The question asks to rationalize the expression $\frac{9}{3\sqrt{7} + 6}$.

Step 1: Identify the expression and the conjugate

The original expression is: $\frac{9}{3\sqrt{7} + 6}$ To rationalize this, we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is $3\sqrt{7} - 6$.

Step 2: Multiply by the conjugate

$\frac{9}{3\sqrt{7} + 6} \times \frac{3\sqrt{7} - 6}{3\sqrt{7} - 6} = \frac{9(3\sqrt{7} - 6)}{(3\sqrt{7} + 6)(3\sqrt{7} - 6)}$

Step 3: Simplify the denominator

The denominator is a difference of squares: $(3\sqrt{7})^2 - 6^2 = 9 \cdot 7 - 36 = 63 - 36 = 27$

Step 4: Expand the numerator

The numerator is: $9(3\sqrt{7} - 6) = 27\sqrt{7} - 54$

Step 5: Combine results

$\frac{27\sqrt{7} - 54}{27}$ We can simplify this further by dividing each term by 27: $= \frac{27\sqrt{7}}{27} - \frac{54}{27} = \sqrt{7} - 2$

Conclusion

The rationalized form is $\sqrt{7} - 2$, which corresponds to option D from the multiple choices.

Would you like a detailed explanation of any part? Here are some further related questions you might find interesting:

1. How do you find the conjugate of a radical expression?
2. What is the significance of rationalizing the denominator?
3. How can we simplify complex fractions involving radicals?
4. What are the properties of difference of squares?
5. How can we rationalize expressions with more than one radical term?

Tip: Always check for opportunities to use conjugates when rationalizing expressions involving square roots in the denominator.