The problem asks us to rationalize the denominator of the given fraction:

$\frac{\sqrt{5} + 2\sqrt{2}}{\sqrt{6}}$

Step-by-Step Solution

Step 1: Multiply numerator and denominator by $\sqrt{6}$

To rationalize the denominator, we eliminate the square root from the denominator by multiplying both numerator and denominator by $\sqrt{6}$:

$\frac{\sqrt{5} + 2\sqrt{2}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

This results in:

$\frac{(\sqrt{5} + 2\sqrt{2}) \cdot \sqrt{6}}{6}$

Step 2: Distribute $\sqrt{6}$ in the numerator

Now, expand the numerator:

$(\sqrt{5} + 2\sqrt{2}) \cdot \sqrt{6} = \sqrt{5} \cdot \sqrt{6} + 2\sqrt{2} \cdot \sqrt{6}$

Using the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, simplify each term:

$\sqrt{5} \cdot \sqrt{6} = \sqrt{30}, \quad 2\sqrt{2} \cdot \sqrt{6} = 2\sqrt{12}.$

So the numerator becomes:

$\sqrt{30} + 2\sqrt{12}.$

Step 3: Simplify $\sqrt{12}$

Since $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$, substitute this back:

$2\sqrt{12} = 2 \cdot 2\sqrt{3} = 4\sqrt{3}.$

Thus, the numerator becomes:

$\sqrt{30} + 4\sqrt{3}.$

Step 4: Write the final expression

Now, the fraction is:

$\frac{\sqrt{30} + 4\sqrt{3}}{6}.$

Final Answer:

$\boxed{\frac{\sqrt{30} + 4\sqrt{3}}{6}}$

Would you like further clarification or additional examples? Let me know!

Here are 5 related questions to explore:

1. How do you rationalize a denominator involving sums or differences, such as $(\sqrt{a} + b)$?
2. Can you explain why multiplying by $\sqrt{6} / \sqrt{6}$ preserves the value of the fraction?
3. What are the general rules for simplifying square roots like $\sqrt{12}$?
4. How would you rationalize a denominator with a binomial, such as $(\sqrt{5} - \sqrt{3})$?
5. How can you verify the final answer is equivalent to the original expression?

Tip: Always simplify square roots as much as possible (e.g., $\sqrt{12} = 2\sqrt{3}$) before writing your final answer.