To rewrite the expression \( 27 \frac{1}{2} \cdot 27 \frac{1}{4} \), we first convert the mixed numbers into improper fractions:
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Converting \( 27 \frac{1}{2} \): \[ 27 \frac{1}{2} = 27 + \frac{1}{2} = \frac{54}{2} + \frac{1}{2} = \frac{55}{2} \]
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Converting \( 27 \frac{1}{4} \): \[ 27 \frac{1}{4} = 27 + \frac{1}{4} = \frac{108}{4} + \frac{1}{4} = \frac{109}{4} \]
Now, we multiply the two improper fractions: \[ \frac{55}{2} \cdot \frac{109}{4} = \frac{55 \cdot 109}{2 \cdot 4} = \frac{5995}{8} \]
Now we convert this back to the form of \( 27 \) raised to a fractional power. First, we express \( 27 \) as a power of 3: \[ 27 = 3^3 \] Thus, we can rewrite \( 27 \frac{1}{2} \cdot 27 \frac{1}{4} \) in terms of \( 3^3 \):
Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \): \[ 27 \frac{1}{2} = 3^{3 + \frac{1}{2}} = 3^{\frac{6}{2} + \frac{1}{2}} = 3^{\frac{7}{2}} \] \[ 27 \frac{1}{4} = 3^{3 + \frac{1}{4}} = 3^{\frac{12}{4} + \frac{1}{4}} = 3^{\frac{13}{4}} \]
Now multiplying these together: \[ 3^{\frac{7}{2}} \cdot 3^{\frac{13}{4}} = 3^{\left(\frac{7}{2} + \frac{13}{4}\right)} \]
To add the exponents, we need a common denominator. The least common multiple of 2 and 4 is 4, so we convert: \[ \frac{7}{2} = \frac{14}{4} \] Now we can add: \[ \frac{14}{4} + \frac{13}{4} = \frac{27}{4} \]
Thus, the final expression becomes: \[ 27 \frac{1}{2} \cdot 27 \frac{1}{4} = 27^{\frac{27}{4}} \]
In summary, the combined expression is \( 27 \) to the power of \( \frac{27}{4} \).
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