To rewrite the expression \( 2712 \cdot 2714 \), we can factor it as follows:
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Notice that \( 2712 \) and \( 2714 \) can be expressed in terms of powers of 27:
\[ 2712 = 27^{3/4} \quad \text{and} \quad 2714 = 27^{1/8} \]
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Then we can rewrite the product:
\[ 2712 \cdot 2714 = 27^{3/4} \cdot 27^{1/8} \]
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To combine the powers, we add the exponents:
\[ = 27^{3/4 + 1/8} \]
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To add the fractions, we need a common denominator. The least common multiple of 4 and 8 is 8:
\[ 3/4 = 6/8 \] \[ 3/4 + 1/8 = 6/8 + 1/8 = 7/8 \]
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So, we have:
\[ 2712 \cdot 2714 = 27^{7/8} \]
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An alternative way to express \( 27^{7/8} \) is to use roots:
\[ 27^{7/8} = \sqrt[8]{27^7} \]
Thus, the rewritten expression can be represented as:
- \( 27^{7/8} \)
- or \( \sqrt[8]{27^7} \)
However, if you're specifically looking for the other representations provided in the responses, it appears that only \( 27^{3/4} \) is directly related to \( 2712 \) and that \( \sqrt[3]{27} \) simplifies to \( 3 \) since \( 27 = 3^3 \).
So the best matching outputs were \( 27^{1/8} = \sqrt[4]{27} \), which makes sense mathematically!
Hence:
- \( 27^{3/4} \)
- \( \sqrt[3]{27} \)
Please select the appropriate response according to the context.
1 answer