It looks like there was an issue with directly using the expression in a specific format. Let me rationalize the denominator manually by multiplying the numerator and the denominator by the conjugate of \(3 - \sqrt{11}\).

### Step-by-Step Rationalization:
The expression we are rationalizing is:

\[
\frac{1}{3 - \sqrt{11}}
\]

To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator, which is \(3 + \sqrt{11}\):

\[
\frac{1}{3 - \sqrt{11}} \times \frac{3 + \sqrt{11}}{3 + \sqrt{11}} = \frac{3 + \sqrt{11}}{(3 - \sqrt{11})(3 + \sqrt{11})}
\]

The denominator simplifies using the difference of squares formula:

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

Thus, the rationalized expression becomes:

\[
\frac{3 + \sqrt{11}}{-2} = -\frac{3 + \sqrt{11}}{2}
\]

So, the rationalized form of the expression is:

\[
-\frac{3 + \sqrt{11}}{2}
\]

Would you like more details or have any questions? Here are five related questions to consider:

1. How does the difference of squares formula help in rationalization?
2. Can this approach be generalized to other similar expressions?
3. How would you handle an expression with a different square root in the denominator?
4. Why do we multiply by the conjugate when rationalizing?
5. What are some real-world applications of rationalizing denominators?

**Tip:** When rationalizing, always look for patterns like the difference of squares to simplify your calculations.

Learn how to rationalize the denominator of the expression 1/(3 - sqrt(11)) step-by-step. Understand the concept of rationalization, the use of conjugates, and the difference of squares formula in simplifying radicals. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation