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The given expression is:

\[
\frac{3}{-9 - \sqrt{3}}
\]

We can simplify this in a few steps. Let's break it down:

1. **Denominator Simplification:** We have \(-9 - \sqrt{3}\) in the denominator, which cannot be combined further as it contains both a constant and a radical term.
   
2. **Rationalizing the Denominator:** To simplify this further, we could rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of \(-9 - \sqrt{3}\) is \(-9 + \sqrt{3}\).

Thus, we multiply both the numerator and denominator by \(-9 + \sqrt{3}\):

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

Now let's compute the denominator:

\[
(-9)^2 - (\sqrt{3})^2 = 81 - 3 = 78
\]

So, the expression becomes:

\[
\frac{3(-9 + \sqrt{3})}{78} = \frac{-27 + 3\sqrt{3}}{78}
\]

We can further simplify this by dividing both terms in the numerator by 78:

\[
= \frac{-27}{78} + \frac{3\sqrt{3}}{78} = -\frac{9}{26} + \frac{\sqrt{3}}{26}
\]

Thus, the simplified expression is:

\[
-\frac{9}{26} + \frac{\sqrt{3}}{26}
\]

Would you like any additional details or explanations on any step?

### Related Questions:
1. How do you multiply by the conjugate of a binomial?
2. Why do we rationalize the denominator in expressions with square roots?
3. Can we further simplify the expression \(\frac{\sqrt{3}}{26}\)?
4. What happens if we have complex numbers in the denominator instead of square roots?
5. How would the result change if we had a different square root term, like \(\sqrt{5}\) instead of \(\sqrt{3}\)?

### Tip:
When rationalizing a denominator, always remember that the goal is to eliminate any square roots by using the conjugate, which ensures the terms simplify neatly.

Simplify the expression: \frac{3}{-9 - \sqrt{3}}

Learn how to simplify the expression \frac{3}{-9 - \sqrt{3}} by rationalizing the denominator using the conjugate method. This step-by-step solution covers key concepts in Algebra, including the difference of squares and handling radical expressions. Suitable for students in Grades 9-11.

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