https://file.aiquickdraw.com/mathbot/file/1726438962131el518apn.jpg

To determine which of the numbers in the image is rational, let's evaluate each expression:

1. **\(\sqrt{\frac{16}{6}}\)**:
   - \(\frac{16}{6} = \frac{8}{3}\), which is not a perfect square.
   - \(\sqrt{\frac{8}{3}}\) is not a rational number because it cannot be simplified to a perfect square.

2. **\(\sqrt{\frac{36}{16}}\)**:
   - \(\frac{36}{16} = \frac{9}{4}\).
   - \(\sqrt{\frac{9}{4}} = \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2}\), which is a rational number.

3. **\(\sqrt{\frac{36}{6}}\)**:
   - \(\frac{36}{6} = 6\), and \(\sqrt{6}\) is irrational.

4. **\(\sqrt{6}\)**:
   - \(\sqrt{6}\) is also irrational.

The only rational number among the options is **\(\sqrt{\frac{36}{16}}\)**, which simplifies to \(\frac{3}{2}\).

Would you like further details or have any other questions?

Here are some related questions to explore:
1. What makes a number rational or irrational?
2. How can you tell if a square root results in a rational number?
3. Can \(\sqrt{\frac{8}{3}}\) be expressed in a simpler form?
4. Why is \(\sqrt{6}\) considered irrational?
5. How do fractions affect the rationality of square roots?

**Tip:** A number is rational if it can be expressed as a fraction of two integers, and a square root is rational if the radicand is a perfect square.

Learn how to determine which square roots of fractions result in rational numbers. This problem explores the concepts of rationality, square roots, and fractions, suitable for students in Grades 8-10.

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