Express in simplest radical form with a rational denominator.
start fraction, 4, plus, square root of, 7, end square root, divided by, 2, minus, square root of, 7, end square root, end fraction
2− 
7
​
 
4+ 
7
​
 
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We are asked to simplify the following expression:

\[
\frac{4 + \sqrt{7}}{2 - \sqrt{7}}
\]

To simplify, we need to rationalize the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2 - \sqrt{7} \) is \( 2 + \sqrt{7} \).

### Step 1: Multiply the numerator and denominator by \( 2 + \sqrt{7} \):

\[
\frac{4 + \sqrt{7}}{2 - \sqrt{7}} \cdot \frac{2 + \sqrt{7}}{2 + \sqrt{7}} = \frac{(4 + \sqrt{7})(2 + \sqrt{7})}{(2 - \sqrt{7})(2 + \sqrt{7})}
\]

### Step 2: Simplify the denominator using the difference of squares formula:

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

So the denominator simplifies to \( -3 \).

### Step 3: Expand the numerator:

\[
(4 + \sqrt{7})(2 + \sqrt{7}) = 4(2) + 4(\sqrt{7}) + \sqrt{7}(2) + \sqrt{7}(\sqrt{7})
\]
\[
= 8 + 4\sqrt{7} + 2\sqrt{7} + 7
\]
\[
= 15 + 6\sqrt{7}
\]

### Step 4: Combine the numerator and denominator:

\[
\frac{15 + 6\sqrt{7}}{-3}
\]

### Step 5: Simplify by dividing both terms in the numerator by \(-3\):

\[
\frac{15}{-3} + \frac{6\sqrt{7}}{-3} = -5 - 2\sqrt{7}
\]

Thus, the expression in simplest radical form with a rational denominator is:

\[
-5 - 2\sqrt{7}
\]

Would you like a detailed explanation of any step or have any questions?

Here are five related questions for practice:
1. Rationalize the denominator: \(\frac{5 + \sqrt{2}}{3 - \sqrt{2}}\)
2. Simplify: \(\frac{7 + \sqrt{3}}{2 + \sqrt{3}}\)
3. Express in simplest form: \(\frac{6 - \sqrt{5}}{2 + \sqrt{5}}\)
4. Rationalize and simplify: \(\frac{1 + \sqrt{6}}{4 - \sqrt{6}}\)
5. Simplify: \(\frac{3 + \sqrt{8}}{5 - \sqrt{8}}\)

**Tip:** When rationalizing the denominator, always multiply by the conjugate to eliminate square roots from the denominator.

Express in simplest radical form with a rational denominator: (4 + √7) / (2 - √7)

Learn how to simplify the expression (4 + √7) / (2 - √7) by rationalizing the denominator using the conjugate and applying the difference of squares. This step-by-step guide is perfect for students in Grades 9-12, covering key algebraic concepts and radical simplifications.

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