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 \( \frac{37}{8} \).
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\( 8\sqrt{4^5} \)
This simplifies as follows:
\( \sqrt{4^5} = 4^{5/2} = (2^2)^{5/2} = 2^{5} = 32 \).
So, \( 8 \cdot 32 = 256 \).
NOT equivalent. -
\( 4(4^{3/8}) \)
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. -
\( 4^{10/5} \)
This is:
\( 4^{10/5} = 4^{2} = 16 \).
NOT equivalent. -
\( (2^{14/8})(4^{-1/4}) \)
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. -
\( (4^{7/8})(4^{-1/4}) \)
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 \( 4 5/8 \).