To simplify the expression \( 13 \cdot 13 \cdot 13 \cdot 13 \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \), we first group the terms.
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The first part \( 13 \cdot 13 \cdot 13 \cdot 13 \) can be rewritten using exponents: \[ 13^4 \]
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The second part \( \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} \) can also be rewritten using exponents: \[ \left(\frac{1}{3}\right)^4 = \frac{1^4}{3^4} = \frac{1}{3^4} \]
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Combining these two simplified parts: \[ 13^4 \cdot \frac{1}{3^4} = \frac{13^4}{3^4} \]
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Finally, this can be expressed in exponential form: \[ \left(\frac{13}{3}\right)^4 \]
Thus, the fully simplified result is: \[ \left(\frac{13}{3}\right)^4 \]