To determine which expressions are equivalent to \( 4 \frac{5}{8} \), we first convert it to an improper fraction or its decimal equivalent.
- Convert \( 4 \frac{5}{8} \):
- \( 4 \frac{5}{8} = 4 + \frac{5}{8} = \frac{32}{8} + \frac{5}{8} = \frac{37}{8} \).
Now let’s evaluate each of the provided options:
A. \( 4 \times (4^{\frac{3}{8}}) \)
- This does not equal \( \frac{37}{8} \).
B. \( \sqrt[8]{4^5} \)
- \( 4^5 = (2^2)^5 = 2^{10} \),
- Thus, \( \sqrt[8]{4^5} = \sqrt[8]{2^{10}} = 2^{10/8} = 2^{5/4} = 4 \times \sqrt[4]{2} \neq \frac{37}{8} \).
C. \( (2^{\frac{14}{8}})(4^{-\frac{1}{4}}) \)
- \( 2^{\frac{14}{8}} = 2^{\frac{7}{4}} \) and \( 4^{-\frac{1}{4}} = (2^2)^{-\frac{1}{4}} = 2^{-\frac{1}{2}} \),
- Thus, \( (2^{\frac{7}{4}})(2^{-\frac{1}{2}}) = 2^{\frac{7}{4} - \frac{2}{4}} = 2^{\frac{5}{4}} \).
- Therefore, this does not equal \( 4 \frac{5}{8} \).
D. \( (4^{\frac{7}{8}})(4^{-\frac{1}{4}}) \)
- Using \( 4 = 2^2 \), we have \( (2^2)^{\frac{7}{8}} \cdot (2^2)^{-\frac{1}{4}} = 2^{\frac{7}{4}} \cdot 2^{-0.5} = 2^{\frac{7}{4} - \frac{2}{4}} = 2^{\frac{5}{4}} \).
- This simplifies to the same as C, which does not equal \( 4 \frac{5}{8} \).
E. \( 4^{10 - \sqrt{5}} \)
- The evaluation of this expression is complex, but without advanced calculation directly, it's clear that this is not in simple fractional form such as \( 4 \frac{5}{8} \) and should also not yield such a simple fraction when evaluated.
Upon evaluating all options, none of them is equal to \( 4 \frac{5}{8} \) or \( \frac{37}{8} \).
Hence, there are no correct responses that are equivalent to \( 4 \frac{5}{8} \). All responses A, B, C, D, and E are not equivalent to \( 4 \frac{5}{8} \).