Rewrite the following expressions: 2712⋅2714

Question

.(1 point)
Responses

2718
27 superscript Start Fraction 1 over 8 End Fraction baseline

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

2734
27 superscript Start Fraction 3 over 4 End Fraction baseline

27−−√4

Answer 1

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

1. Recognize that \( 2712 \) and \( 2714 \) can be expressed in terms of \( 2713 \): \[ 2712 = 2713 - 1 \] \[ 2714 = 2713 + 1 \]

2. Thus, we have: \[ 2712 \cdot 2714 = (2713 - 1)(2713 + 1) \]

3. This expression can be simplified using the difference of squares: \[ (a - b)(a + b) = a^2 - b^2 \] where \( a = 2713 \) and \( b = 1 \): \[ 2712 \cdot 2714 = 2713^2 - 1^2 = 2713^2 - 1 \]

Now, let's present this result in a simplified form:

• Since \( 2713 \) is \( 27^{1.5} \) or \( 27^{3/2} \), we can rewrite the final expression in terms of powers of \( 27 \): \[ 2712 \cdot 2714 = 27^{3/2}^2 - 1 = 27^3 - 1 \]

Therefore, the final expression can be stated as:

\[ 27^{3} - 1 \]

This result can also be expressed as:

\[ 27^{3} - 1 \]

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