Question

Rewrite the following expressions: 27 1/2⋅27 1/4
.(1 point)
Responses

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

2718
27 superscript Start Fraction 1 over 8 End Fraction baseline

2734
27 superscript Start Fraction 3 over 4 End Fraction baseline

27−−√4

Answer 1

To rewrite the expression \( 27 \frac{1}{2} \cdot 27 \frac{1}{4} \), we start by converting the mixed numbers to improper fractions:

1. \( 27 \frac{1}{2} = \frac{27 \cdot 2 + 1}{2} = \frac{54 + 1}{2} = \frac{55}{2} \)
2. \( 27 \frac{1}{4} = \frac{27 \cdot 4 + 1}{4} = \frac{108 + 1}{4} = \frac{109}{4} \)

Now, we calculate the product:

\[
27 \frac{1}{2} \cdot 27 \frac{1}{4} = \left(\frac{55}{2}\right) \cdot \left(\frac{109}{4}\right) = \frac{55 \cdot 109}{8}
\]

However, the multiple-choice options you provided suggest a simpler expression can be derived from the exponents or roots of 27. Notable is the fact that \( 27 = 3^3 \), which may help us express 27 in different forms.

Given the choices, let's check the roots and exponents of 27:

- \( 27^{- \frac{1}{2}} \) would equal \( \frac{1}{\sqrt{27}} \)
- \( 27^{\frac{1}{8}} \) suggests an eighth root
- \( 27^{\frac{3}{4}} \) suggests a three-quarters root
- \( 27^{- \frac{1}{4}} \) suggests a reciprocal of a fourth root

Considering the multiplication of the fractional exponents:

\[
27^{\frac{1}{2} + \frac{1}{4}} = 27^{\frac{2}{4} + \frac{1}{4}} = 27^{\frac{3}{4}}
\]

Thus, the equivalent expression for \( 27 \frac{1}{2} \cdot 27 \frac{1}{4} \) is:

**Response: \( 27^{\frac{3}{4}} \)**.