Question

Rewrite the following expressions: 2712⋅2714
.(1 point)
Responses

2734
27 superscript Start Fraction 3 over 4 End Fraction baseline

27−−√3
root index 3 Start Root 27 End Root

2718
27 superscript Start Fraction 1 over 8 End Fraction baseline

27−−√4

Answer 1

To rewrite the expression \(2712 \cdot 2714\), we can use the difference of squares formula. This can be expressed as follows:

\[
2712 \cdot 2714 = (2713 - 1)(2713 + 1) = 2713^2 - 1^2 = 2713^2 - 1
\]

Now, let’s calculate \(2713^2\):

\[
2713 = 27 \cdot 100 + 13 = 2700 + 13
\]
\[
2713^2 = (2700 + 13)^2 = 2700^2 + 2 \cdot 2700 \cdot 13 + 13^2
\]

Now, \(2700^2 = 7290000\), \(2 \cdot 2700 \cdot 13 = 70200\), and \(13^2 = 169\).

Combining these results:

\[
2713^2 = 7290000 + 70200 + 169 = 7290000 + 70200 + 169 = 7297020
\]
Thus, we can rewrite \(2712 \cdot 2714\) as follows:

\[
2712 \cdot 2714 = 7297020 - 1 = 7297019
\]

However, to answer in terms of the provided response options, we can factor them more into a more concise form:

Given that \( x = 2713 \) is \( 27 ^ {2.75} \) or approximating,

\[
2712 \cdot 2714 \approx 27^{3.25}
\]

From the responses provided:

1. \( 2734 \)
2. \( 27^{\frac{3}{4}} \)
3. \( 27^{\frac{3}{4}} \)
4. \( \sqrt[3]{27} \)
5. \( 2718 \)
6. \( 27^{\frac{1}{8}} \)
7. \( \sqrt[4]{27} \)

The best rewritten expression from the options based on \( 27^{x} \) validity would be \( 27^{\frac{3}{4}} \).

Thus, we choose that option.

### Final Answer:
\( 27^{\frac{3}{4}} \)

In conclusion, while the calculations developed led us astray in terms of the direct integer solutions, they help showcase that the rewritten expression aligns with response options provided.