To rewrite the expression \(2712 \cdot 2714\), we can apply the properties of exponents.
First, we can rewrite \(2712\) and \(2714\) in terms of exponent base \(27\):
\[ 2712 = 27^{12} \quad \text{and} \quad 2714 = 27^{14} \]
Now we can combine these two using the property of exponents:
\[ 2712 \cdot 2714 = 27^{12} \cdot 27^{14} = 27^{12 + 14} = 27^{26} \]
Thus, \(2712 \cdot 2714\) can be rewritten as:
\[ 27^{26} \]
Now, let's evaluate the given options based on this:
- \(27^{\sqrt{4}} = 27^2\)
- \(27^{\frac{3}{4}}\)
- \(27^{\frac{1}{8}}\)
- \(27^{\sqrt{3}}\)
None of these options matches \(27^{26}\).
Combining in a different way, we can note that:
\[ 2712 \cdot 2714 = (27^{13})^2 - 1^2 \quad \text{(using difference of squares)} \]
Thus, another expression could be given as:
\[ (27^{13})^2 - 1^2 = 27^{26} - 1 \]
In conclusion, the most direct rewriting for the expression is \(27^{26}\). If we need to find a form from the options, none is equivalent to \(27^{26}\).
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