Question

Choose all the expressions that are equivalent to 4 5/8
(1 point)
Responses

4(4 3/8)
4 times open paren 4 raised to the 3 eighths power close paren

45−−√8
8th root of 4 to the 5th power

(2 14/8)(4 −1/4)
open paren 2 raised to the 14 eighths power close paren times open paren 4 raised to the negative 1 fourth power close paren

(4 7/8)(4 −1/4)
open paren 4 raised to the 7 eighths power close paren times open paren 4 raised to the negative 1 fourth power close paren

410−−−√5

Answer 1

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.

1. **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 \(\frac{37}{8}\):

1. **\(4(4 \frac{3}{8})\)**
- Convert \(4 \frac{3}{8}\) to an improper fraction:
\[
4(4 \frac{3}{8}) = 4 \times \frac{35}{8} = \frac{140}{8}
\]
- Not equivalent.

2. **\(4^{\frac{5}{8}}\) (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 \(4 \frac{5}{8}\).

3. **\(45^{-\frac{1}{8}}\)**
- \( \sqrt[8]{4^5} = (4^5)^{\frac{1}{8}} = 4^{\frac{5}{8}}\) which requires more context.

4. **\((2^{\frac{14}{8}})(4^{-\frac{1}{4}})\)**
- 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 \(\frac{37}{8}\).

5. **\((4 \frac{7}{8})(4^{-\frac{1}{4}})\)**
- 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.

6. **\(410 - \sqrt{5}\)**
- 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 \(4 \frac{5}{8}\).