To rewrite the expression \( 27 \frac{1}{2} \cdot 27 \frac{1}{4} \), we first convert the mixed numbers to improper fractions:
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Convert \( 27 \frac{1}{2} \): \[ 27 \frac{1}{2} = 27 + \frac{1}{2} = \frac{54}{2} + \frac{1}{2} = \frac{55}{2} \]
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Convert \( 27 \frac{1}{4} \): \[ 27 \frac{1}{4} = 27 + \frac{1}{4} = \frac{108}{4} + \frac{1}{4} = \frac{109}{4} \]
Now, multiply the two improper fractions: \[ 27 \frac{1}{2} \cdot 27 \frac{1}{4} = \frac{55}{2} \cdot \frac{109}{4} = \frac{55 \cdot 109}{2 \cdot 4} = \frac{5995}{8} \]
Next, we need to express \( 27 \) in terms of a power: Since \( 27 = 3^3 \), we can express our result in terms of powers of \( 3 \): \[ 27 \frac{1}{2} \cdot 27 \frac{1}{4} = (3^3)^{\frac{55}{2}} \cdot (3^3)^{\frac{109}{4}} = 3^{3 \cdot \frac{55}{2}} \cdot 3^{3 \cdot \frac{109}{4}} \]
Now, we apply the property of exponents \( a^m \cdot a^n = a^{m+n} \): \[ 3^{3 \cdot \frac{55}{2} + 3 \cdot \frac{109}{4}} \]
Now we need to find a common denominator to combine the exponents. The least common multiple of \( 2 \) and \( 4 \) is \( 4 \): Convert \( \frac{55}{2} \) to have a denominator of \( 4 \): \[ \frac{55}{2} = \frac{110}{4} \]
So, \[ 3^{3 \cdot \frac{110}{4} + 3 \cdot \frac{109}{4}} = 3^{\frac{330 + 327}{4}} = 3^{\frac{657}{4}} \]
Thus, the combined expression is: \[ 27^{\frac{657}{4}} \]
So, the answer is: The combined expression is \( 27^{\frac{657}{4}} \) .
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