Question

To simplify the expression \( (43 \cdot 7^{-2})^{-579} \cdot 4^{-13} \) using the properties of integer exponents and generate an equivalent expression with only positive exponents, we can follow these steps:

1. Use the property \( (a^m)^n = a^{m \cdot n} \) to handle the exponent on the first term.

\[
(43 \cdot 7^{-2})^{-579} = 43^{-579} \cdot (7^{-2})^{-579}
\]

Simplifying \( (7^{-2})^{-579} \):

\[
(7^{-2})^{-579} = 7^{2 \cdot 579} = 7^{1158}
\]

Now, we have:

\[
(43 \cdot 7^{-2})^{-579} = 43^{-579} \cdot 7^{1158}
\]

2. Next, we combine it with \( 4^{-13} \):

\[
43^{-579} \cdot 7^{1158} \cdot 4^{-13}
\]

We can also convert \( 4^{-13} \) using the property:

\[
4^{-13} = \frac{1}{4^{13}}
\]

This allows us to rewrite the entire expression as:

\[
\frac{7^{1158}}{43^{579} \cdot 4^{13}}
\]

3. Now we convert \( 4^{13} \) into base powers of \( 2 \):

\[
4^{13} = (2^2)^{13} = 2^{26}
\]

Substituting this back into the expression gives:

\[
\frac{7^{1158}}{43^{579} \cdot 2^{26}}
\]

Now we have our final expression with only positive exponents:

\[
\frac{7^{1158}}{43^{579} \cdot 2^{26}}
\]

The expression is now simplified and does not contain any negative exponents. Since the problem asks for a simplified fraction and doesn't specify numerical evaluation beyond this structure, this is our final answer:

\[
\frac{7^{1158}}{43^{579} \cdot 2^{26}}
\]