To determine which of the expressions are equivalent to \(4 \frac{5}{8}\), we should express \(4 \frac{5}{8}\) in a more manageable form.
- Convert \(4 \frac{5}{8}\) to an improper fraction: \[ 4 \frac{5}{8} = \frac{4 \times 8 + 5}{8} = \frac{32 + 5}{8} = \frac{37}{8} \]
Now, let’s evaluate each provided expression to see if it simplifies to \(\frac{37}{8}\):
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\(4(4 \frac{3}{8})\)
- Convert \(4 \frac{3}{8}\) to an improper fraction: \[ 4(4 \frac{3}{8}) = 4 \times \frac{35}{8} = \frac{140}{8} \]
- Not equivalent.
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\(4^{\frac{5}{8}}\) (interpreted from "4 times open paren 4 raised to the 3 eighths power close paren")
- This expression doesn’t immediately appear to relate to \(\frac{37}{8}\). It doesn't simplify to a simple fraction and seems irrelevant to \(4 \frac{5}{8}\).
-
\(45^{-\frac{1}{8}}\)
- \( \sqrt[8]{4^5} = (4^5)^{\frac{1}{8}} = 4^{\frac{5}{8}}\) which requires more context.
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\((2^{\frac{14}{8}})(4^{-\frac{1}{4}})\)
- This can simplify as follows: \[ 2^{\frac{14}{8}} = 2^{\frac{7}{4}}, \quad 4^{-\frac{1}{4}} = (2^2)^{-\frac{1}{4}} = 2^{-1/2} = \frac{1}{\sqrt{2}} \]
- But this still doesn't reduce directly to \(\frac{37}{8}\).
-
\((4 \frac{7}{8})(4^{-\frac{1}{4}})\)
- Convert \(4 \frac{7}{8}\) to an improper fraction: \[ 4(4 \frac{7}{8}) = 4 \times \frac{39}{8} = \frac{156}{8} \]
- As above – not equivalent.
-
\(410 - \sqrt{5}\)
- Given as an evaluation error; let's treat this as undefined unless evaluated.
None of the expressions simplify or directly produce \(\frac{37}{8}\) or convert equivalently to \(4 \frac{5}{8}\).
Hence, the final answer is that none of these expressions are equivalent to \(4 \frac{5}{8}\).