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To rationalize the denominator of the expression:

\[
\frac{17}{3 + \sqrt{11}}
\]

we will multiply both the numerator and the denominator by the conjugate of the denominator, \(3 - \sqrt{11}\). The idea is to eliminate the square root from the denominator.

### Step-by-Step Process:

1. **Multiply numerator and denominator by the conjugate:**

\[
\frac{17}{3 + \sqrt{11}} \times \frac{3 - \sqrt{11}}{3 - \sqrt{11}} = \frac{17(3 - \sqrt{11})}{(3 + \sqrt{11})(3 - \sqrt{11})}
\]

2. **Simplify the denominator using the difference of squares formula:**

\[
(3 + \sqrt{11})(3 - \sqrt{11}) = 3^2 - (\sqrt{11})^2 = 9 - 11 = -2
\]

3. **Multiply out the numerator:**

\[
17(3 - \sqrt{11}) = 17 \cdot 3 - 17 \cdot \sqrt{11} = 51 - 17\sqrt{11}
\]

4. **Write the entire expression:**

\[
\frac{51 - 17\sqrt{11}}{-2}
\]

5. **Simplify further by dividing both terms in the numerator by \(-2\):**

\[
\frac{51}{-2} - \frac{17\sqrt{11}}{-2} = -\frac{51}{2} + \frac{17\sqrt{11}}{2}
\]

Thus, the rationalized form of the expression is:

\[
-\frac{51}{2} + \frac{17\sqrt{11}}{2}
\]

Would you like any more details or explanations on this process?

### Related Questions:
1. What is the purpose of rationalizing the denominator in algebra?
2. Can you explain the difference of squares formula in more detail?
3. What are the benefits of using conjugates in rationalization problems?
4. How would this process change if the denominator were \(3 - \sqrt{11}\)?
5. What other types of expressions benefit from using the conjugate technique?

### Tip:
When multiplying by conjugates, remember that the middle terms involving square roots always cancel out, leaving a simpler expression to work with.

Rationalize the denominator: 17 / (3 + √11)

Learn how to rationalize the denominator of 17 / (3 + √11) using the conjugate technique. This step-by-step solution covers key algebraic concepts such as the difference of squares formula and simplifying radicals. Suitable for students in Grades 10-12.

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