To determine which expressions are equivalent to \( 4 \frac{5}{8} \), we will first convert \( 4 \frac{5}{8} \) into an improper fraction.
- Convert \( 4 \frac{5}{8} \) to an improper fraction: \[ 4 \frac{5}{8} = \frac{4 \cdot 8 + 5}{8} = \frac{32 + 5}{8} = \frac{37}{8} \]
Now let's evaluate each of the options given:
A. \( 4(4^{3/8}) \)
- This simplifies to \( 4 \times 4^{3/8} = 4^{1 + 3/8} = 4^{(8/8) + (3/8)} = 4^{11/8} \).
- This does not equal \( 4^{37/8} \) so this is NOT equivalent.
B. \( \sqrt[8]{4^5} \)
- This equals \( 4^{5/8} \), which is not equal to \( \frac{37}{8} \) or \( 4^{11/8} \). This is NOT equivalent.
C. \( (2^{14/8})(4^{-1/4}) \)
- First, simplify: \( 2^{14/8} = 2^{7/4} = (2^7)/(2^4) = 128/16 = 8 \).
- \( 4^{-1/4} = (2^2)^{-1/4} = 2^{-1/2} = \frac{1}{\sqrt{2}} \).
- Combining those gives: \( 8 \cdot \frac{1}{\sqrt{2}} \), which does not equate to \( 4 \frac{5}{8} \). This is NOT equivalent.
D. \( (4^{7/8})(4^{-1/4}) \)
- This can be simplified by combining the exponents: \[ 4^{7/8} \cdot 4^{-1/4} = 4^{7/8 - 2/8} = 4^{5/8}. \]
- This does not equal \( \frac{37}{8} \) or \( 4^{11/8} \) either. This is NOT equivalent.
E. \( 4 \cdot \sqrt[10]{5} \)
- This does not give us any expression based on the form of \( 4 \frac{5}{8} \). This is NOT equivalent.
Given the analysis of the options, none of the expressions A, B, C, D, or E are equivalent to \( 4 \frac{5}{8} \). Therefore, the correct response would be that there are no equivalent expressions among the options provided.