To simplify the expression \((143 \cdot 27^{-7})^{034}\) using the properties of integer exponents, we first note that \(034\) is simply \(34\). So we can rewrite the expression as:
\[ (143 \cdot 27^{-7})^{34} \]
Next, we can apply the property of exponents that states \((a \cdot b)^n = a^n \cdot b^n\). This gives us:
\[ (143)^{34} \cdot (27^{-7})^{34} \]
Next, we handle the term \((27^{-7})^{34}\) using the property \((a^m)^n = a^{m \cdot n}\):
\[ (27^{-7})^{34} = 27^{-7 \cdot 34} = 27^{-238} \]
Now, we can put everything together:
\[ (143)^{34} \cdot 27^{-238} \]
To eliminate the negative exponent and write it in terms of positive exponents, we use the property \(a^{-n} = \frac{1}{a^n}\):
\[ (143)^{34} \cdot \frac{1}{27^{238}} = \frac{(143)^{34}}{27^{238}} \]
Now, we have an equivalent expression with only positive exponents:
\[ \frac{(143)^{34}}{27^{238}} \]
Next, we check if these numbers can be simplified further.
Final Result:
\[ \frac{(143)^{34}}{27^{238}} \]
This is the simplified expression in the form of a fraction with no remaining exponents.
Thus, the final answer is:
\[ \frac{(143)^{34}}{27^{238}} \]
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