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.