Question

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

8√4^5
8th root of 4 to the 5th power

4(438)
4 times open paren 4 raised to the 3 eighths power close paren

410−−−√5
5th root of 4 to the 10th power

(2148)(4−14)
open paren 2 raised to the 14 eighths power close paren times open paren 4 raised to the negative 1 fourth power close paren

(478)(4−14)

Answer 1

To find the equivalent expressions to 4 5/8, we first convert 4 5/8 into an improper fraction:

\[
4 \frac{5}{8} = \frac{4 \times 8 + 5}{8} = \frac{32 + 5}{8} = \frac{37}{8}
\]

Now we will check each of the given expressions to see if they are equivalent to \( \frac{37}{8} \).

1. **\( 8\sqrt{4^5} \)**
This simplifies as follows:
\( \sqrt{4^5} = 4^{5/2} = (2^2)^{5/2} = 2^{5} = 32 \).
So, \( 8 \cdot 32 = 256 \).
NOT equivalent.

2. **\( 4(4^{3/8}) \)**
We can simplify this:
\( 4 = 4^{1} \), thus:
\( 4(4^{3/8}) = 4^{1 + 3/8} = 4^{11/8} \).
\( 4^{11/8} \) is equivalent to \( \frac{4^{11}}{8} = \frac{(2^2)^{11}}{8} = \frac{2^{22}}{2^3} = 2^{19} \).
NOT equivalent.

3. **\( 4^{10/5} \)**
This is:
\( 4^{10/5} = 4^{2} = 16 \).
NOT equivalent.

4. **\( (2^{14/8})(4^{-1/4}) \)**
Simplifying:
\( 4^{-1/4} = (2^2)^{-1/4} = 2^{-2/4} = 2^{-1/2} = \frac{1}{\sqrt{2}} \).
Thus:
\( (2^{14/8})(\frac{1}{\sqrt{2}}) = (2^{7/4})(\frac{1}{\sqrt{2}}) = (2^{7/4})(2^{-1/2}) = 2^{7/4 - 2/4} = 2^{5/4} \).
NOT equivalent.

5. **\( (4^{7/8})(4^{-1/4}) \)**
This simplifies as follows:
\( 4^{-1/4} = 4^{-2/8} \), therefore:
\( (4^{7/8})(4^{-2/8}) = 4^{(7/8 - 2/8)} = 4^{5/8} \).
NOT equivalent.

Thus, **none of the expressions** provided are equivalent to \( 4 \frac{5}{8} \) or \( \frac{37}{8} \). All options given are not equivalent to \( 4 5/8 \).