Express as a fraction in simplest form with a rational denominator:

start fraction, minus, 2, divided by, minus, 10, minus, square root of, 13, end square root, end fraction

−10−13​−2​

To express the fraction 

\[
\frac{-2}{-10 - \sqrt{13}}
\]

in simplest form with a rational denominator, we need to rationalize the denominator. To do this, we multiply the numerator and denominator by the conjugate of the denominator, which is \(-10 + \sqrt{13}\). This will remove the square root in the denominator.

### Step 1: Multiply by the Conjugate
\[
\frac{-2}{-10 - \sqrt{13}} \times \frac{-10 + \sqrt{13}}{-10 + \sqrt{13}} = \frac{-2(-10 + \sqrt{13})}{(-10 - \sqrt{13})(-10 + \sqrt{13})}
\]

### Step 2: Expand the Numerator and Simplify
The numerator becomes:
\[
-2(-10 + \sqrt{13}) = 20 - 2\sqrt{13}
\]

### Step 3: Expand the Denominator Using the Difference of Squares
The denominator is:
\[
(-10 - \sqrt{13})(-10 + \sqrt{13}) = (-10)^2 - (\sqrt{13})^2 = 100 - 13 = 87
\]

### Step 4: Write the Final Expression
So, we have:

\[
\frac{20 - 2\sqrt{13}}{87}
\]

This is already in its simplest form.

### Final Answer
\[
\frac{20 - 2\sqrt{13}}{87}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you rationalize a denominator with multiple terms?
2. What is the conjugate of a binomial with a square root?
3. Can you rationalize the denominator of \(\frac{3}{5 + \sqrt{7}}\)?
4. Why is rationalizing the denominator important in simplifying expressions?
5. What happens if both terms in the denominator have square roots?

**Tip**: Always use the conjugate when rationalizing a binomial denominator to eliminate square roots efficiently.

Express as a fraction in simplest form with a rational denominator: \[\frac{-2}{-10 - \sqrt{13}}\]

This guide explains how to simplify the fraction \(\frac{-2}{-10 - \sqrt{13}}\) by rationalizing the denominator. Learn the step-by-step process using algebraic concepts like the difference of squares. Suitable for Grades 9-11.

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