To rewrite the expression \( 27 \frac{1}{2} \cdot 27 \frac{1}{4} \), we start by converting the mixed numbers to improper fractions:
- \( 27 \frac{1}{2} = \frac{27 \cdot 2 + 1}{2} = \frac{54 + 1}{2} = \frac{55}{2} \)
- \( 27 \frac{1}{4} = \frac{27 \cdot 4 + 1}{4} = \frac{108 + 1}{4} = \frac{109}{4} \)
Now, we calculate the product:
\[ 27 \frac{1}{2} \cdot 27 \frac{1}{4} = \left(\frac{55}{2}\right) \cdot \left(\frac{109}{4}\right) = \frac{55 \cdot 109}{8} \]
However, the multiple-choice options you provided suggest a simpler expression can be derived from the exponents or roots of 27. Notable is the fact that \( 27 = 3^3 \), which may help us express 27 in different forms.
Given the choices, let's check the roots and exponents of 27:
- \( 27^{- \frac{1}{2}} \) would equal \( \frac{1}{\sqrt{27}} \)
- \( 27^{\frac{1}{8}} \) suggests an eighth root
- \( 27^{\frac{3}{4}} \) suggests a three-quarters root
- \( 27^{- \frac{1}{4}} \) suggests a reciprocal of a fourth root
Considering the multiplication of the fractional exponents:
\[ 27^{\frac{1}{2} + \frac{1}{4}} = 27^{\frac{2}{4} + \frac{1}{4}} = 27^{\frac{3}{4}} \]
Thus, the equivalent expression for \( 27 \frac{1}{2} \cdot 27 \frac{1}{4} \) is:
Response: \( 27^{\frac{3}{4}} \).
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