https://file.aiquickdraw.com/mathbot/file/1726449477511jyp30p59.jpg

The expression shown in the image is:

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

The task is to rationalize the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 + \sqrt{11}\) is \(1 - \sqrt{11}\).

Let's go through the steps:

1. Multiply both the numerator and the denominator by \(1 - \sqrt{11}\):

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

2. In the denominator, apply the difference of squares formula:

\[
(1 + \sqrt{11})(1 - \sqrt{11}) = 1^2 - (\sqrt{11})^2 = 1 - 11 = -10
\]

3. Now, simplify the numerator:

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

4. Therefore, the rationalized expression becomes:

\[
\frac{3 - 3\sqrt{11}}{-10}
\]

We can simplify this by dividing both terms in the numerator by \(-10\):

\[
\frac{3}{-10} - \frac{3\sqrt{11}}{-10} = -\frac{3}{10} + \frac{3\sqrt{11}}{10}
\]

Thus, the final simplified form is:

\[
-\frac{3}{10} + \frac{3\sqrt{11}}{10}
\]

Would you like further details or have any other questions?

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Here are 5 related questions for deeper understanding:

1. What is the conjugate of a binomial involving square roots?
2. How do you apply the difference of squares formula to rationalize denominators?
3. Why is it important to rationalize the denominator in some cases?
4. Can you generalize the process of rationalizing any denominator of the form \(a + \sqrt{b}\)?
5. What happens if you don't rationalize a denominator in a fraction?

**Tip:** Always check if the denominator contains a radical before performing rationalization—it simplifies many expressions!

Rationalize the denominator in the expression 3/(1 + √11). Make sure to simplify your answer. Use exact values. Enter the numerator and denominator separately.

Learn how to rationalize the denominator of the expression 3/(1 + √11) using the difference of squares. This problem covers key concepts in simplifying radical expressions and using exact values. Suitable for students in Grades 9-12.

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