Apply the Properties of Integer Exponents to generate equivalent expressions to (4(3) ⋅ 7(−2)(−5) 7(9) ⋅ 4(−13) with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

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.

simplify that

To simplify the fraction \( \frac{823543}{1048576} \), we first check if the numerator and denominator have any common factors.

1. **Factorize the numerator**:
- The numerator \( 823543 \) is equal to \( 7^7 \).

2. **Factorize the denominator**:
- The denominator \( 1048576 \) is equal to \( 4^{10} = (2^2)^{10} = 2^{20} \).

Thus, the expression can be rewritten as:
\[
\frac{7^7}{2^{20}}
\]

Since \( 7 \) and \( 2 \) are coprime numbers (they have no common factors), the fraction \( \frac{7^7}{2^{20}} \) is already in simplest form.

Therefore, the simplified result of the expression \( \frac{823543}{1048576} \) is:
\[
\frac{7^7}{2^{20}}
\]
Or if you prefer it as a fraction:
\[
\frac{823543}{1048576}
\]

If you need a decimal approximation, you could compute that as:
\[
7^7 = 823543 \quad \text{and} \quad 2^{20} = 1048576 \quad \text{(there's no simplification)}
\]
So the answer remains:
\[
\frac{823543}{1048576}
\]
as the simplest form.