases with the same exponents**:
- For the base \( 4 \):
\[
4^3 \cdot 4^{-13} = 4^{3 + (-13)} = 4^{-10}
\]
- For the base \( 7 \):
\[
7^{-2} \cdot 7^{9} = 7^{-2 + 9} = 7^{7}
\]
2. **Rewrite the entire expression**:
Now we have:
\[
4^{-10} \cdot 7^{7}
\]
3. **Convert negative exponents to positive**:
To express \( 4^{-10} \) with a positive exponent, we use the property \( a^{-n} = \frac{1}{a^{n}} \):
\[
4^{-10} = \frac{1}{4^{10}}
\]
Putting this back together, we have:
\[
\frac{7^{7}}{4^{10}}
\]
4. **Final expression**:
The simplified expression is:
\[
\frac{7^{7}}{4^{10}}
\]
5. **Calculating values**:
Now we should compute \( 7^7 \) and \( 4^{10} \):
- \( 7^7 = 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 \cdot 7 = 823543 \)
- \( 4^{10} = (2^2)^{10} = 2^{20} = 1048576 \) (since \( 4^{10} = 4^5 \cdot 4^5 = 1024 \cdot 1024 \))
Hence, the expression becomes:
\[
\frac{823543}{1048576}
\]
This fraction is already in simplest form, as both \( 823543 \) and \( 1048576 \) have no common factors.
Generating the final simplified expression:
\[
\frac{7^7}{4^{10}} = \frac{823543}{1048576}
\]
This is the final answer.