To determine which expressions are equivalent to \( 4 \frac{5}{8} \), we first convert \( 4 \frac{5}{8} \) into an improper fraction or a decimal for easier comparison:
\[ 4 \frac{5}{8} = 4 + \frac{5}{8} = \frac{32}{8} + \frac{5}{8} = \frac{37}{8} \]
Now let's analyze each of the given expressions:
-
\( 8\sqrt{4^5} \):
- \( \sqrt{4^5} = 4^{5/2} = (2^2)^{5/2} = 2^{5} = 32 \)
- Therefore, \( 8 \sqrt{4^5} = 8 \cdot 32 = 256 \)
-
\( 4(4^{3/8}) \):
- This simplifies to \( 4 \cdot 4^{3/8} = 4^{1 + 3/8} = 4^{11/8} = \frac{(4^8)(4^3)}{4^8} = \frac{64 \cdot 64}{64} = 64 \)
- Thus, it does not equal \( 37/8 \).
-
\( 5\sqrt{4^{10}} \):
- \( \sqrt{4^{10}} = 4^5 = 1024 \)
- Therefore, \( 5\sqrt{4^{10}} = 5 \cdot 1024 = 5120 \)
-
\( (2^{14/8})(4^{-1/4}) \):
- Simplifying, this equals \( 2^{14/8} = 2^{7/4} \) and \( 4^{-1/4} = 2^{-1/2} = \frac{1}{\sqrt{2}} \)
- Thus, \( 2^{7/4} \cdot 2^{-1/2} = 2^{(7/4) - (2/4)} = 2^{5/4} = \frac{4\sqrt{2}}{2} \approx 2.828 \), which does not equal \( 37/8 \).
-
\( (4^{7/8})(4^{-1/4}) \):
- Simplifying, we have \( 4^{7/8} \cdot 4^{-1/4} = 4^{(7/8) - (2/8)} = 4^{5/8} \)
From analyzing all the expressions, the only ones that are not equal to \( 4\frac{5}{8} \) are:
- \( 8\sqrt{4^5} \)
- \( 5\sqrt{4^{10}} \)
- \( (2^{14/8})(4^{-1/4}) \)
Thus, none of the options mentioned above are equivalent to \( 4\frac{5}{8} \) or \( \frac{37}{8} \).