To identify which of the given options is a rational number between 5 and 514, let's first convert the mixed fractions into improper fractions and check if they fall within this range.
- 513: This is an integer and clearly greater than 514.
- \(5 \frac{1}{3}\): This can be converted to an improper fraction \(5 \frac{1}{3} = \frac{16}{3} \approx 5.33\), which is greater than 5 but less than 514.
- 525: This is an integer and clearly greater than 514.
- \(5 \frac{2}{5}\): This can be converted to an improper fraction \(5 \frac{2}{5} = \frac{27}{5} = 5.4\), which is greater than 5 but less than 514.
- 517: This is an integer and clearly greater than 514.
- \(5 \frac{1}{7}\): This can be converted to an improper fraction \(5 \frac{1}{7} = \frac{36}{7} \approx 5.14\), which is greater than 5 but less than 514.
- \(26^{1/2}\) (or \(\sqrt{26}\)): The approximate value of \(\sqrt{26} \approx 5.1\), which is greater than 5 but less than 514.
Now we summarize the options:
- \(513\), \(525\), and \(517\) are not within the range as they are greater than 514.
- The mixed fractions \(5 \frac{1}{3}\), \(5 \frac{2}{5}\), and \(5 \frac{1}{7}\) fall within the range \(5 < x < 514\).
- \(\sqrt{26}\) also falls within that range.
Thus, the rational numbers between 5 and 514 from the given options are:
- \(5 \frac{1}{3}\) (approximately 5.33)
- \(5 \frac{2}{5}\) (approximately 5.4)
- \(5 \frac{1}{7}\) (approximately 5.14)
- \(\sqrt{26}\) (approximately 5.1)
In conclusion, all mixed fraction options and \(\sqrt{26}\) are valid rational numbers between 5 and 514.