Express as a fraction in simplest form with a rational denominator:
start fraction, minus, 1, divided by, minus, 2, minus, square root of, 7, end square root, end fraction
−2− 
7
​
 
−1
​


We need to express the given expression as a fraction with a rational denominator. The given expression is:

\[
\frac{-1}{-2 - \sqrt{7}}
\]

### Step 1: Rationalize the denominator
To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \(-2 - \sqrt{7}\) is \(-2 + \sqrt{7}\).

\[
\frac{-1}{-2 - \sqrt{7}} \times \frac{-2 + \sqrt{7}}{-2 + \sqrt{7}} = \frac{-1 \cdot (-2 + \sqrt{7})}{(-2 - \sqrt{7})(-2 + \sqrt{7})}
\]

### Step 2: Simplify the numerator
The numerator becomes:

\[
-1 \cdot (-2 + \sqrt{7}) = 2 - \sqrt{7}
\]

### Step 3: Simplify the denominator
Use the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\), where \(a = -2\) and \(b = \sqrt{7}\):

\[
(-2 - \sqrt{7})(-2 + \sqrt{7}) = (-2)^2 - (\sqrt{7})^2 = 4 - 7 = -3
\]

### Step 4: Combine the result
Now, the expression becomes:

\[
\frac{2 - \sqrt{7}}{-3}
\]

### Step 5: Simplify further
Distribute the negative sign in the denominator:

\[
\frac{2 - \sqrt{7}}{-3} = \frac{-2 + \sqrt{7}}{3}
\]

Thus, the simplified expression is:

\[
\frac{-2 + \sqrt{7}}{3}
\]

### Final Answer:
\[
\frac{-2 + \sqrt{7}}{3}
\]

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Do you want more details or have any questions? Here are some related questions for practice:

1. Rationalize the denominator: \(\frac{1}{2 + \sqrt{5}}\).
2. Express \(\frac{-3}{1 + \sqrt{2}}\) in simplest form.
3. Rationalize the denominator: \(\frac{4}{5 - \sqrt{3}}\).
4. Simplify the expression: \(\frac{2}{\sqrt{7} - 1}\).
5. Express \(\frac{5}{\sqrt{2} + 3}\) in simplest form.

**Tip**: When rationalizing the denominator, always multiply by the conjugate, as it eliminates the square root in the denominator.

Express as a fraction in simplest form with a rational denominator: -1 / (-2 - sqrt(7))

This tutorial shows how to rationalize the denominator and express -1 / (-2 - sqrt(7)) in its simplest form using the difference of squares formula. Follow the detailed steps to eliminate the square root and simplify the expression. Suitable for high school students in Grades 9-11.

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