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To simplify the given expression \(\frac{-9}{6 - \sqrt{13}}\), we need to rationalize the denominator. This can be done by multiplying both the numerator and the denominator by the conjugate of the denominator, which is \(6 + \sqrt{13}\).

### Step-by-step solution:

1. **Multiply by the conjugate:**
   Multiply the fraction by \(\frac{6 + \sqrt{13}}{6 + \sqrt{13}}\):
   \[
   \frac{-9}{6 - \sqrt{13}} \times \frac{6 + \sqrt{13}}{6 + \sqrt{13}} = \frac{-9(6 + \sqrt{13})}{(6 - \sqrt{13})(6 + \sqrt{13})}
   \]

2. **Simplify the denominator:**
   Use the difference of squares formula:
   \[
   (6 - \sqrt{13})(6 + \sqrt{13}) = 6^2 - (\sqrt{13})^2 = 36 - 13 = 23
   \]

3. **Simplify the numerator:**
   Distribute \(-9\) across \(6 + \sqrt{13}\):
   \[
   -9(6 + \sqrt{13}) = -54 - 9\sqrt{13}
   \]

4. **Write the fraction:**
   Now, the fraction is:
   \[
   \frac{-54 - 9\sqrt{13}}{23}
   \]

So, the simplified form of the expression is:
\[
\frac{-54 - 9\sqrt{13}}{23}
\]

Would you like further details or have any questions about this solution?

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Here are 5 follow-up questions based on this problem:
1. What is the purpose of using the conjugate when rationalizing a denominator?
2. How does the difference of squares formula simplify this process?
3. Could the numerator be factored further?
4. How would you handle an expression with a binomial in the numerator as well?
5. How do you identify which conjugate to multiply by?

**Tip:** Always check if you can factor out common terms after simplifying a radical expression.

Express as a fraction in simplest form with a rational denominator: -9 / (6 - √13)

Learn how to rationalize and simplify the expression -9 / (6 - √13) by multiplying the numerator and denominator by the conjugate of the denominator, using the difference of squares method. Step-by-step guide with detailed solution for Grades 10-12.

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