To determine which of the given options is a rational number between 5 and \( 5 \frac{1}{4} \), we first convert \( 5 \frac{1}{4} \) into an improper fraction for easier comparison.
\[ 5 \frac{1}{4} = \frac{21}{4} \]
Now we'll examine each option to see if it lies between 5 and \( 5 \frac{1}{4} \):
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\( \sqrt{26} \): \[ \sqrt{26} \approx 5.099 \quad \text{(since \( 5^2 = 25 \) and \( 6^2 = 36 \))} \] \( \sqrt{26} \) is greater than \( 5 \frac{1}{4} \) (which is 5.25).
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\( 5 \frac{1}{8} \): \[ 5 \frac{1}{8} = 5.125 \] \( 5.125 \) is between 5 (5.0) and \( 5 \frac{1}{4} \) (5.25). This one is valid.
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\( 5 \frac{2}{5} \): \[ 5 \frac{2}{5} = 5 + 0.4 = 5.4 \] \( 5.4 \) is greater than \( 5 \frac{1}{4} \) (5.25).
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\( 5 \frac{1}{3} \): \[ 5 \frac{1}{3} \approx 5.333 \] \( 5.333 \) is also greater than \( 5 \frac{1}{4} \) (5.25).
After evaluating all options, the only rational number that lies between 5 and \( 5 \frac{1}{4} \) is:
\( 5 \frac{1}{8} \).