The problem is asking to find the rational form of the expression:

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

To rationalize this expression, we will multiply both the numerator and the denominator by the conjugate of the denominator.

Steps:

1. Identify the conjugate of the denominator: The denominator is $\sqrt{7} - 2\sqrt{3}$, so the conjugate is $\sqrt{7} + 2\sqrt{3}$.

2. Multiply the expression by the conjugate of the denominator: Multiply both the numerator and denominator by $\sqrt{7} + 2\sqrt{3}$:

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

3. Expand both the numerator and the denominator:

   • Numerator: Using the distributive property: $(3\sqrt{3} + \sqrt{7})(\sqrt{7} + 2\sqrt{3}) = 3\sqrt{3}\cdot \sqrt{7} + 3\sqrt{3}\cdot 2\sqrt{3} + \sqrt{7}\cdot \sqrt{7} + \sqrt{7}\cdot 2\sqrt{3}$ Simplify each term: $= 3\sqrt{21} + 6\cdot 3 + 7 + 2\sqrt{21}$ $= 3\sqrt{21} + 18 + 7 + 2\sqrt{21}$ Combine like terms: $= 5\sqrt{21} + 25$

   • Denominator: The denominator is a difference of squares: $(\sqrt{7} - 2\sqrt{3})(\sqrt{7} + 2\sqrt{3}) = \sqrt{7}^2 - (2\sqrt{3})^2$ Simplify: $= 7 - 4\cdot 3 = 7 - 12 = -5$

4. Final Expression: After simplifying, the expression becomes: $\frac{5\sqrt{21} + 25}{-5} = -\frac{5\sqrt{21} + 25}{5} = -\sqrt{21} - 5$

So, the rationalized form of the given expression is:

$-\sqrt{21} - 5$

Let me know if you would like any further explanation.

Relative Questions:

1. What is the importance of using the conjugate when rationalizing denominators?
2. How does the difference of squares apply in simplifying expressions with square roots?
3. What are the steps to handle complex conjugates in general rationalization problems?
4. Can this method be used to simplify expressions with cube roots or higher radicals?
5. What other types of algebraic techniques are useful for rationalizing irrational denominators?

Tip: Always simplify square root terms separately before multiplying them in rationalization problems.