Let's simplify the expression \( (14^3 \cdot 27^{-7})^{034} \).
First, we'll rewrite the expression with positive exponents and simplify it:
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Rewrite \( 27^{-7} \): \[ 27^{-7} = \frac{1}{27^7} \]
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So, we can rewrite our expression: \[ (14^3 \cdot 27^{-7})^{034} = (14^3 \cdot \frac{1}{27^7})^{034} = \left(\frac{14^3}{27^7}\right)^{034} \]
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Now use the property of exponents that states \((a \cdot b)^n = a^n \cdot b^n\): \[ \left(\frac{14^3}{27^7}\right)^{034} = \frac{(14^3)^{034}}{(27^7)^{034}} \]
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Apply the power of a power property which states \( (a^m)^n = a^{m \cdot n} \): \[ (14^3)^{034} = 14^{3 \cdot 34} = 14^{102} \] \[ (27^7)^{034} = 27^{7 \cdot 34} = 27^{238} \]
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The expression now simplifies to: \[ \frac{14^{102}}{27^{238}} \]
Now to simplify this expression further, we should keep in mind that our result needs to be in simplified fraction form with no remaining exponents.
Since we do not know the values of \( 14 \) and \( 27 \) without further evaluating:
Using the fact that \( 27 = 3^3 \), we can express \( 27^{238} \) in terms of powers of 3: \[ 27^{238} = (3^3)^{238} = 3^{714} \]
Thus, the expression can be rewritten as: \[ \frac{14^{102}}{3^{714}} \]
This is our final answer in simplified fraction form and with positive exponents: \[ \frac{14^{102}}{3^{714}} \]