We are given the expression:

$\frac{7\sqrt{2} + 3\sqrt{3}}{4\sqrt{2} - 2\sqrt{2}}$

First, simplify the denominator:

$4\sqrt{2} - 2\sqrt{2} = (4 - 2)\sqrt{2} = 2\sqrt{2}$

So, the expression becomes:

$\frac{7\sqrt{2} + 3\sqrt{3}}{2\sqrt{2}}$

Now, we can break this into two separate fractions:

$\frac{7\sqrt{2}}{2\sqrt{2}} + \frac{3\sqrt{3}}{2\sqrt{2}}$

Let's simplify each fraction separately:

1. For the first term: $\frac{7\sqrt{2}}{2\sqrt{2}} = \frac{7}{2}$

2. For the second term: $\frac{3\sqrt{3}}{2\sqrt{2}} = \frac{3}{2} \cdot \frac{\sqrt{3}}{\sqrt{2}} = \frac{3}{2} \cdot \sqrt{\frac{3}{2}} = \frac{3}{2} \cdot \sqrt{\frac{3 \times 2}{2 \times 2}} = \frac{3}{2\sqrt{2}}$

Therefore, the expression simplifies to:

$\frac{7}{2} + \frac{3}{2\sqrt{2}} \cdot \sqrt{3}$

Express this as $p + q\sqrt{r}$:

$\frac{7}{2} + \frac{3}{2}\sqrt{\frac{3}{2}}$

Thus, the expression in the required form is:

$p = \frac{7}{2}, q = \frac{3}{2}, r = 3$

So, the final expression in the form $p + q\sqrt{r}$ is:

$\frac{7}{2} + \frac{3}{2}\sqrt{3}$

This is the simplest form of the given expression.

Would you like any further details or clarifications?

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Here are 8 related questions:

1. How do you rationalize the denominator of an expression?
2. What are the steps to combine like terms involving square roots?
3. How do you express a number in the form $p + q\sqrt{r}$?
4. What does it mean for a number to be rational?
5. How do you add or subtract radical expressions?
6. How do you multiply radical expressions?
7. How can you simplify the expression $\frac{\sqrt{a}}{\sqrt{b}}$?
8. What is the importance of simplifying expressions in algebra?

Tip: When dealing with radicals, always look for opportunities to simplify before performing arithmetic operations.