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:
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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:
\[ (43 \cdot 7^{-2})^{-579} = 43^{-579} \cdot 7^{1158} \]
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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:
\[ \frac{7^{1158}}{43^{579} \cdot 4^{13}} \]
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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:
\[ \frac{7^{1158}}{43^{579} \cdot 2^{26}} \]
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