Question

Rewrite the following expressions:

2712⋅2714


(2 points)
The combined expression is 27 to the power of
(write as a fraction)

Answer 1

To rewrite the expression \( 2712 \cdot 2714 \) in terms of powers of \( 27 \), we can express the numbers in terms of \( 27 \):

\[
2712 = 27 \times 100 + 12
\]
\[
2714 = 27 \times 100 + 14
\]

However, a more useful approach is to observe that:

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

Now, let’s compute \( 2713^2 \):

\[
2713 = 27 \cdot 100 + 13
\]

But we can also directly compute \( 2713^2 \) since we are interested in the expression concerning powers of \( 27 \).

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

For the original expression, rewrite \(2712 \cdot 2714\) as:

\[
(27^2 \cdot (100)^2 + 13 \cdot 27^2) - 1
\]

Now we can relate this to \(27^2\) and finalize as a fraction:

Let me simplify it correctly, we should realize that \(2712 \cdot 2714 = (2700 + 12)(2700 + 14)\)

On final remarks:

\[
2712 \cdot 2714 \approx 27^{\frac{2}{1}} (which indicates power of 27 with proper expression, though direct calculation could be done for exact value yielding powers, here indicating it yields proper net expression.)
\]

Thus, the final fraction we find is:

\( 27^{\frac{3}{1}} \)

So the power of 27 related to express as fractions similarly maintains a close relation here.

Thus rewriting it succinctly, we would summarize:
The combined expression is \( 27^{\frac{3}{1}} \).