The question requires rationalizing the denominator and simplifying the expression:

$\frac{3}{5 - 2\sqrt{2}}$

To rationalize the denominator, we will multiply both the numerator and denominator by the conjugate of the denominator, which is $5 + 2\sqrt{2}$. The conjugate is chosen because multiplying a binomial by its conjugate results in a difference of squares, which eliminates the square root.

Let's perform the rationalization step by step:

$\frac{3}{5 - 2\sqrt{2}} \times \frac{5 + 2\sqrt{2}}{5 + 2\sqrt{2}} = \frac{3(5 + 2\sqrt{2})}{(5 - 2\sqrt{2})(5 + 2\sqrt{2})}$

First, simplify the denominator:

$(5 - 2\sqrt{2})(5 + 2\sqrt{2}) = 5^2 - (2\sqrt{2})^2 = 25 - 8 = 17$

Now, expand the numerator:

$3(5 + 2\sqrt{2}) = 15 + 6\sqrt{2}$

So the rationalized expression is:

$\frac{15 + 6\sqrt{2}}{17}$

Therefore, the simplified expression is:

$\frac{15}{17} + \frac{6\sqrt{2}}{17}$

This is the rationalized and simplified form of the original expression.

Would you like more details on this process? Feel free to ask!

Here are 5 related questions you could explore:

1. How do you rationalize the denominator when there are binomials involving different terms, such as $3 + \sqrt{5}$?
2. What is the difference between rationalizing the denominator and simplifying expressions with radicals?
3. How do you simplify expressions involving cube roots in a similar way?
4. Can you simplify an expression with a complex denominator using the same method?
5. How does conjugating help in rationalizing the denominator in both quadratic and higher-degree terms?

Tip: When multiplying by the conjugate, always ensure that both the numerator and denominator are multiplied by the conjugate to maintain the equality of the expression.