The problem is to simplify the radical expression $\sqrt{\frac{9}{6}}$.

Solution:

1. Simplify the fraction under the square root: $\frac{9}{6} = \frac{3}{2}.$

2. The expression becomes: $\sqrt{\frac{3}{2}}.$

3. Use the property of square roots: $\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}.$

4. Rationalize the denominator: Multiply numerator and denominator by $\sqrt{2}$: $\frac{\sqrt{3}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{6}}{2}.$

Final Answer:

$\frac{\sqrt{6}}{2}.$

Let me know if you need more explanation!

Here are 5 related questions to expand your understanding:

1. How do you simplify $\sqrt{\frac{8}{3}}$?
2. What is the purpose of rationalizing the denominator in radical expressions?
3. Can $\sqrt{\frac{9}{6}}$ also be expressed as a decimal? If so, what is it?
4. How would you simplify $\sqrt{\frac{12}{5}}$?
5. What property of radicals allows $\sqrt{\frac{a}{b}}$ to be written as $\frac{\sqrt{a}}{\sqrt{b}}$?

Tip: Always simplify fractions first before applying the square root for cleaner calculations!