Question

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

27^1/8

27 superscript Start Fraction 1 over 8 End Fraction baseline

root index 4 Start Root 27 End Root

27^3/4
root index 3 Start Root 27 End Root

Answer 1

To rewrite the expression \( 2712 \cdot 2714 \), we can factor it as follows:

1. Notice that \( 2712 \) and \( 2714 \) can be expressed in terms of powers of 27:

\[
2712 = 27^{3/4} \quad \text{and} \quad 2714 = 27^{1/8}
\]

2. Then we can rewrite the product:

\[
2712 \cdot 2714 = 27^{3/4} \cdot 27^{1/8}
\]

3. To combine the powers, we add the exponents:

\[
= 27^{3/4 + 1/8}
\]

4. To add the fractions, we need a common denominator. The least common multiple of 4 and 8 is 8:

\[
3/4 = 6/8
\]
\[
3/4 + 1/8 = 6/8 + 1/8 = 7/8
\]

5. So, we have:

\[
2712 \cdot 2714 = 27^{7/8}
\]

6. 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.