To simplify the expression \((143 \cdot 27^{-7})^{034}\), we can start by applying the properties of integer exponents.
Step 1: Simplify the exponent
The exponent \(034\) is simply \(34\), so we rewrite:
\[ (143 \cdot 27^{-7})^{34} \]
Step 2: Apply the power of a product property
Using the power of a product property, we can distribute the exponent:
\[ (143)^{34} \cdot (27^{-7})^{34} \]
Step 3: Apply the power of a power property
Now, applying the power of a power property to the second term:
\[ (27^{-7})^{34} = 27^{-7 \cdot 34} = 27^{-238} \]
So our expression becomes:
\[ 143^{34} \cdot 27^{-238} \]
Step 4: Convert to positive exponents
We can rewrite \(27^{-238}\) as:
\[ \frac{143^{34}}{27^{238}} \]
Now, we have:
\[ (143 \cdot 27^{-7})^{034} = \frac{143^{34}}{27^{238}} \]
Step 5: Apply Exponent to Base 27
Next, we rewrite the base \(27\) using its prime factorization. Since \(27 = 3^3\), we can express it as:
\[ 27^{238} = (3^3)^{238} = 3^{3 \cdot 238} = 3^{714} \]
Thus, the expression becomes:
\[ \frac{143^{34}}{3^{714}} \]
Final result:
The simplified form of the original expression in terms of positive exponents is:
\[ \frac{143^{34}}{3^{714}} \]
As a result, the expression without remaining exponents is:
\[ \frac{143^{34}}{3^{714}} \]
This is your final simplified expression.