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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.

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Learn to rationalize the expression 9/(3√7 + 6) by multiplying with the conjugate. Follow a detailed step-by-step solution to simplify the radical in the denominator using algebraic methods. Suitable for students in Grades 10-12.

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